Nonlinear Analysis 73 (2010) 1966–1972 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na On a backward Cauchy problem associated with continuous spectrum operator Nguyen Huy Tuan a,∗ , Dang Duc Trong b , Pham Hoang Quan a a Department of Mathematics, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Viet Nam b Department of Mathematics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Viet Nam article abstract info Article history: Received 12 March 2009 Accepted 12 May 2010 The nonlinear backward Cauchy problem ut + Au(t ) = f (u(t )), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator, which has a continuous spectrum and f is a Lipschitz function being given is regularized by the well-posed problem The new error estimates of the regularized solution are obtained This work extends to the nonlinear case earlier results by the authors [7,1] and by Denche and Bessila [8,13] © 2010 Elsevier Ltd All rights reserved MSC: 35K05 35K99 47J06 47H10 Keywords: Nonlinear parabolic problem Backward problem Semigroup of operator Ill-posed problem Contraction principle Introduction Recently, in 2008, in [1] we have considered the nonlinear parabolic backward in time problem ut + Au = f (u(t )), < t < T, u(T ) = ϕ, (1) (2) where A is a self-adjoint operator, having a discrete spectrum on a Hilbert space H, with the inner product , and the norm , such that −A generates a compact contraction semi-group on H This problem is well known to be severely ill-posed and regularization methods for it are required We have established, under the hypothesis that f is a global Lipschitzian function from H to H, the existence of a unique solution for the approximated problem ut + f (A)u = f (u (t )), < t < T, u (T ) = ϕ, (3) (4) with < < and f (A) being chosen later under some better conditions We assume that (1)–(2) has a unique smooth solution u(t ) Under this assumption, we obtain an error estimate on the smoothing of u, generalizing a previous work of in [2] This error is given a form in Holder type u(t ) − u (t ) ≤ M β( )t /T ∗ Corresponding author E-mail address: tuanhuy_bs@yahoo.com (N.H Tuan) 0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.na.2010.05.025 N.H Tuan et al / Nonlinear Analysis 73 (2010) 1966–1972 1967 We note that stability estimates of the Holder type for the nonlinear heat-parabolic equation backwards in time have been established in some recent papers, such as [3–7,1] These results give no information on the continuous dependence of the solution on the data at t = And so, the convergence of the approximate solution is very slow when t is in the neighborhood of zero To our knowledge, the case where the operator A has a discrete spectrum has been treated in many recent papers, such as [8,9] Once more, we also note that there are many works on the linear homogeneous case of the backward Cauchy problem, but the literature on the nonlinear case of the problem is quite scarce So, it is not easy to regularize the nonlinear problems Recently, the nonhomogeneous and nonlinear backward problem in Banach space has been considered by Hetrict and Hughes [10,11] As we know, the position operator usually has a continuous spectrum, much like the momentum operator in an infinite space But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems, especially bound states, tend to have a discrete (quantized) spectrum—that is where the name quantum mechanics comes from The formal scattering theory has a strong overlap with the theory of continuous