The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed system of integral differential equations and a non- linear scalar integral differential equation. It wi...The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed system of integral differential equations and a non- linear scalar integral differential equation. It will be shown that there exist six standing wave solutions （（u（x,t）,w（x,t）） = （U（x）,W（x）） to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave so- lutions u（x,t） = U（x） to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.展开更多
This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial ...This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial data is in the following Sobolev spaces H 2, L 1∩H 2 with ∫(u 0, A 0)dx≠0, or L 1∩H 2 with ∫(u 0, A 0)dx=0, respectively. Most of the decay rates in these estimates are optimal. Moreover, the author proves various uniform stability results, like sup t>0 ‖(w, E, r)(t))‖ YC‖(w 0, E 0)‖ X, where X and Y are Sobolev spaces. It should be pointed out that the decay estimates of the solutions for the case (u 0, A 0)∈L 1∩H 2 follow from the uniform stability estimates. The author utilizes the Fourier splitting method invented by Professor Schonbek and the new elaborate global energy estimates.展开更多
文摘The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed system of integral differential equations and a non- linear scalar integral differential equation. It will be shown that there exist six standing wave solutions （（u（x,t）,w（x,t）） = （U（x）,W（x）） to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave so- lutions u（x,t） = U（x） to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.
文摘This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial data is in the following Sobolev spaces H 2, L 1∩H 2 with ∫(u 0, A 0)dx≠0, or L 1∩H 2 with ∫(u 0, A 0)dx=0, respectively. Most of the decay rates in these estimates are optimal. Moreover, the author proves various uniform stability results, like sup t>0 ‖(w, E, r)(t))‖ YC‖(w 0, E 0)‖ X, where X and Y are Sobolev spaces. It should be pointed out that the decay estimates of the solutions for the case (u 0, A 0)∈L 1∩H 2 follow from the uniform stability estimates. The author utilizes the Fourier splitting method invented by Professor Schonbek and the new elaborate global energy estimates.
