Một Số Bài Toán Không Chỉnh Cho Phương Trình Đạo Hàm Riêng.pdf

SỞ KHOA HỌC VÀ CÔNG NGHỆ TP HỒ CHÍ MINH VIỆN KHOA HỌC VÀ CÔNG NGHỆ TÍNH TOÁN BÁO CÁO TỔNG KẾT MỘT SỐ BÀI TOÁN KHÔNG CHỈNH CHO PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG Đơn vị thực hiện PTN Khoa học Môi Trường Chủ nh[.]

SỞ KHOA HỌC VÀ CƠNG NGHỆ TP HỒ CHÍ MINH VIỆN KHOA HỌC VÀ CƠNG NGHỆ TÍNH TỐN BÁO CÁO TỔNG KẾT MỘT SỐ BÀI TỐN KHƠNG CHỈNH CHO PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG Đơn vị thực hiện: PTN Khoa học Môi Trường Chủ nhiệm nhiệm vụ : PGS TS Nguyễn Huy Tuấn TP HỒ CHÍ MINH, THÁNG 06/2020 SỞ KHOA HỌC VÀ CƠNG NGHỆ TP HỒ CHÍ MINH VIỆN KHOA HỌC VÀ CƠNG NGHỆ TÍNH TỐN BÁO CÁO TỔNG KẾT MỘT SỐ BÀI TỐN KHƠNG CHỈNH CHO PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG Viện trưởng: Đơn vị thực hiện: PTN Khoa học Môi Trường Chủ nhiệm nhiệm vụ : PGS TS Nguyễn Huy Tuấn Nguyễn Kỳ Phùng Nguyễn Huy Tuấn TP HỒ CHÍ MINH, THÁNG 06/2020 Một số tốn khơng chỉnh cho phương trình đạo hàm riêng MỤC LỤC Trang MỞ ĐẦU ĐƠN VỊ THỰC HIỆN KẾT QUẢ NGHIÊN CỨU I Báo cáo khoa học II II Tài liệu khoa học xuất 21 III Chương trình giáo dục đào tạo 22 IV Hội nghị, hội thảo V File liệu TÀI LIỆU THAM KHẢO 23 CÁC PHỤ LỤC 27 PHỤ LỤC 1: Bài báo số PHỤ LỤC 2: Bài báo số PHỤ LỤC 3: Bài báo số PHỤ LỤC 4: Bằng Thạc sĩ Viện Khoa học Công nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng MỞ ĐẦU Đề tài nghiên cứu thuộc lĩnh vực tốn ngược tốn đặt khơng chỉnh phương trình đạo hàm riêng mà đối tượng khảo sát toán xác định điều kiện đầu tốn xác định hàm nguồn cho phương trình khuếch tán với đạo hàm cấp khơng ngun Bài tốn xác định điều kiện đầu toán xác định hàm nguồn cho phương trình khuếch tán với đạo hàm cấp khơng ngun có nhiều ứng dụng khác lĩnh vực khoa học công nghệ, vật lý, học, y học, xử lý ảnh, kinh tế Đó toán mà liệu thu thập phương pháp vật lý thông qua thiết bị đo đạc chuyên dụng Trong thực tế, ta đo đạc liệu đầu vào ln có sai số Vì tính ổn định nghiệm tốn khơng thỏa, tức sai số nhỏ liệu đo đạc dẫn đến sai số lớn nghiệm, gây khó khăn việc việc tính tốn số Vì thế, chúng tơi ta cần phải có phương pháp để “chỉnh hóa” tốn Các loại toán xác định điều kiện đầu toán xác định hàm nguồn cho cho phương trình khuếch tán với đạo hàm cấp khơng ngun nhiều nhà tốn học ngồi nước quan tâm thời gian gần Từ thập niên 50 kỉ trước, cơng việc chỉnh hóa tốn khơng chỉnh nghiên cứu nhiều nhà khoa học giới Một kết phương trình parabolic ngược thời gian cơng trình John [15] cơng bố năm 1955 John đề xuất phương pháp số để giải tốn Cauchy cho phương trình truyền nhiệt ngược thời gian, chứng minh phương pháp ổn định tập hàm số dương bị chặn Từ năm 1963, sau Tikhonov [29] đưa phương pháp chỉnh hóa tiếng ơng, tốn khơng chỉnh toán ngược trở thành ngành riêng vật lý tốn học khoa học tính toán Năm 1967, Lattes Lions [16] đưa phương pháp tựa đảo (quasi-reversibility) R.Ewing [10] dùng phương pháp phương trình Sobolev để chỉnh hóa tốn Sau đó, Showalter [28] đề xuất phương pháp tựa biên (quasi-boundary value) hay gọi phương pháp QBV Chúng ta kể thêm số kết tiếng toán parabolic ngược thời gian tuyến tính [3,7,9,10,22] Tiếp theo đây, chúng tơi xin đề cập đến tình hình nghiên cứu ngồi nước khoảng thời gian gần đây: Tình hình nghiên cứu nước: Thời gian gần có số kết tốn khơng chỉnh cho phương trình đạo hàm riêng, điển hình là: + GS TSKH Đinh Nho Hào (Viện toán học Việt Nam) với phương pháp phương pháp làm nhuyễn (mollification) [12], phương pháp tựa biên [11], + TS Nguyễn Thành Long (Đại học Khoa học Tự nhiên TP.HCM) công bố vài cơng trình tốn ngược thời gian phi tuyến, dùng phương pháp Quasireversibility [20] Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng + PGS TS Phạm Hồng Qn [33,35,38,39,40] nghiên cứu phương pháp chỉnh hóa như: phương pháp tựa biên,… để khảo sát loại tốn cauchy cho phương trình parabolic phương trình elliptic + TS Trần Nhân Tâm Quyền (Đại học Đà Nẵng) [48,49] nghiên cứu phương pháp chỉnh hóa Tikhonov cho toán xác định hệ số cho phương trình elliptic + TS Nguyễn Văn Đức (Đại Học Vinh) [50] nghiên cứu phương pháp hàm lồi để chỉnh hóa số tốn parabolic ngược thời gian + GS TS Đặng Đức Trọng, PGS TS Nguyễn Huy Tuấn (Đại học Khoa học Tự nhiên TP.HCM) [42,43,44,45,46,47] có nhiều kết nghiên cứu việc chỉnh hóa cho phương trình đạo hàm riêng phương pháp chặt cụt, phương pháp tựa khả đảo,… Tình hình nghiên cứu nước: Từ thập niên 50 kỉ trước, cơng việc chỉnh hóa tốn khơng chỉnh nghiên cứu nhiều nhà khoa học giới Một kết phương trình parabolic ngược thời gian cơng trình John [15] cơng bố năm 1955 John đề xuất phương pháp số để giải toán Cauchy cho phương trình truyền nhiệt ngược thời gian, chứng minh phương pháp ổn định tập hàm số dương bị chặn Từ năm 1963, sau Tikhonov [29] đưa phương pháp chỉnh hóa tiếng ơng, tốn khơng chỉnh tốn ngược trở thành ngành riêng vật lý tốn học khoa học tính tốn Năm 1967, Lattes Lions [16] đưa phương pháp tựa đảo (quasi-reversibility) R.Ewing [10] dùng phương pháp phương trình Sobolev để chỉnh hóa tốn Sau đó, Showalter [28] đề xuất phương pháp tựa biên (quasi-boundary value) hay gọi phương pháp QBV Chúng ta kể thêm số kết tiếng tốn parabolic ngược thời gian tuyến tính [3,7,9,10,22] Trong khoảng vài chục năm gần đây, có nhiều nhà khoa học giới quan tâm, tìm hiểu nghiên cứu lĩnh vực Chẳng hạn K.Ames… [1,2], A.S Carasso … [5,6] , Denche Bessila [8] , J Lee D Sheen[18,19], Marban Palencia [21], Santo Prizzi [26], T.I Seidman [27], W.B Muniz [23] Theo hiểu biết chúng tôi, loại tốn khơng chỉnh cho phương trình đạo hàm riêng sau cịn “bài tốn mở” tiếp tục nghiên cứu Trong nhiệm vụ này, nghiên cứu số phương pháp chỉnh hóa để khảo sát tốn ngược Các tốn có nhiều ý nghĩa lĩnh vực khoa học, kỹ thuật Chúng nghiên cứu toán mở sau đây: Loại 1: Bài toán ngược cho phương trình khuếch tán với bậc khơng ngun Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng Loại 2: Bài tốn ngược khơng chỉnh cho phương trình hệ phương trình parabolic, elliptic sóng Loại 3: Bài toán ngược với liệu nhiễu ngẫu nhiên Mục tiêu đề tài đưa phương pháp chỉnh hóa nhằm tìm nghiệm xấp xỉ tốn Chúng tơi chứng minh tốn xấp xỉ chỉnh Sau đó, chúng tơi khảo sát tốc độ hội tụ nghiệm chỉnh hóa nghiệm xác Lời cảm ơn đến ICST Nghiên cứu tài trợ Viện Khoa học Cơng nghệ tính tốn Thành phố Hồ Chí Minh (ICST phố Hồ Chí Minh) với tên dự án số tốn khơng chỉnh cho phương trình đạo hàm riêng Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng ĐƠN VỊ THỰC HIỆN Phịng thí nghiệm: Khoa học Môi Trường Chủ nhiệm nhiệm vụ: PGS TS Nguyễn Huy Tuấn Thành viên nhiệm vụ: Nguyễn Huy Tuấn (chủ nhiệm nhiệm vụ) Lê Đình Long (Thư kí Khoa học) Võ Văn Âu (Thành viên chính) Hồ Thị Kim Vân (Thanh viên chính) Nguyễn Hữu Cần (Thành viên chính) Cơ quan phối hợp: Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng KẾT QUẢ NGHIÊN CỨU I BÁO CÁO KHOA HỌC Trong nhiệm vụ này, khảo sát tìm nghiệm cho tốn sau: Trước hết, chúng tơi giới thiệu số tính chất tốn tử ( −) /2 , xem [18] Giá trị riêng toán tử fractional Laplacian: Giá trị riêng ( −) /2 số thực Một họ giá trị riêng {k }k=1 thõa mãn  1  2  3   , and k →  as k →  Chúng tơi có {k }k=1 giá trị riêng vector riêng phụ thuộc vào trị riêng toán tử fractional Laplacian  với điều kiện biên Dirichlet biên  : −k ( x) = kk ( x), x  ,  k ( x) = 0, on , Thì chúng tơi định nghĩa tốn tử ( −) /2   k =0 k =0 (−) /2 u :=  ckk ( x) = − ck k /2k ( x) với ánh xạ từ H 0 () vào L2 () Cho     Bằng H 0 () đặt không gian hàm thuộc g  L2 () với tính chất sau   (1 +  )  | g k =1 k sau‖ g‖ H  ( ) = k |2  , g k =  g ( x )k ( x)dx Chúng định nghĩa    (1 +  )  | g k =1 k k |2 Nếu  = H  () L2 () Bây sử dụng phương pháp tách biến để tìm nghiệm phương trình (1) Giả sử nghiệm phương  trình (1) định nghĩa sau u ( x, t ) =  u k (t )k ( x), with uk (t ) = u (., t ), k ( x) k =1 Thì việc mở rộng hàm riêng đinh nghĩa phương pháp chuỗi Fourier.