spectra In this paper, we shall use new methods to extend the continuous dependence results of [12,13] to more general nonlinear problems We also improve some related results given in [8,13,7,1] with two objectives First, the present work is a first step in the nonlinear backward Cauchy problem, in which the operator A has a continuous spectrum Thus, for some related questions on homogeneous parabolic equations backwards in time, as in the case A where has a continuous spectrum, we refer the reader to [12,13] As far as we know, there are few nonlinear backward Cauchy problems until now Second, we give some new error estimates, which are not of the Holder type The major object of this paper is to provide a quite simple and convenient new regularization method Meanwhile, some more faster convergence error estimates are given Especially, the convergence of the approximate solution at t = is also proved This paper is organized as follows In the next section, for ease of the reading, we summarize some well-known facts in semigroup of operator The stability estimates of the regularized solution will be presented in Section 2 The basic results In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis We denote by {Eλ , λ ≥ 0} the spectral resolution of the identity associated to A ∞ We denote by S (t ) = e−tA = e−t λ dEλ ∈ L(H ), t ≥ 0, the C0 -semigroup generated by −A Some basic definitions are listed in the following theorem See Chap 2, Theorem 6.13, in [14] For this family of operators we have: S (t ) ≤ 1, for all t ≥ 0; the function t −→ S (t ), t > 0, is analytic; for every real r ≥ and t > 0, the operator S (t ) ∈ L(H , D (Ar )); for every integer k ≥ and t > 0, S (k) (t ) = Ak S (t ) ≤ c (k)t −k ; for every x ∈ D (Ar ), r ≥ we have S (t )Ar x = Ar S (t )x Theorem Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space X over K Then there exists exactly one spectral family {Eλ } such that +∞ λdEλ u Au = (5) for all u ∈ D(A) In this connection, u ∈ D(A) iff the integral (5) exists, i.e., +∞ λ2 d Eλ u < ∞ Definition Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space H over K and let f , g : R → K be a piecewise continuous function We set +∞ D(f (A)) = |f (λ)|2 d Eλ u u∈H : independent of w, v ∈ H , t ∈ R Then problem (6) has a unique solution uα ∈ C ([0, T ]; H ) Proof of Theorem First, we consider the following function for λ > Rα (λ, 0) = (αλ + e−λT )−1 It is easy to prove that for < α < eT then Rα (λ, 0) ≤ Rα ln αT ,0 T = T α −1 ln(Teα −1 ) −1 (7) For ≤ t ≤ s ≤ T , then Rα (λ, T + t − s) = exp((s − t − T )λ)(αλ + e−λT ) t −s T (αλ + e−λT ) ≤ exp((s − t − T )λ)(αλ + e−λT ) t −s T (e−λT ) ≤ (Rα (λ, 0)) ≤α t −s T s−t −T T s−t −T T s−t T T −1 ln(Teα −1 ) t −s T = M (α, t )M −1 (α, s) (8) where t M (α, t ) = α T T −1 ln(Teα −1 ) t T −1 , t ∈ [0, T ] For w ∈ C ([0, T ]; H ), we define the operator F by ∞ F (w)(t ) = ∞ T Rα (λ, t )dEλ ϕ − Rα (λ, T + t − s)dEλ f (w(s))ds (9) t Now we prove that for all w, v ∈ C ([0, T ]; H ) the following inequality holds F m (w)(t ) − F m (v)(t ) ≤ (kT α −1 C )m |||w − v|||, m! (10) N.H Tuan et al / Nonlinear Analysis 73 (2010) 1966–1972 1969 where C = max{T , 1} and |||.