Một trường hợp tổng quát cho các số hạng phi tuyến của bài toán

Một phần của tài liệu Tính giải được và các tính chất của nghiệm của một số bài toán biên phi tuyến (2) (Trang 135)

của bài toán

Với cùng phương pháp đã thực hiện, nếu xem xét bài toán (3.0.1) – (3.0.4) với các số hạng phi tuyến tổng quát gần giống với các số hạngFr,r 2 fp,q, p0, q0,r0, p1,q1,r1g,

thì kết quả vẫn còn đúng. Chi tiết hơn, chúng tôi giới thiệu lớp hàmLpsau đây:

Với mỗi pmà1 < p <∞, gọiLplà tập các hàm f : R ! Rliên tục, đơn điệu tăng trênRthỏa các điều kiện sau đây:

(A) Tồn tại các hằng số dươngC1,C01,C2,sao cho (i) x f(x) C1jxjp C10 8x 2R,

(ii) jf(x)j C2(1+jxjp 1) 8x 2 R;

(B) Nếup 2,thì

(i) 8M>0,9KM >0 :jf(x) f(y)j KMjx yj,8x,y 2[ M,M],

(ii) 9Cp >0 :(x y) (f(x) f(y)) Cpjx yjp,8x,y 2R;

(C) Nếu1< p <2,thì tồn tại hằng số dươngdp, sao cho

Chú ý rằngLpcòn có các tính chất sau (i) Fp 2 Lp, 1< p<∞,

(ii) f +g, αf 2 Lp,8f,g 2 Lp,8α >0, 1< p<∞,

(iii) λf + (1 λ)g2 Lp,8f,g2 Lp, 0 λ 1, 1< p <∞,(Lplà tập lồi), (iv) f +g 2 Lp, 8f 2 Lp,8g2 Lq, 2 q p <∞.

Và có thể nghiệm lại rằng hàm f xác định bởi f(x) = Ajxjp 2x+Bjxjq 2x+Cx+ D,trong đó2 q < p<∞, A>0,B>0,C>0,D 2Rlà các hằng số, cũng thuộcLp. Ta ký hiệu [p 1] để chỉ phần nguyên của p 1, với 1 < p < ∞, ký hiệu Lep = Lp\C[p 1](R;R), với1< p <∞.Khi đóLepcũng chứa tất cả các hàmFp, 1< p<∞.

Xét bài toán 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : utt x(µ(x,t)ux) +KΨp(u) +λΨq(ut) = F(x,t),0< x<1, 0<t<T, ( 1)iµ(i,t)ux(i,t) = KiΨpi(u(i,t)) +λiΨqi(ut(i,t)) +gi(t) Z t 0 ki(t s)Ψri(u(i,s))ds,i =0, 1, u(x, 0) = u˜0(x), ut(x, 0) = u˜1(x), (3.4.1) trong đóΨr 2 Ler, r 2 fp,q,p0, q0,r0, p1, q1, r1g, với các hằng số p,q, p0, q0, r0, p1,q1,

r1,K,λ,Ki,λivà các hàm cho trướcµ,F, ˜u0, ˜u1 gi,kithỏa các điều kiện như đã nêu cho bài toán (3.0.1) – (3.0.4), ta thu được các kết quả tương tự.

Kết luận chương 3

Bài toán (3.0.1) – (3.0.4) đã được xét trong [T1] với f(u,ut) là tuyến tính, tức là f(u,ut) =Ku+λut, vớiK 0,λ >0là các hằng số cho trước. Trong đó, chúng tôi đã thiết lập các kết quả về tồn tại toàn cục, về tính duy nhất, tính trơn của nghiệm yếu và

khai triển tiệm cận của nghiệm theo hai tham sốK,λ. Trong các chứng minh, phương

pháp Faedo-Galerkin liên kết với các đánh giá tiên nghiệm, hội tụ yếu, phương pháp đơn điệu và tính compact đã được áp dụng.

Vận dụng tất cả các phương pháp và kỹ thuật nói trên với sự cải tiến thích hợp, chương 3 đã mở rộng các kết quả trong bài báo [T1] cho trường hợp f(u,ut) = Kjujp 2u+

cho phương trình sóng phi tuyến dạng (3.0.1) - dạng phương trình đã và đang được quan tâm nghiên cứu bởi nhiều nhà toán học do những ứng dụng của nó trong nghiên cứu dao động của các vật liệu đàn hồi nhớt.

Với lớp các hàm phi tuyếnLep, 1 < p < ∞,các kết quả cho bài toán (3.0.1) – (3.0.4) vẫn còn đúng nếu các số hạng phi tuyếnFr, r 2 fp,q, p0, q0, r0, p1, q1,r1g được thay bởi các hàmΨr 2Ler,r2 fp,q,p0,q0,r0, p1,q1,r1g.

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