Đó là, chúng tơi nhân hai vế phương trình (1) k (x) lấy tích phân hàm theo biến x Sử dụng công thức Green k  = , chúng tơi có hệ phương trình với điều kiện đầu giá trị đầu cho phương trình vi phấn cấp phân số với hệ số Fourier uk (t) Chúng nghiên cứu tốn tìm (u,f) thỏa phương trình khuếch tán với đạo hàm Caputo sau:   u   − uxx =  (t ) f(x), (x,t)   (0,T),  t (x,t)   (0,T) u(x,t) = 0, u( x,T) = g( x), x    (1) Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng Chúng tơi tìm cơng thức hàm nguồn f (x) sau Tiếp theo, áp dụng phương pháp tách biến để biểu diễn nghiệm dạng chuỗi Fourier sau: %  u ( x, t ) =  u p (t ) p ( x), u p (t ) = u (., t ),  p ( x)   Từ phương pháp tách biến, chúng tơi p =1 đưa phương trình (1) phương trình vi phân sau:   Dt u p (t ) =  p u p (t ) + Fp (t ),  t  T ,  u p (0) = u ( x,0),  p ( x)  Trong Fp (t ) = F ( x, t ),  p ( x)  Như báo hai tác giả Sakamoto Masahiro Yamamoto, công thức nghiệm phương trinh vi phân phía biểu diễn sau: t u p (t ) = E ,1 ( − pt )u p (0) +  (t − s) −1E , ( −p (t − s ) ) Fp ( s)ds   Từ điều kiện u(x,0) = điều kiện Fp ( s) = R( s)  f ( x),  p ( x)  biết T  h( x),  p ( x) = u p (T ) =  (T − s ) −1E , ( − p (T − s ) ) R ( s) ds  f ( x ),  p ( x )  Bằng phép biến đổi đơn giản, kết luận f ( x) =  h( x), p ( x)  p ( x)   p =1 T  (T − s)  −1 E , (− p (T − s) ) R(s)ds Bằng cách đặt, G (s,  p ) = (T − s ) −1 E , ( − p (T − s ) ) Chúng tơi có cơng thức hàm nguồn biểu diển sau:   h ( x ),  p ( x)   p ( x) f ( x ) =  p =1 T  G (s,  ) R(s)ds p Định lý 1: Tính khơng chỉnh tốn tìm lại hàm nguồn Chúng tơi định nghĩa tốn từ tuyến tính K : L2 () → L2 () sau:  T p =1 Kf ( x) = [ G ( s,  p ) R( s )ds]  f ( x),  p ( x)   p ( x) =  k ( x,  ) f ( )d   Trong  T p =1 k ( x,  ) = [ G (s,  p ) R(s )ds] p ( x) p ( ) Vì k ( x,  ) = k ( , x) , chúng tơi biết tốn tử K tốn tử tuyến tính tự liên hợp Kế tiếp, chứng minh tính compact Chúng tơi định nghĩa tốn tử hữu hạn chiều K N sau: N T p =1 K N f ( x ) = [ G ( s,  p ) R( s)ds]  f ( x),  p ( x )   p ( x) Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Một số tốn khơng chỉnh cho phương trình đạo hàm riêng Từ địng nghĩa phía trên, chúng tơi có kết sau ‖ K N f − Kf ‖ =     ‖ R‖ p = N +1 ‖ R‖ C2 ([0,T ])   [ G (s,  p = N +1 C ([0,T ]) p  T | f ( x),  p  |  f ( x),  ) R ( s)ds]2 | f ( x ),  p ( x ) |2 ( x) |2  N p = N +1 Điều dẫn đến p p ( x) |2 ‖ R‖ C2 [0,T ] ‖ K N f − Kf ‖   N ‖ R‖ C[0,T ] ‖ f ‖ 2L2 ( ) = ‖ f ‖ L2 ( ) N Chính điều này, ‖ K N − K‖ tiến dần N tiến dần vô Cũng thế, K tốn tử compact Tiếp theo, giá trị kì dị tốn tử tuyến tính tự liên hợp compact K T cho sau:  p =  G (s ,  p ) R( s )ds vecto riêng  p sở trực giao khơng gian L2 () Chính lí trên, tốn tìm lại hàm nguồn biểu diễn dạng phương trình tốn tử sau: K f(x)=h(x) Theo Kirsch, biết tốn đặt khơng chỉnh Để làm rõ cho vấn đề này, trình bày  ( x) ví dụ minh họa, cách chọn liệu đầu vào thời điểm cuối hm ( x) = m , m công thức hàm nguồn ứng với  ( x)  m ,  p ( x)  m   h ( x ),  ( x )   m ( x) p m f m ( x) =  T  p ( x) = T = T p =1  G (s,  p ) R( s)ds  G ( s,  p ) R(s)dsp ( x) m  G (s, m ) R(s)ds 0 Dĩ nhiên chúng tơi chọn liệu xác đầu vào thời điểm cuối g=0 Công thức nghiệm f (x) = Sai số theo chuẩn không gian L2 () ứng với hai liệu đầu vào  ( x) sau: ‖ h m − g‖ L ( ) =‖ m ‖ L ( ) = Điều dẫn đến : m m lim ‖ h m − g‖ L2 (  ) = lim m →+ m →+ m = 0, (2) Và dẫn đến sai số nghiệm hai hàm nguồn không gian L2 () sau: m ( x) m ‖ ‖ f − f ‖ L2 (  ) = ‖ T m  G (s , m ) R (s )ds L2 (  ) = 0 ‖ R‖ C[0,T ] T Sử dụng bất đẳng thức sau  G ( s, m ) R( s )ds  ‖ f m − f‖ T m  G (s, m ) R( s) ds m , có:  L2 (  ) m  ‖ R‖ C[0,T ] Chúng tơi có kết luận sau: Viện Khoa học Cơng nghệ Tính tốn TP Hồ Chí Minh Page Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 68 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Definition 4.1 For each \varepsilon > 0, a function v \varepsilon is said to be a weak solution of (14) if \bigl( \bigl( \bigr) \bigr) v \varepsilon \in L2 0, T ; H01 (\Omega ) \cap L\infty 0, T ; L2 (\Omega ) and it holds that d \varepsilon \langle v , \psi \rangle - A (15) dt \int \nabla v \varepsilon \cdot \nabla \psi dx - \rho \varepsilon \langle v \varepsilon , \psi \rangle \Bigr) \Bigr\rangle \bigl\langle \Bigl\langle \Bigl( \bigr\rangle = e\rho \varepsilon (t - T ) F \cdot , t; e\rho \varepsilon (T - t) v \varepsilon , \psi + P\beta \varepsilon v \varepsilon , \psi for all \psi \in H01 (\Omega ) \Omega Let \BbbS n be the space generated by \phi , \phi , , \phi n for n = 1, 2, , where in general \{ \phi j \} is a Schauder basis of H (\Omega ) (so it can be the eigenfunctions mentioned in Example 3.3), then let vn\varepsilon (x, t) = (16) n \sum \varepsilon Vjn (t) \phi j (x) j=1 be the weak solution of the following approximate problem, corresponding to (14): \int \langle (vn\varepsilon )t , \psi \rangle - A \nabla vn\varepsilon \cdot \nabla \psi dx - \rho \varepsilon \langle vn\varepsilon , \psi \rangle (17) \Omega \Bigl\langle \Bigl( \Bigr) \Bigr\rangle \bigl\langle \bigr\rangle = e\rho \varepsilon (t - T ) F \cdot , t; e\rho \varepsilon (T - t) vn\varepsilon , \psi + P\beta \varepsilon vn\varepsilon , \psi , for all \psi \in \BbbS n , with the final condition (18) vn\varepsilon (T ) = vf\varepsilon n = n \sum \bigl( Vf\varepsilon \bigr) jn \phi j \rightarrow u\varepsilon f strongly in L2 (\Omega ) as n \rightarrow \infty j=1 To derive the nonlinear ordinary differential equations with respect to the time argument for Vjn (t), it follows from (17) with using \psi = \phi j that for \leq j \leq n, \Bigl\langle \Bigl( \Bigr) \Bigr\rangle \bigl\langle \bigl( \varepsilon \bigr) \bigr\rangle \varepsilon Vjn t - (A + \rho \varepsilon )Vjn = e\rho \varepsilon (t - T ) F \cdot , t; e\rho \varepsilon (T - t) vn\varepsilon , \phi j + P\beta \varepsilon vn\varepsilon , \phi j , \varepsilon and Vjn (T ) = (Vf\varepsilon )jn By using the Newton Liebniz formula, one has \int T \bigl( \varepsilon \bigr) \varepsilon \varepsilon (19) Vjn (t) = Vf jn - (A + \rho \varepsilon ) Vjn (s) ds t \int - T \Bigl[ \Bigl\langle \Bigl( \Bigr) \Bigr\rangle \bigl\langle \bigr\rangle \Bigr] e\rho \varepsilon (s - T ) F \cdot , s; e\rho \varepsilon (T - s) vn\varepsilon , \phi j + P\beta \varepsilon vn\varepsilon (s) , \phi j ds t Lemma 4.2 Suppose that (6) holds For any fixed n \in \BbbN and for each \varepsilon > 0, the \varepsilon system (17) (18) has a unique solution Vjn \in C([0, T ]) Proof The proof of this lemma is standard Here we sketch out some important steps because it seems pertinent to see more detailed impact of \rho \varepsilon on all the analysis We define the norm in the Banach space Y = C ([0, T ] ; \BbbR n ) as follows: \| c\| Y := sup n \sum t\in [0,T ] j=1 | cj (t)| with c = (cj )1\leq j\leq n Copyright © by SIAM Unauthorized reproduction of this article is prohibited REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM 69 Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php By virtue of (19), we can define a Volterra-type integral equation and then set the operator \scrG : C ([0, T ] ; \BbbR n ) \rightarrow C ([0, T ] ; \BbbR n ) by \scrG (V \varepsilon ) (t) = H \varepsilon - T \int K \varepsilon (s, V \varepsilon ) ds, t \varepsilon where in the vector form, V and H indicate Vj\varepsilon and (Vf\varepsilon )j , respectively, and K \varepsilon stands for the right-hand side of (19) under