Global behavior for Whitham type equations on a finite interval

Kaikina, Elena
September 2008
Calculus of Variations & Partial Differential Equations;Sep2008, Vol. 33 Issue 1, p113
Academic Journal
We study the initial-boundary value problem for nonlinear nonlocal equations on a finite interval $$\left\{\begin{array}{ll}u_{t}+\lambda |u|^{\sigma}u+\mathbb {K}u = 0,\quad (x, t) \in (0, a) \times \mathbf {R}^{+},\\ u(x, 0) = u_{0}(x),\quad \quad \quad \quad x \in [0, a],\quad \quad \quad \quad \quad (0.1)\\ u(a, t) = 0, \quad \quad \quad \quad \quad \quad t > 0 \end{array}\right.$$ where ? > 0 and pseudodifferential operator $$\mathbb {K}$$ is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the asymptotic representation in the case of the large initial data.


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