Abstract:

In this paper, we consider the convergence problem and dispersive blow-up for the modified Kawahara equation. Firstly, we prove that $u(x,t)\rightarrow u_0(x),$ a.e. $x\in\mathbb{R}$ as $t\rightarrow 0$ by the Fourier restriction norm method, high-low frequency technique and Strichartz estimate, respectively. Here $u(x,t)$ is the solution of the modified Kawahara equation, and the initial value $u_0(x)\in H^{s}(\mathbb{R})$ $(s\geq\frac{1}{4})$. Secondly, using the Fourier restriction norm method, we show that $u(x,t)\rightarrow U(t)u_0(x)$ as $t\rightarrow 0$ with $u_0(x)\in H^{s}(\mathbb{R})$ $(s>0)$. Finally, we establish the dispersive blow-up of the modified Kawahara equation.

Key words: Modified Kawahara equation, Pointwise convergence, Uniform convergence, Dispersive blow-up