the integration in time Observe that when summing (19) with respect to j up to n, we have (20) n \sum j=1 Vj\varepsilon (t) = n \sum \bigl( Vf\varepsilon j \bigr) j=1 n \int T \sum - - (A + \rho \varepsilon ) \varepsilon n \int \sum j=1 T Vj\varepsilon (s) ds t \Bigr) \Bigr\rangle \bigl\langle \Bigl\langle \Bigl( \Bigl[ \bigr\rangle \Bigr] e\rho \varepsilon (s - T ) F \cdot , s; e\rho \varepsilon (T - s) vn\varepsilon , \phi j + P\beta \varepsilon vn\varepsilon (s) , \phi j ds t j=1 For V \varepsilon \in C ([0, T ] ; \BbbR n ) and W \varepsilon \in C ([0, T ] ; \BbbR n ) we have the following estimates With the aid of the Lipschitz assumption (A3 ) we easily get (21) n \bigm| \Bigl\langle \Bigl( \Bigr) \Bigl( \Bigr) \Bigr\rangle \bigm| \sum \bigm| \bigm| | Vk\varepsilon - Wk\varepsilon | , \bigm| F \cdot , s; e\rho \varepsilon (T - s) vn\varepsilon - F \cdot , s; e\rho \varepsilon (T - s) wn\varepsilon , \phi j \bigm| \leq CLF e\rho \varepsilon (T - s) k=1 and in the same vein, using (6) implies that (22) n \sum \bigm| \bigl\langle \beta \varepsilon \bigr\rangle \bigl\langle \bigr\rangle \bigm| \bigm| P\varepsilon (s) , \phi j - P\beta \varepsilon wn\varepsilon (s) , \phi j \bigm| \leq CC1 log (\gamma (T, \beta )) | Vk\varepsilon - Wk\varepsilon | k=1 Grouping (21) and (22), it follows from (1) that the following estimate can be obtained: | \scrG (V \varepsilon ) - \scrG (W \varepsilon )| \bigr) \bigl( \leq C (T - t) M + \rho \varepsilon + nLF + C1 log (\gamma (T, \beta )) \| V \varepsilon - W \varepsilon \| Y , and furthermore, by induction we deduce | \scrG m (V \varepsilon ) - \scrG m (W \varepsilon )| m \bigr) m \bigl( (T - t) \leq C m M + \rho \varepsilon + LF + C1 log (\gamma (T, \beta )) \| V \varepsilon - W \varepsilon \| Y , m! where we denote by \scrG m (V \varepsilon ) = \scrG (\scrG \scrG (V \varepsilon )) Since for each \varepsilon > and n \in \BbbN , there exists m0 \in \BbbN such that m0 (T - t) m0 ! \bigl( \bigr) m0 C m0 M + \rho \varepsilon + LF + C1 log (\gamma (T, \beta )) < 1, then \scrG m0 is a contraction mapping from C ([0, T ] ; \BbbR n ) onto itself By the Banach fixed-point argument, there exists a unique solution V \varepsilon in Y such that \scrG m0 (V \varepsilon ) = V \varepsilon Combining this with the fact that \scrG m0 (\scrG (V \varepsilon )) = \scrG (\scrG m0 (V \varepsilon )) = \scrG (V \varepsilon ), the integral equation \scrG (V \varepsilon ) = V \varepsilon admits a unique solution in C ([0, T ] ; \BbbR n ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 70 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php From here on, we state the existence result in the following theorem Theorem 4.3 For each \varepsilon > 0, the regularized problem (14) \bigl( has a weak solution \bigr) v \varepsilon in the sense of Definition 4.1 Moreover, it satisfies v \varepsilon \in C [0, T ] ; L2 (\Omega ) and \bigl( \bigr) \prime vt\varepsilon \in L2 (0, T ; H (\Omega ) ) Proof We now give some a priori estimates for the solution of the problem (14) When doing so, we choose \psi = vn\varepsilon in (17) to get (23) d \varepsilon 2 \| v \| - A \| \nabla vn\varepsilon \| [L2 (\Omega )]d - \rho \varepsilon \| vn\varepsilon \| L2 (\Omega ) dt n L (\Omega ) \Bigl\langle \Bigl( \Bigr) \Bigr\rangle \bigl\langle \bigr\rangle = e\rho \varepsilon (t - T ) F \cdot , t; e\rho \varepsilon (T - t) vn\varepsilon , vn\varepsilon + P\beta \varepsilon vn\varepsilon , vn\varepsilon \underbrace{} \underbrace{} \underbrace{} \underbrace{} :=I4 :=I3 Note from the resulting structural condition of F in (A3 ) that \bigm| \Bigl( \bigm| \Bigr) \bigm| \bigm| e\rho \varepsilon (t - T ) \bigm| F x, t; e\rho \varepsilon (T - t) vn\varepsilon - F (x, t; 0)\bigm| \leq LF | vn\varepsilon | ; one thus has e2\rho \varepsilon (t - T ) 2LF 2\rho \varepsilon (t - T ) e \geq - 2LF I3 \geq - \bigm\| \Bigl( \Bigr) \bigm\| \bigm\| \bigm\| \bigm\| F \cdot , t; e\rho \varepsilon (T - t) vn\varepsilon \bigm\| L (\Omega ) \| F (\cdot , t; 0)\| L2 (\Omega ) - - LF \varepsilon \| \| L2 (\Omega ) 2 (1 + LF ) \| vn\varepsilon \| L2 (\Omega ) Similarly, based on the structural definition of P\beta \varepsilon in (6), it yields \bigl\langle \Bigr) \bigm\| \bigr\rangle \Bigl( \bigm\| \varepsilon \bigm\| P\beta \varepsilon vn\varepsilon \bigm\| 2 P\beta \varepsilon vn\varepsilon , vn\varepsilon \geq - + \| v \| n L (\Omega ) L (\Omega ) \bigr) \bigl( 2 \geq - C1 log2 (\gamma (T, \beta )) + \| vn\varepsilon \| L2 (\Omega ) Then, (23) can be estimated by d \varepsilon e2\rho \varepsilon (t - T ) \| \| L2 (\Omega ) + \| F (\cdot , t; 0)\| L2 (\Omega ) dt LF \bigl( \bigr) 2 \geq 2M \| \nabla vn\varepsilon \| [L2 (\Omega )]d + 2\rho \varepsilon - (1 + LF ) - C12 log2 (\gamma (T, \beta )) - \| vn\varepsilon \| L2 (\Omega ) , where we have used the assumption (A3 ) Hereby, for each \varepsilon > we choose 2\rho \varepsilon = LF + C12 log2 (\gamma (T, \beta )) + > 0, then integrate the resulting estimate from t to T to obtain \int e - 2T \rho \varepsilon T 2 \varepsilon \| (T )\| L2 (\Omega ) + \| F (\cdot , s; 0)\| L2 (\Omega ) ds LF t \int T 2 \geq \| vn\varepsilon (t)\| L2 (\Omega ) + M \| \nabla vn\varepsilon (s)\| [L2 (\Omega )]d ds t Since vn\varepsilon (T ) \rightarrow u\varepsilon f in L2 (\Omega ) (cf (18)), we can find an \varepsilon -independent constant c\= > such that \int T 2 \| vn\varepsilon (t)\| L2 (\Omega ) + M \| \nabla vn\varepsilon \| [L2 (\Omega )]d ds \leq c \= t Copyright © by SIAM Unauthorized reproduction of this article is prohibited REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM 71 Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php As a by-product, we have (24) \bigl( \bigl( \bigr) \bigr) vn\varepsilon is bounded in L\infty 0, T ; L2 (\Omega ) and in L2 0, T ; H01 (\Omega ) Observe that \Bigr) \Bigl( \bigl( \bigr) \prime (vn\varepsilon )t + A\Delta vn\varepsilon - \rho \varepsilon vn\varepsilon = e\rho \varepsilon (t - T ) F x, t; e\rho \varepsilon (T - t) vn\varepsilon + P\beta \varepsilon vn\varepsilon \in H (\Omega ) , which provides \Bigl( \bigl( \bigr) \prime \Bigr) (vn\varepsilon )t is bounded in L2 0, T ; H (\Omega ) (25) Thanks to the Banach Alaoglu theorem, the uniform bounds with respect to n, as obtained in (24) (25), imply that one can extract a subsequence (which we relabel with the index n if necessary) such that for each \varepsilon > 0, \bigl( \bigr) (26) vn\varepsilon \rightarrow v \varepsilon weakly - \ast in L\infty 0, T ; L2 (\Omega ) , (27) \bigl( \bigr) vn\varepsilon \rightarrow v \varepsilon weakly in L2 0, T ; H01 (\Omega ) , (28) \Bigl( \bigl( \bigr) \prime \Bigr) (vn\varepsilon )t \rightarrow vt\varepsilon weakly in L2 0, T ; H (\Omega ) Furthermore, by the Aubin Lions\bigl( compactness theorem in combination with the \bigr) \prime Gelfand triple H01 (\Omega ) \subset L2 (\Omega ) \subset H (\Omega ) , one gets from (26) and (28) that (29) vn\varepsilon \rightarrow v \varepsilon strongly in L2 (QT ) and so a.e in QT for a further subsequence Note also that due to (6), one has for each \varepsilon > 0, (30) P\beta \varepsilon vn\varepsilon \rightarrow P\beta \varepsilon v \varepsilon strongly in L2 (QT ) and so a.e in QT for a further subsequence In the same manner, one has for each \varepsilon > 0, \Bigl( \Bigr) \Bigl( \Bigr) (31) F e\rho \varepsilon (T - t) vn\varepsilon \rightarrow F e\rho \varepsilon (T - t) v \varepsilon strongly in L2 (QT ) From here on, by grouping (26) (28) and (29) (31) we can pass to the limit in (17) to show that v \varepsilon satisfies the problem (14) in the