Local well-posedness and small data scattering for energy super-critical nonlinear wave equations

Abstract

In this work, we consider the following nonlinear wave equations ${\mathrm{\partial }}_{tt}u-\mathrm{\Delta }u+|u{|}^{p}u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N.$ We prove that when $p>\frac{4}{N-2}$ and $\begin{array}{rl}& 3\le N\le 9;\text{or}N\ge 10,\\ & p<\frac{N^2-4N+1-\sqrt{N^4-8N^3-14N^2+56N-31}}{4(N-1)}.\end{array}$ The Cauchy problem is locally well-posed in ${\dot{H}}^{s_c}\left({\mathbb{R}}^N\right)\times \dot{\phantom{a}}{H}^{s_c-1}\left({\mathbb{R}}^N\right)$ with $s_c=\frac{N}{2}-\frac{2}{p}$. Moreover, the small data theory holds under the same restriction.

COMMUNICATED BY:

• Ming Mei

Keywords:

• Energy super-critical
• nonlinear wave equations
• local well-posedness
• scattering
• small initial data

2010 Mathematics Subject Classification:

• 35B40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are partially supported by NSFC (National Natural Science Foundation of China) [grant numbers 11771325 and 11571118].