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Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations
Authors:Shuang Ping Tao  Shang Bin Cui
Institution:(1) Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China;(2) Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China
Abstract:This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation $$
\frac{{\partial u}}
{{\partial t}} + a\frac{{u^{2} \partial u}}
{{\partial x}} + \beta \frac{{\partial ^{3} u}}
{{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}}
{{\partial x^{5} }} = 0,
$$ (x,t) ∈ R 2. We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function u 0(x) ∈ H s (R) with s ≥ 1/4, and a global solution exists if s ≥ 2. Supported by NWNU-KJCXGC-212 and NWNU-QNJSJJ
Keywords:Kawahara equation  Initial value problem  Solution  Local existence  Global existence
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