weak sense (15) On top of that, due to (24) and (25), we have \bigl( \bigr) v \varepsilon \in C [0, T ] ; L2 (\Omega ) , \bigl( \bigr) \prime where we have applied the embeddings H01 (\Omega ) \subset L2 (\Omega ) \subset H (\Omega ) and H (0, T ) \subset C [0, T ] Now, it remains to verify the terminal data In fact, we take a function \vargamma \in C [0, T ] with \vargamma (0) = and \vargamma (T ) = As a consequence of the convergence (25), one has \int \int T T \langle (vn\varepsilon )t , \psi \rangle \vargamma dt \rightarrow \langle vt\varepsilon , \psi \rangle \vargamma dt for all \psi \in L2 (\Omega ) , Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 72 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO and by integration by parts together with the Newton Liebniz formula, it becomes \int T \int T (32) - \langle vn\varepsilon , \psi \rangle \vargamma t dt + \langle vn\varepsilon (T ) , \psi \rangle \vargamma (T ) \rightarrow - \langle v \varepsilon , \psi \rangle \vargamma t dt + \langle v \varepsilon (T ) , \psi \rangle \vargamma (T ) 0 for all \psi \in L (\Omega ) Consequently, the weak convergence (27) allows us to obtain \langle vn\varepsilon (T ) , \psi \rangle \rightarrow \langle v \varepsilon (T ) , \psi \rangle for all \psi \in H01 (\Omega ) from (32) Combine this convergence with the fact already known that vn\varepsilon (T ) converges strongly to u\varepsilon f in L2 (\Omega ); see (18) We thus get \langle vn\varepsilon (T ) , \psi \rangle \rightarrow \langle u\varepsilon f , \psi \rangle for all \psi \in H01 (\Omega ) Due to the uniqueness of the limit, it reveals that \langle v \varepsilon (T ) , \psi \rangle = \langle u\varepsilon f , \psi \rangle for all \psi \in H01 (\Omega ) and thus v \varepsilon (T ) = u\varepsilon f a.e in \Omega Now we show the positivity and boundedness of the solution to the regularized problem (14) In the following theorem, if the measured inputs of the concentrations are positive and essentially bounded in a spatial environment, their distributions that obey the proposed approximation remain the same properties therein by a suitable choice of the auxiliary parameter \rho \varepsilon In other words, the behavior of the regularized solution strictly depends on the way \rho \varepsilon is taken Theorem 4.4 Let v \varepsilon be a weak solution of the problem (14) as deduced in Theorem 4.3 For each \varepsilon > 0, suppose that \leq u\varepsilon f \in L\infty (\Omega ) and F (x, t; 0) \equiv for a.e (x, t) \in QT Moreover, for all real-valued constant C > we assume P\beta \varepsilon C = Q\beta \varepsilon C \geq Then, \leq v \varepsilon \leq \| u\varepsilon f \| L\infty (\Omega ) for a.e (x, t) \in QT Proof Let v \varepsilon := v \varepsilon ,+ - v \varepsilon , - , where f + := max \{ f, 0\} and f - := max \{ - f, 0\} In (15), we now take the test function \psi = - v \varepsilon , - Then, by (A3 ), (A4 ), and (6) we have (33) \bigm\| \bigm\| \bigm\| d \bigm\| \bigm\| v \varepsilon , - \bigm\| 2 \geq M \bigm\| \nabla v \varepsilon , - \bigm\| [L2 (\Omega )]d L (\Omega ) dt \bigm\| \bigm\| + (\rho \varepsilon - LF - C1 log (\gamma (T, \beta )) - 1) \bigm\| v \varepsilon , - \bigm\| L2 (\Omega ) , inspired very much the way we have estimated (23) Choosing \rho \varepsilon = LF + C1 log (\gamma (T, \beta )) + > and observing that v \varepsilon , - | t=T = 0, we integrate (33) from t to T to get \| v \varepsilon , - \| L2 (\Omega ) \leq 0, which indicates the positivity of \varepsilon v + To prove the upper bound, we take the test function \psi = (v \varepsilon - B) in (15) where \varepsilon B \geq \| uf \| L\infty (\Omega ) Thus, we arrive at \bigm\| d \bigm\| \bigm\| \varepsilon + \bigm\| (34) \bigm\| (v - B) \bigm\| dt L (\Omega ) \bigm\| \bigm\| \bigm\| \bigm\| \Bigl\langle \Bigr\rangle \bigm\| \bigm\| \varepsilon + \bigm\| + \bigm\| + \varepsilon \geq M \bigm\| \nabla (v \varepsilon - B) \bigm\| + \rho ( v - B) + \rho B, (v - B) \bigm\| \bigm\| \varepsilon \varepsilon [L2 (\Omega )]d L2 (\Omega ) \Bigl\langle \Bigr\rangle \Bigl\langle \Bigr\rangle + + + + P\beta \varepsilon (v \varepsilon - B) , (v \varepsilon - B) + P\beta \varepsilon B, (v \varepsilon - B) \Bigl\langle \Bigl( \Bigr) \Bigr\rangle + + e\rho \varepsilon (t - T ) F e\rho \varepsilon (T - t) v \varepsilon , (v \varepsilon - B) \underbrace{} \underbrace{} :=I5 Here, taking into account the structural condition of F we get \bigm| \Bigl\langle \Bigr\rangle \bigm| \bigm| + \bigm| I5 \geq - LF \bigm| | v \varepsilon | , (v \varepsilon - B) \bigm| \biggl( \bigm\| \bigm\| \Bigl\langle \Bigr\rangle \biggr) \bigm\| + \bigm\| + \geq - LF \bigm\| (v \varepsilon - B) \bigm\| + B, (v \varepsilon - B) L (\Omega ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM 73 At this stage, we proceed as in the proof of the positivity By choosing \rho \varepsilon = + C1 log (\gamma (T, \beta )) + LF > 0, it follows from (34) that (v \varepsilon - B) = 0, provided that + \varepsilon (v - B) | t=T = Hence, we complete the proof of the theorem 4.2 Convergence analysis We are now going to derive the convergence rates obtained when the regularized solution u\varepsilon \beta of (10) (11) is applied to approximate the solution u of (12) (2) in the presence of noise on the final data Note that in the previous subsection we only write u\varepsilon as the regularized solution since the parameter \varepsilon is already fixed Instead, we denote in this part u\varepsilon \beta due to the choice of the regularization parameter \beta (\varepsilon ) which plays a vital role in this analysis Although Example 3.3 shows that C1 = T1 , for an arbitrary C1 > we need C1 T \leq in our main results to gain strong convergences At some points, this is in the same spirit of the terminology small solution defined in [9] 4.2.1 Statement of the results Here we state our main results as Theorems 4.5 and 4.6; their solid proofs are deferred to subsections 4.2.2 and 4.2.3, respectively Moreover, the proof of Corollary 4.7 is given in subsection 4.2.4 In the following, let \gamma (t, \beta ) for t \in [0, T ] and \beta := \beta (\varepsilon ) be as in section We choose (35) lim \gamma C1 T (T, \beta ) \varepsilon = K \in (0, \infty ) \varepsilon \rightarrow 0+ Theorem 4.5 (error estimates for < t \leq T ) Assume that the problem (12) (2) admits a unique solution (36) u \in C ([0, T ] ; \BbbW ) , \beta where the precise structure of \BbbW depends on the choice of the operator \bigl( Q\varepsilon in (5).