||| is sup norm in C ([0, T ]; H ) In fact, for m = 1, using (8), the Lipschitz property of f and noting that Rα (λ, T + t − s) ≤ α t −s T t −s T T −1 ln(Teα −1 ) < α , we have 2 ∞ T F (w)(t ) − F (v)(t ) Rα (λ, T + t − s)dEλ (f (w(s)) − f (v(s)))ds = t ∞ T T d Eλ (f (w(s)) − f (v(s))) (Rα (λ, T + t − s))2 ds ≤ T −t ≤ 2 ds t T −t T f (w(s)) − f (v(s)) ds α t (T − t )k α ≤ ds ∞ T d Eλ (f (w(s)) − f (v(s))) α = t t w(s) − v(s) ds ≤ (kTC )2 α −2 |||w − v||| |2 (11) Suppose that (11) holds for m = j We prove that (11) holds for m = j + In fact, we have F (w)(t ) − F (v)(t ) j +1 2 ∞ T j +1 Rα (λ, T + t − s)dEλ (f (F w)(s) − f (F v)(s))ds j = t j (T − t )k T j F (w)(s) − F j (v)(s) ds α t (kT α −1 C )2j+2 |||w − v|||2 ≤ (j + 1)! ≤ Therefore, by the induction principle, we have (10) for all w, v ∈ C ([0, T ]; H ) We consider F : C ([0, T ]; H ) → C ([0, T ]; H ) (kT α −1 C )m Since limm→∞ = 0, there exists a positive integer number m0 such that F m0 is a contraction It follows that the m! equation F m0 (w) = w has a unique solution uα ∈ C ([0, T ]; H ) We claim that F (uα ) = uα In fact, one has F (F m0 (uα )) = F (uα ) Hence F m0 (F (uα )) = F (uα ) By the uniqueness of the fixed point of F m0 , one has F (uα ) = uα , i.e., the equation F (w) = w has a unique solution uα ∈ C ([0, T ]; H ) Now we have the following theorem in which, we show that the stability magnitude of our method is less than order one in the previous methods Theorem Let α be as in Theorem Then the (unique) solution of problem (1)–(2) depends continuously (in C ([0, T ]; H )) on ϕ , this means that, if u and v are two solutions of problem (6) corresponding to the final value ϕ and ω respectively then u(t ) − v(t ) ≤ √ t T t −T 2 2ek (T −t ) α T T −1 ln(Teα −1 ) −1 ϕ−ω Proof of Theorem It is well known that, for all t ∈ [0, T ], ∞ u(t ) − v(t ) = ∞ T Rα (λ, t )dEλ (ϕ − ω) − Rα (λ, T + t − s)dEλ (f (u(s)) − f (v(s)))ds t Using Lemmas and and the Lipchitz property of f we get ∞ u(t ) − v(t ) ≤ 2α −2 M (α, t ) d Eλ (ϕ − ω) ∞ T T (Rα (λ, T + t − s))2 ds +2 t d Eλ (f (w(s)) − f (v(s))) ds t T ≤ 2α −2 M (α, t ) ϕ − ω + 2k2 (T − t ) M (α, t )M −2 (α, s) u(s) − v(s) t ds 1970 N.H Tuan et al / Nonlinear Analysis 73 (2010) 1966–1972 Hence T M −2 (α, t ) u(t ) − v(t ) M −2 (α, s) u(s) − v(s) ≤ 2α −2 ϕ − ω + 2k2 (T − t ) 2 ds t Applying Gronwall’s inequality, we have M −2 (α, t ) u(t ) − v(t ) (T −t )2 α −2 ϕ − ω ≤ 2e2k Hence, we get u(t ) − v(t ) ≤ √ t −T 2 2ek (T −t ) α T T −1 ln(Teα −1 ) t T −1 ϕ−ω This inequality follows the solution of the problem (6) depending continuously on ϕ and Theorem is proved Remark In [7] (see p 238), and in [1], the authors give better stability estimates than the latter discussed methods They t T show that the stability estimate is of order T having the form of T −1 In [13] (see Theorem 2.2 page 2), the authors give another stability bound α(1+ln α ) In our paper, we give a better estimation of the stability order, which is t C α T −1 T −1 ln(Teα −1 ) t T −1 Comparing our results with the above related results, we see that the order of the error, introduced by small changes in the final value ϕ , is less than the order given in [4,8,13,7,1] This is among of the best advantages of our method Theorem Let u ∈ C ([0, T ]; H ) be a solution of (1)–(2) Assume that u has the eigenfunction expansion u(t ) = satisfying ∞ λ2 e2t λ d Eλ u(t ) < ∞, ∞ dEλ u(t ) (12) for every t ∈ (0, T ] Then, for any α ∈ (0, Te), k2 T u(t ) − uα (t ) ≤ N exp t α T T −1 ln(Teα −1 ) t T −1 , ∀t ∈ (0, T ], where ∞ N = sup λ2 e2t λ d Eλ u(t ) , t ∈[0,T ] and uα is the unique solution of Problem (6) Proof of Theorem The function u(t ), uα (t ) has the expansion ∞ u(t ) = ∞ T eλ(T −t ) dEλ ϕ − t T ∞ uα (t ) = ∞ Rα (λ, t )dEλ ϕ − eλ(s−t ) dEλ f (u(s))ds, Rα (λ, T + t − s)dEλ f (uα (s))ds t Hence, we get ∞ u(t ) − uα (t ) = (eT