Global well-posedness for nonlinear wave equations with supercritical source and damping terms

TL;DR

This paper establishes the global existence and uniqueness of weak solutions for a class of nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus, extending previous results to larger source exponents.

Contribution

It proves global well-posedness for nonlinear wave equations with supercritical source terms, allowing the source exponent to exceed the critical threshold, which was not previously established.

Findings

Proves global existence of weak solutions for supercritical source terms.

Allows source exponent p to be larger than 6, surpassing traditional limits.

Demonstrates well-posedness on a three-dimensional torus.

Abstract

We prove the global well-posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus $\mathbb T^3$ of the prototype \begin{align*} &u_{tt}-\Delta u+|u_t|^{m-1}u_t=|u|^{p-1}u, \;\; (x,t) \in \mathbb T^3 \times \mathbb R^+ ; \notag\\ &u(0)=u_0 \in H^1(\mathbb T^3)\cap L^{m+1}(\mathbb T^3), \;\; u_t(0)=u_1\in L^2(\mathbb T^3), \end{align*} where $1\leq p\leq \min\{ \frac{2}{3} m + \frac{5}{3} , m \}$. Notably, $p$ is allowed to be larger than $6$.