\bigr) For a suitable choice of the operator P\beta \varepsilon in (6), we consider u\varepsilon \beta \in C [0, T ] ; L2 (\Omega ) as a solution of (13) (11) corresponding to the measured data u\varepsilon f Then the following error estimate holds: \int T \sqrt{} \bigm\| \varepsilon \bigm\| \bigm\| \bigm\| \varepsilon \bigm\| \nabla u\beta (\cdot , s) - \nabla u (\cdot , s)\bigm\| d ds \bigm\| u\beta (\cdot , t) - u (\cdot , t)\bigm\| + 2M L (\Omega ) [L (\Omega )] t \Bigl( \Bigr) \surd \leq \gamma - C1 t (T, \beta ) K + 2T C0 \gamma C1 T - (T, \beta ) \| u\| C([0,T ];\BbbW ) eT C2 for t \in (0, T ) and Ci > (i \in \{ 0, 1, 2\} ) independent of \varepsilon Theorem 4.6 (error estimate for t = 0) Under the assumptions of Theorem 4.5, we assume further that \bigl( \bigr) (37) u \in C ([0, T ] ; \BbbW ) \cap C 0, T ; L2 (\Omega ) Then, for \varepsilon > small enough we can find a unique t\varepsilon \in (0, T ) such that \Bigl[ \Bigl( \Bigr) \surd \bigm\| \varepsilon \bigm\| C1 T - \bigm\| u\beta (\cdot , t\varepsilon ) - u (\cdot , 0)\bigm\| \leq K + 2T C \gamma (T, \beta ) \| u\| eT C2 C([0,T ];\BbbW ) L (\Omega ) \Bigr] + \| ut \| C(0,T ;L2 (\Omega )) \surd , C1 log (\gamma (T, \beta )) where Ci > (i \in \{ 0, 1, 2\} ) are independent of \varepsilon Corollary 4.7 Under the assumptions of Theorem 4.5, one has for any < t < T , u\varepsilon \beta is strongly convergent to u in L2 (t, T ; Lr (\Omega )) for some r > with the same rate as in Theorem 4.5 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 74 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 4.2.2 Proof of Theorem 4.5 For an auxiliary parameter \rho \beta > 0, we put w\beta \varepsilon (x, t) := e\rho \beta (t - T ) [u\varepsilon \beta (x, t) - u (x, t)] Then, we compute that (38) \partial w\beta \varepsilon + A\Delta w\beta \varepsilon - \rho \beta w\beta \varepsilon \partial t \bigl[ \bigl( \bigr) \bigr] = P\beta \varepsilon w\beta \varepsilon + e\rho \beta (t - T ) Q\beta \varepsilon u + e\rho \beta (t - T ) F x, t; u\varepsilon \beta - F (x, t; u) This equation is associated with the zero Dirichlet boundary condition w\beta \varepsilon = on \partial \Omega \times (0, T ) and the following terminal condition: w\beta \varepsilon (x, T ) = u\varepsilon \beta f (x) - uf (x) for x \in \Omega Multiplying (38) by w\beta \varepsilon and then integrating the resulting equation over \Omega , we arrive at \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| d \bigm\| \bigm\| w\beta \varepsilon \bigm\| 2 (39) - A \bigm\| \nabla w\beta \varepsilon \bigm\| [L2 (\Omega )]d - \rho \beta \bigm\| w\beta \varepsilon \bigm\| L2 (\Omega ) L (\Omega ) dt \bigl\langle \bigr\rangle \bigl\langle \bigr\rangle \bigl\langle \bigl( \bigr) \bigr\rangle = P\beta \varepsilon w\beta \varepsilon , w\beta \varepsilon + e\rho \beta (t - T ) Q\beta \varepsilon u, w\beta \varepsilon + e\rho \beta (t - T ) F u\varepsilon \beta - F (u) , w\beta \varepsilon \underbrace{} \underbrace{} \underbrace{} \underbrace{} \underbrace{} \underbrace{} :=\scrI :=\scrI :=\scrI To investigate the convergence analysis, we need to bound from below the righthand side of (39) Relying on the structural property of the operator P\beta \varepsilon (cf (6)), \scrI can be estimated by \bigm\| \bigm\| (40) \scrI \geq - C1 log (\gamma (T, \beta )) \bigm\| w\beta \varepsilon \bigm\| L (\Omega ) with the aid of H\"older's inequality Using the Young inequality and the structural property of the operator Q\beta \varepsilon (cf (5)), \scrI can be estimated by \scrI \geq - C02 \gamma - (T, \beta ) \| u\| \BbbW - (41) \bigm\| \bigm\| \bigm\| w\beta \varepsilon \bigm\| 2 L (\Omega ) From now on, also taking into account the Lipschitz constant LF and choosing an appropriate Young inequality, we get the estimate of \scrI as follows: (42) \bigm\| \bigm\| \bigm\| \bigl( \bigr) e2\rho \beta (t - T ) \bigm\| \bigm\| F u\varepsilon \beta - F (u)\bigm\| 2 - 2L2F \bigm\| w\beta \varepsilon \bigm\| [L2 (\Omega )]N L (\Omega ) 8L \biggl( F \biggr) \bigm\| \bigm\| \geq - + 2L2F \bigm\| w\beta \varepsilon \bigm\| L2 (\Omega ) \scrI \geq - Plugging (40), (41), and (42) into (39), and then integrating the resulting estimate from t to T , we obtain, after some rearrangement, that \bigm\| \varepsilon \bigm\| \bigm\| w\beta (T )\bigm\| 2 (43) + (T - t) C02 \gamma - (T, \beta ) \| u\| \BbbW L (\Omega ) \int T \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \nabla w\beta \varepsilon (s)\bigm\| 2 d ds \geq \bigm\| w\beta \varepsilon (t)\bigm\| L2 (\Omega ) + 2M [L (\Omega )] t + 2L2F \varepsilon u\beta \in L2 by putting \rho \beta = C1 log (\gamma (T, \beta )) + \bigl( > \bigr) Note here that the existence of 0, T ; H01 (\Omega ) has already been obtained in subsection 4.1 Due to (A4 ) the first norm on the left-hand side of (43) is bounded from above by \varepsilon By the back-substitution w\beta \varepsilon (x, t) := e\rho \beta (t - T ) [u\varepsilon \beta (x, t) - u (x, t)] and the choice of \rho \beta , we thus conclude that Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM (44) 75 \int T \bigm\| \varepsilon \bigm\| \varepsilon \bigm\| \bigm\| \bigm\| \nabla u\beta (\cdot , s) - \nabla u (\cdot , s)\bigm\| 2 d ds \bigm\| u\beta (\cdot , t) - u (\cdot , t)\bigm\| 2 + 2M L (\Omega ) [L (\Omega )] t \Bigl( \Bigr) \leq \gamma 2C1 (T - t) (T, \beta ) \varepsilon + (T - t) C02 \gamma - (T, \beta ) \| u\| C([0,T ];\BbbW ) e2(T - t)C2 , where we have denoted C2 := (45) + 2L2F Together with the \varepsilon -dependent blow-up rate of \gamma in (35), this ends the proof of the theorem 4.2.3 Proof of Theorem 4.6 It is clear that in Theorem 4.5 the convergence does not hold at t = Taking a number t\varepsilon \in (0, T ), we prove that for each \varepsilon > 0, there exists t\varepsilon > such that u\varepsilon \beta (x, t = t\varepsilon ) is a good approximation candidate of u (x, t = 0) Indeed, if the source condition (37) holds true, we get \bigm\| \bigm\| \varepsilon \bigm\| u\beta (\cdot , t\varepsilon ) - u (\cdot , 0)\bigm\| L (\Omega ) \bigm\| \bigm\| \varepsilon \varepsilon \bigm\| \leq u\beta (\cdot , t ) - u (\cdot , t\varepsilon )\bigm\| L2 (\Omega ) + \| u (\cdot , t\varepsilon ) - u (\cdot , 0)\| L2 (\Omega ) \Bigl( \Bigr) \surd \varepsilon \leq \gamma - C1 t (T, \beta ) K + 2T C0 \gamma C1 T - (T, \beta ) \| u\| C([0,T ];\BbbW ) eT C2 + t\varepsilon \| ut \| C(0,T ;L2 (\Omega )) Observe that the error bound \| u\varepsilon \beta (\cdot , t\varepsilon ) - u (\cdot , 0) \| L2 (\Omega ) is essentially decided by \bigl( \bigr) \varepsilon the infimum of 21 \gamma - C1 t (T, \beta ) + t\varepsilon with respect to t\varepsilon > We find that the term \varepsilon \gamma - C1 t (T, \beta ) is decreasing and t\varepsilon obviously possesses a linear growth Therefore, for every \beta := \beta (\varepsilon ) > there exists a unique t\varepsilon \in (0, T ) such that \Biggl\{ lim+ t\varepsilon = 0, \varepsilon \rightarrow (46) \varepsilon t\varepsilon = \gamma - C1 t (T, \beta ) , and the second equation can be rewritten as log (t\varepsilon ) = - C1 log (\gamma (T, \beta )) t\varepsilon (47) Using the elementary inequality log (a) > - a - for all a > 0, it follows from (47) that \sqrt{} \varepsilon t < C1 log (\gamma (T, \beta )) Henceforth, for t\varepsilon sufficiently small we complete the proof of the theorem 4.2.4 Proof of Corollary 4.7 In this part, we rely on the Gagliardo Nirenberg interpolation inequality for functions having zero trace to derive the error estimate Essentially, it reads as \int T \bigm\| \varepsilon \bigm\| \bigm\| u\beta (\cdot , s) - u (\cdot , s)\bigm\| r ds (48) L (\Omega ) t \bigm\| \bigm\| 2\alpha \leq C\Omega \bigm\| u\varepsilon \beta - u\bigm\| C([t,T ];L2 (\Omega )) \int t T \bigm\| \bigl( \varepsilon \bigm\| \bigr) \bigm\| \nabla u\beta - u (\cdot , s)\bigm\| 2(1 - \alpha ) ds, [L2 (\Omega )]d Copyright © by SIAM Unauthorized reproduction of this article is prohibited 76 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where C\Omega > is a generic constant that only depends on the geometry of \Omega , and the involved parameters should hold with r > and < \alpha < satisfying d - \alpha (1 - \alpha ) (d - 2) > and = + r 2d r 2d \bigl( \bigr) Note that (48) is available because of the existence of u\varepsilon \beta \in L2 0, T ; H01 (\Omega ) \cap \bigl( \bigr) \bigl( \bigr) L\infty 0, T ; L2 (\Omega ) leading to C [0, T ] ; L2 (\Omega ) ; see subsection 4.1, and the compact embedding H (\Omega ) \subset Lr (\Omega ) The special case of (48) in two-and three-dimensional versions (d = 2, 3) is the well-known Ladyzhenskaya inequality Using H\" older's inequality we