λ − αλ + e−λT ) e−λt dEλ ϕ − Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds + t ∞ = ∞ T αλRα (λ, t ) eT λ dEλ ϕ − t esλ dEλ f (u(s))ds ∞ T Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds + t ∞ t eλ(s−t −T ) dEλ f (u(s))ds ∞ T = ∞ T ∞ T α Rα (λ, t )λet λ dEλ u(t ) + αλRα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds t (13) N.H Tuan et al / Nonlinear Analysis 73 (2010) 1966–1972 1971 then ∞ u(t ) − uα (t ) ≤ 2M (α, t ) λ2 e2t λ d Eλ u(t ) ∞ T + (T − t )M (α, t ) ∞ ds t ≤ 2M (α, t ) d Eλ (f (w(s)) − f (v(s))) M −2 (α, s) λ2 e2t λ d Eλ u(t ) T M −2 (α, s) (f (w(s)) − f (v(s))) + (T − t )M (α, t ) ds t So, we obtain T M −2 (α, t ) u(t ) − uα (t ) M −2 (α, s) u(s) − uα (s) ds ≤ N + k2 (T − t ) t Using Gronwall’s inequality, we get u(t ) − uα (t ) 2T ≤ N ek 2t T 2t α T T −1 ln(Teα −1 ) −2 This completes the proof of Theorem Remark One superficial advantage of this method is that there is an error estimation in the original time t = 0, which is not given in many recently known results in [7,1] We have the following estimate u(0) − uα (0) ≤ NT ekT + ln T α −1 This error is similar to Theorem 2.6, page in [13] If f (u) = 0, we have ∞ u(0) = et λ dEλ u(t ) Taking the derivative of u, we get ∞ u (0) = − λe−t λ dEλ u(t ) Then, we get ∞ u (0) = Au(0) = λ2 e2λt d Eλ u(t ) Hence, the condition (12) is natural and acceptable In [7] (see p 241), and in [1] the error estimates between the exact solution and the approximate solution are of t the same order U (α, t ) = C α T And in this article, the convergence rate is in a slightly different form than given in [4, t t −1 V (α,t ) 8,13,1], defined by V (α, t ) = Dα T T −1 ln(Teα −1 ) T We note that limα→0 U (α,t ) = Hence, this error is the optimal error estimate to our knowledge Moreover, we also have limα→0 (limt →0 U (α, t )) = C and limα→0 (limt →0 V (α, t )) = limα→0 D T −1 ln(Teα −1 ) −1 = So, it is easy to see that if the time t is near to the original time t = 0, the convergence of the approximation solution is very slow This implies that the methods such as Quasi-boundary value and stabilized quasireversibility studied in [7,1], are not useful to derive the error estimations in the case where t is in the neighborhood of zero This also proves that our method give a better approximation than the previous case which we know Comparing (13) with the results obtained in [8,13,7,1], we realize that (13) is sharp and the best known estimate This is a generalization of many previous results Acknowledgements The authors would like to thank Professor Ravi P Agarwal for his valuable help in the presentation of this paper The authors are also grateful to the anonymous referees for their valuable comments leading to the improvement of our paper References [1] D.D Trong, N.H Tuan, Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems, Electron J Differential Equations (84) (2008) 1–12 [2] K Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems, in: Symposium on Non-Well Posed Problems and 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backwards in time, as in the case A where has a continuous spectrum, we refer the reader... nonlinear backward problem in Banach space has been considered by Hetrict and Hughes [10,11] As we know, the position operator usually has a continuous spectrum, much like the momentum operator in an... class of abstract parabolic ill-posed problems, Bound Value Probl 2006 (2006) Article ID 37524, 18 pages A Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,