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