can write (48) as (49) \int T t \bigm\| \varepsilon \bigm\| \bigm\| u\beta (\cdot , s) - u (\cdot , s)\bigm\| r ds L (\Omega ) \bigm\| \bigm\| 2\alpha \leq C\Omega (T - t) \bigm\| u\varepsilon \beta - u\bigm\| C([t,T ];L2 (\Omega )) \alpha \Biggl( \int t T \Biggr) - \alpha \bigm\| \bigl( \varepsilon \bigm\| \bigr) \bigm\| \nabla u\beta - u (\cdot , s)\bigm\| d ds [L (\Omega )] We remark that in (49) we are only able to get the convergence until the near zero point of time, i.e., it merely holds for < t < T Accordingly, it is straightforward to obtain the rate in Lr from (62) Thus, we complete the proof of the corollary Discussions 5.1 Some remarks on the system (1) Having completed main results for the semilinear case (12), it now suffices to provide some amendable remarks surrounding the general system (1) and its regularization (10) Uniqueness result It is discernible that the regularized problem may have many solutions but those regularized solutions (if they exist) must converge to a unique true solution Here we introduce collectively important steps, included in Lemma 5.1, to prove the uniqueness result for the time-reversed system (1) with the zero Dirichlet boundary condition Then, from now on we will not come back to this issue in future publications for the regularization of this system The technique we follow is mainly from [21, Chapter 6], which was used to study the large-time behavior of solutions to a linear class of initial-boundary value parabolic equations Detailed proofs of the following results can be inspired from [27] for the observations in the semilinear case (12) with H\"older nonlinearities and the nonlinear Robin-type boundary condition Setting the function space \bigl( \bigr) \bigl( \bigr) \bigl( \bigr) WT (\Omega ) := C [0, T ] ; H01 (\Omega ) \cap W 2,\infty (\Omega ) \cap L\infty 0, T ; H (\Omega ) \cap C 0, T ; L2 (\Omega ) , we denote by PT (\Omega ) the set of functions in WT (\Omega ) such that they vanish on the boundary \partial \Omega and at the moments t \in \{ 0, T \} , i.e., PT (\Omega ) := \{ u \in WT (\Omega ) : u| \partial \Omega = 0, u| t=T = 0, u| t=0 = 0\} Then, for \eta > we set (50) \lambda (t) = t - T - \eta Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM 77 In what follows, this function plays a prime factor to prove the backward uniqueness result According to solid proofs in [27], it is also worth noting that Lemma 5.1 is m essentially a Carleman estimate with the weight \lambda - k ; see [31] for the observation in this spirit Lemma 5.1 Assume the diffusion aij (x, t, \cdot , \cdot ) \in C (QT ) for \leq i, j \leq N is such that it satisfies the strict ellipticity condition and the mapping (p, q) \mapsto \rightarrow a (x, t; p; q) \bigl[ \bigr] N d For any v \in [PT (\Omega )]N , for any is sesquilinear for (p, q) \in [L2 (\Omega )]N \times L2 (\Omega ) positive m and any positive real k, one has \bigm\| - m \bigm\| \bigm\| \lambda k (\nabla \cdot (a (v; \nabla v) \nabla v) - vt )\bigm\| 2 [L (QT )]N \bigm\| \bigm\| \bigm\| m m m \bigm\| \bigm\| \lambda - k - v \bigm\| \geq - D \bigm\| \lambda - k \nabla v \bigm\| [L2 (Q )]N d , [L (QT )]N T k where D depends only on the bounds of \partial t a Moreover, if < T \leq \mu for < \mu \leq \mu and < \eta \leq \eta sufficiently small, there exists a positive K independent of m such that \bigm\| m \bigm\| K \bigm\| \lambda - k (\nabla \cdot (a (v; \nabla v) \nabla v) - vt )\bigm\| [L2 (Q )]N (51) T \bigm\| \bigm\| - m - \bigm\| \bigm\| m \geq \bigm\| \lambda k v \bigm\| [L2 (Q )]N + \bigm\| \lambda - k \nabla v \bigm\| [L2 (Q )]N d T T for m sufficiently large Let u and v be the two solutions of the backward problem (1) (2) in [WT (\Omega )]N The difference system for w = u - v reads as (52) wt + \nabla \cdot ( - a (x, t; w; \nabla w) \nabla w) = F (x, t; u; \nabla u) - F (x, t; v; \nabla v) + \nabla \cdot (a (x, t; u; \nabla u) \nabla u) - \nabla \cdot (a (x, t; v; \nabla v) \nabla v) - \nabla \cdot (a (x, t; w; \nabla w) \nabla w) , endowed with the zero Dirichlet boundary condition and the zero terminal condition Under the assumptions that a, F are Lipschitz-continous with respect to the nonlinear arguments p, q and that a satisfies the strict ellipticity condition, we can find a positive constant C such that from (52) the following differential inequality holds: \Bigl( \Bigr) 2 (53) | wt + \nabla \cdot ( - a (x, t; w; \nabla w) \nabla w)| \leq C | w| + | \nabla w| Observe that w \in [PT (\Omega )]N , we can obtain the uniqueness result in [PT (\Omega )]N for (1) by using (51), (53) and by choosing appropriately small values of \mu and \eta Nonlocal diffusion We could ameliorate the existence result (cf Theorem 4.3) when the diffusion a in the system (1) is of the following physical types: \bullet a = a(x, t) typically accounting for the anisotropic diffusion and possibly taxis processes; \bigm| \int \bigm| \bigr\} \bigl\{ \bullet a = a (t; u) = max \theta , \theta + \bigm| \Omega u (x, t) dx\bigm| + \theta for some \theta , \theta , \theta > The diffusion in this form is controlled by the local movements of species involved in the evolution equation (see, e.g., [1, 52] for the concrete biological motivation of this equation); \int 2 \bullet a = a(t; \| \nabla u\| L2 (\Omega ) ) = \theta + \Omega | \nabla u| dx for some \theta > indicating a Kirchhofftype diffusion model for, e.g., flows through porous media Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 78 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Using the same argument in Theorem 4.3, it is worth mentioning that the convergence results obtained in (27) and (29) are sufficient to passing to the limit in the diffusion term involving the aforementioned forms Consequently, the existence result remains true in these cases for any spatial dimensions d However, this technique is not valid for the p-Laplacian equation inspired from the power-law type of Ohm's law in conductivity of electricity, which reads as \Bigl( \Bigr) p - (54) ut - \nabla \cdot | \nabla u| \nabla u = F (u) for p \geq 2, due to the failure of passage to the limit When d = 1, there is a possibility of proving this solvability by the embedding H01 (\Omega ) \subset L\infty (\Omega ) Since we use the boundedness of the diffusion term as a key point in the convergence analysis, a slight improvement of Theorems 4.5 and 4.6 can be obtained when a is dependent of the gradient In fact, assuming the source condition (compared to (37)) \bigl( \bigr) (55) u \in C ([0, T ] ; \BbbW ) \cap L\infty (0, T ; W 1,\infty (\Omega )) \cap C 0, T ; L2 (\Omega ) , one could suppose that M \geq \eta \| \nabla u\| L\infty (QT ) for some \eta > sufficiently small, somewhat similar to the concept of large diffusion in terms of A, to gain similar error bounds Technically, the reason behind this assumption is to preserve the positivity of the gradient term in (43) In some physical problems, the small diffusion a would fit this circumstance because M now can be taken sufficiently large and then choosing M large is possible Locally Lipschitz-continuous nonlinearities From now on, we extend the convergence analysis when the source term F locally depends on u and \nabla u In this scenario, we need the estimate (4) for the cut-off function F\ell introduced in Remark 2.1 Essentially, there are two main difficulties in the proofs \bullet When exploring the difference equation in the proof of Theorem 4.5 we confront the difference term F\ell \varepsilon (u\varepsilon \beta ; \nabla u\varepsilon \beta ) - F (u; \nabla