کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6417117 | 1338528 | 2015 | 51 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs](/preview/png/6417117.png)
First we present a generalized implicit function theorem for abstract equations of the type F(λ,u)=0. We suppose that u0 is a solution for λ=0 and that F(λ,â ) is smooth for all λ, but, mainly, we do not suppose that F(â ,u) is smooth for all u. We state conditions such that for all λâ0 there exists exactly one solution uâu0, that u is smooth in a certain abstract sense, and that the data-to-solution map λâ¦u is smooth. Then we apply this to time-periodic solutions of first-order hyperbolic systemsâtuj+aj(x,λ)âxuj+bj(t,x,λ,u)=0 and second-order hyperbolic equationsât2uâa(x,λ)2âx2u+b(t,x,λ,u,âtu,âxu)=0. Here we have to prevent small divisors from coming up. Moreover, we need smooth dependence of bj and b on t to get smooth dependence of the solution on λ. This is completely different to the case of parabolic PDEs.
Journal: Journal of Differential Equations - Volume 259, Issue 11, 5 December 2015, Pages 6287-6337