u) Thus, estimating \scrI in (39) would be problematic \bullet This moment the constant C2 in (44) and given by (45) would depend on \ell \varepsilon Observe that the behavior of \ell \varepsilon should be increasing (when \varepsilon \rightarrow 0) as it approximates the source function F in (3) Therefore, this parameter must be formulated in a clear manner to ensure the convergence of our QR scheme These issues really need to be elucidated because, as particularly mentioned in subsection 1.1, the local Lipschitz continuity of F is encountered in most of the significant equations in real-life applications Here we sketch out some essential ideas that we can adapt to the proof of Theorem 4.5 Note that here we need the aid of the source condition (55) First, we choose the cut-off parameter \ell \varepsilon > such that \ell \varepsilon \geq \| u\| L\infty (0,T ;W 1,\infty (\Omega )) (56) This way we solve the first issue because F\ell \varepsilon (x, t; u; \nabla u) = F (x, t; u; \nabla u); cf (3) Taking into account the Lipschitz constant LF (\ell \varepsilon ) > and choosing an appropriate Young inequality, we get the estimate of \scrI as follows: \bigm\| \bigm\| \bigl( \bigr) e2\rho \beta (t - T ) M \bigm\| 2L2F (\ell \varepsilon ) \bigm\| \bigm\| F\ell \varepsilon u\varepsilon \beta ; \nabla u\varepsilon \beta - F\ell \varepsilon (u; \nabla u)\bigm\| 2 \bigm\| w\beta \varepsilon \bigm\| 2 - \varepsilon L (\Omega ) L (\Omega ) 8LF (\ell ) M \Bigr) L2 (\ell \varepsilon ) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| M \Bigl( \bigm\| \bigm\| w\beta \varepsilon \bigm\| 2 \bigm\| w\beta \varepsilon \bigm\| 2 \geq - + \bigm\| \nabla w\beta \varepsilon \bigm\| [L2 (\Omega )]d - F L (\Omega ) L (\Omega ) M (57) \scrI \geq - Copyright © by SIAM Unauthorized reproduction of this article is prohibited 79 REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Henceforth, (43) remains the same when we put \rho \beta = C1 log(\gamma (T, \beta )) + M4+1 + With this choice, the constant C2 in (45) is \varepsilon -dependent and given by (58) C2 (\ell \varepsilon ) := L2F (\ell \varepsilon ) M M + L2F (\ell \varepsilon ) + M Now observe (44) with this new C2 in (58) and have in mind that the error estimate at t = (cf Theorem 4.6) is of the order \scrO (log - (\gamma (T, \beta ))) We only need \varepsilon (T - t)L2 F (\ell ) M in such a way that its growth does not to find a fine control of the term e ruin the logarithmic rate of convergence To so, our strategy is the following: We take \sqrt{} M \varrho := \varrho (\beta ) = (59) log (log\kappa (\gamma (T, \beta ))) > T for some \varepsilon -independent constant \kappa > being selected later Then, we have lim \varrho (\beta ) = \infty (60) \varepsilon \rightarrow 0+ \bigl( \beta \bigr) If we choose \Lambda \beta := sup L - = \varrho (\beta ) and we also obtain F \{ ( - \infty , \varrho (\beta )]\} , then LF \Lambda (T - t)L2 F (61) e (\Lambda \beta ) M \leq log\kappa (\gamma (T, \beta )) \beta Note also that by (60), L - F \{ ( - \infty , \varrho (\beta )]\} \not = and \Lambda \in (0, \infty ) is well-defined \beta Moreover, we can prove that lim\varepsilon \rightarrow 0+ \Lambda = \infty Indeed, we suppose that there exists C > such that \Lambda \beta \leq C for \beta near the zero Since LF is nondecreasing with \bigl( point \bigr) respect to \ell \varepsilon , it holds that LF (C)\bigl( \geq LF\bigr] \Lambda \beta = \varrho (\beta ), which contradicts the fact already known (60) Now, for \ell \varepsilon \in 0, \Lambda \beta we deduce that e \varepsilon (T - t)L2 F (\ell ) M \leq log\kappa (\gamma (T, \beta )) , resulted from (61) This also indicates that we have identified a fine upper bound of the \ell \varepsilon -dependent Lipschitz constant LF , and the error estimate (44) now becomes (62) \int \bigm\| \bigm\| \varepsilon \bigm\| u\beta (\cdot , t) - u (\cdot , t)\bigm\| 2 + 2M L (\Omega ) T \bigm\| \bigm\| \varepsilon \bigm\| \nabla u\beta (\cdot , s) - \nabla u (\cdot , s)\bigm\| 2 d ds [L (\Omega )] t \Bigl( \Bigr) \leq log2\kappa (\gamma (T, \beta )) \gamma 2C1 (T - t) (T, \beta ) \varepsilon + 2T C02 \gamma - (T, \beta ) \| u\| C([0,T ];\BbbW ) e2T C3 , where C3 := M4+1 is no longer dependent of \ell \varepsilon Similar to proof of Theorem 4.6, we inherit from (62) to gain the error estimate at t = with the \scrO (log\kappa - (\gamma (T, \beta ))) Hence, together with (62) we choose \bigl\{ order \bigr\} \kappa := \kappa (t) = C1 t, 12 > to complete the convergence analysis in this case On top of this, the choice of the cut-off parameter can be summarized by (56) and (59), working with sufficiently small values of \varepsilon No-flux boundary condition Since our problem (1) (2) is also present in population dynamics, the zero Neumann condition should be analyzed In this case, we associate the regularized problem (13) with the boundary condition - a\nabla u\varepsilon \cdot n = 0, Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 80 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO taking the place of the zero Dirichlet boundary condition in (11) Under this setting, the techniques used in the proofs of our main results can be applied in the same manner, focusing on the same structure of the weak formulation we have in (15) (where the test function \psi now belongs to the closed subspace of H (\Omega ) that satisfies the zero Neumann boundary condition) and the key equation (39) for the convergence analysis Accordingly, the rates of convergence derived in Theorems 4.5 and 4.6 remain unchanged Moreover, the strong convergence on the boundary is confirmed for < t < T by the following trace inequality: \int T \bigm\| \varepsilon \bigm\| \bigm\| u\beta (\cdot , s) - u (\cdot , s)\bigm\| 2 ds [L (\partial \Omega )]N t \Biggl( \Biggr) \int T \bigm\| \varepsilon \bigm\| \bigm\| \bigl( \varepsilon \bigm\| \bigr) \bigm\| \bigm\| \bigm\| \bigm\| \leq C\Omega u\beta - u \nabla u\beta - u (\cdot , s) N d ds , N + [C([t,T ];L (\Omega ))] [L (\Omega )] t which yields the same rate as in Theorem 4.5 5.2 Possible future generalizations of above results Gevrey class It is worth noting that the property (7) remains true up to a compact Riemannian manifold, which is generally called the Sturm Liouville decomposition As a prominent example, the standard eigenelements for a d-torus d \BbbT d = \BbbR d / (2\pi \BbbZ ) are \phi p (x) = d \prod j=1 e2\pi ipj xj , \mu p = d \sum (2\pi pj ) , pj \in \BbbN , \leq j \leq d, i = \surd - j=1 In this scenario, Gevrey classes are popular in microlocal analysis for the propagation of wavefront set and in the study of analytic regularity for nonlinear evolution equations with periodic boundary data A famous result of the Gevrey solvability for nonlinear analytic parabolic equations is recalled in an example of Appendix A Here, our discussions focus on the preasymptotic error bounds for approximation numbers of periodic Gevrey-type spaces of analytic functions with connection to the Galerkin method d For < \alpha , p, q < \infty , we denote by \BbbG p,q \alpha (\BbbT ) the Gevrey space that consists of all \infty d functions in C (\BbbT ) and satisfies \left( \right) 1/2 \Bigl( \Bigr) \sum q \| u\| \BbbG p,q exp 2\alpha \| k\| p u \^k < \infty , d := \alpha (\BbbT ) k\in \BbbZ d where u \^k denotes the Fourier coefficient of u with respect to the frequency vector k \in \BbbZ d By this definition, the norm \| eM T ( - \Delta ) u\| L2 (\BbbT d ) in Example 3.3 is essentially \| u\| \BbbG 2,2 (\BbbT d ) For q \in (0, 1), this space is the classical Gevrey classes that contain MT nonanalytic functions, while for q \geq all functions are real-analytic therein In approximation theory for Hilbert spaces, approximation numbers represent the worst-case error obtained when approximating a class of functions by projecting them onto the optimal finite-dimensional subspace The basic reason lies in the informationbased complexity that requires the rank n \in \BbbN of the optimal projection operator to be sufficiently large (n > 2d ) to gain the classical error bounds, which is not substantially practical for high dimensions Therefore, approximation numbers can be an excellent candidate to handle this context In a nutshell, the connection between such approximation numbers and Galerkin schemes for a classical variational problem, Copyright © by SIAM Unauthorized reproduction of this article is prohibited REGULARIZATION OF A QUASI-LINEAR PARABOLIC PROBLEM 81 Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where a parabolic problem can be involved, is clearly present in [36, subsection 1.5] with references cited therein for a background of Gevrey classes Definition 5.2 (approximation numbers) Let X and Y be two Banach spaces The norm of an operator \scrA : X \rightarrow Y is denoted by \| \scrA \| X\rightarrow Y The nth approximation number (n \in \BbbN ) of an operator \scrT : X \rightarrow Y is defined by an (\scrT : X \rightarrow Y ) := inf rank(\scrA ) This rigorous estimate is almost identical to the preasymptotic estimate for approximation numbers of the classical embedding Id : \bigl( \bigr) \bigl( \bigr) (\BbbT d ) obviously contains smoother functions than H \BbbT d \rightarrow L2 \BbbT d , albeit \BbbG 2,2 MT \bigl( d \bigr) H \BbbT On the other hand, the approximation numbers for the embedding Id : \bigl( \bigr) \BbbG 2,2 (\BbbT d ) \rightarrow H \BbbT d are asymptotically identical when \leq n \leq d, while for the MT embedding Id : W 1,\infty (\BbbT d ) \rightarrow L2 (\BbbT d ), they are completely identical whenever n \leq 2d Eventually, all these numbers indicate that there is a possibility to choose a combination of an optimal dimensional subspace and a linear finite element algorithm such that an approximate numerical solution by Galerkin methods is a good candidate in L2 and H for the true solution satisfying (36) Consequently, the worst-case a priori error for the (low) n-dimensional subspace in this context behaves like that of the standard finite element methods (FEMs) 5.3 Concluding remarks We have extended a modified QR method for backward quasi-linear parabolic systems with noise Several rates of convergence have been derived, especially the rigorous error estimates in Lr (\Omega ) (r \geq 2) and H (\Omega ), albeit many open questions remain unsolved Although the spectral method that takes into account Duhamel's principle is not used, settings for filter regularized operators still rely on existence of the space \BbbW , which usually plays a role as a class of Gevrey spaces in the existing trend of regularization for time-reversed nonlinear parabolic equations Our present contribution gives rise to some further interesting questions Typically, our present error estimates are not expected to be applied in the stochastic setting, but they can be designed to obtain an approximate solution in the finite element framework In this sense, our theoretical analysis can be a key ingredient to establish regularized multiscale FEM schemes which deal with models in certain complex domains because spatial environments where population densities take place are usually not nice (e.g., porous media) Other open perspectives include the effective iterative QR method and also the presence of the Robin-type boundary condition describing, e.g., the surface reaction in more complex scenarios Appendix A Applications to existing models Here, we examine four types of backward problems arising in many physical applications to show the applicability of our theoretical analysis In order to show existing arguments on the a priori information (55) where \BbbW stands for a class of Gevrey spaces demonstrated in Example 3.3, we specify below the possible regularity assumptions for different models chosen from simple to complex, based on the analysis of the forward models Note that \leq d \leq are only considered due to the practical meaning Copyright © by SIAM Unauthorized reproduction of this article is prohibited 82 HUY TUAN NGUYEN, VO ANH KHOA, AND VAN AU VO Downloaded 01/03/19 to 128.6.218.72 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php A.1 Fisher Burgers equation In a finite interval [0, l] with periodic boundary condition, one concerns the following equation: ut + Cuux = Duxx + Bu (1 - u) , with B, C, D being positive constants, for simplicity This problem is performed as a combination between the classical Fisher and Burgers equations Here we can further consider the real-analytic cases with respect to u of the nonlinear F which imply several types of modeling interactions between particles We know that in [20] the authors proved the local weak solvability of the forward problem In this sense, if the initial condition is sufficiently smooth, viz u0 \in H (\Omega ), then we obtain a unique solution u \in \BbbG 2,2 for any t \in [0, T \ast ] with T \ast t sufficiently small This not only verifies that the Gevrey regularity on the true solution could be valid in some certain models, but also agrees with the mild restriction of time in the convergence results A.2 \bfitp -Laplacian equation In a bounded domain with a H\" older boundary, we take into account (54) with the zero Dirichlet boundary condition Looking at [43], we can obtain the classical solution in L\infty (0, T ; W01,\infty (\Omega )) when u0 \in W 1,\infty (\Omega ) Together with the Fisher Burgers equation, we remark that these forward problems have interesting phenomena including, e.g., profiles of extinction and blow-up in finite time, the instantaneous shrinking of the support from the diffusion coefficient Depending on the situation one may need appropriate choices of the auxiliary parameter \rho \varepsilon involved in the regularized problem to keep track of the arisen phenomena Therefore, rigorous analysis of the regularized problem (10) (11) will be considered in forthcoming works A.3 Gray Scott Klausmeier model Based on the one-dimensional setting with \Omega = \BbbR in [41], we set u = (u1 , u2 ) with u1 > to guarantee the positive-definite diffusion a(u) Then the closed-form nonlinearities are \biggl( \biggr) \biggl( \biggr) 2u1 Cu1x + A (1 - u1 ) - u1 u22 a (u) = , F (u, ux ) = , D - Bu2 + u1 u22 where the involved parameters A, B, C, D are positive This model describes the interaction between water u1 and plant biomass u2 in semiarid landscapes The local well-posedness in H (\BbbR ) (cf [41, Theorem 2.2]) enjoys the possibility of taking W 1,\infty in (55) due to the embedding W 1,1 (\BbbR ) \subset L\infty (\BbbR ) A.4 Shigesada Kawasaki Teramoto model In a three-dimensional setting with no-flux boundary condition, we consider \biggl( \biggr) a10 + 2a11 u1 + a12 u2 a12 u1 a (u) = , a21 u2 a20 + 2a22 u2 + a21 u1 where the nonnegative coefficients aij satisfy 8a11 \geq a12 and 8a22 \geq a21 to fulfill the positive-definiteness of diffusion The source term is taken as the Lotka Volterra functions, which reads as \biggl( \biggr) (b10 - b11 u1 - b12 u2 ) u1 F (u) = , (b20 - b21 u1 - b22 u2 ) u2 where the coefficients bij are nonnegative Copyright © by SIAM Unauthorized reproduction of this article is prohibited