Infinite energy solutions for critical wave equation with fractional damping in unbounded domains

TL;DR

This paper establishes the existence of infinite-energy solutions for a critical wave equation with fractional damping in unbounded domains, extending previous bounded domain results and introducing new Lyapunov functionals for better regularity analysis.

Contribution

It extends known results to unbounded domains, proves well-posedness and attractor existence for critical non-linearity, and introduces a novel Lyapunov functional for regularity.

Findings

Existence of infinite-energy solutions in unbounded domains.

Well-posedness and attractor existence for critical quintic non-linearity.

Development of a new Lyapunov functional enabling enhanced regularity analysis.

Abstract

This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of $\mathbb{R}^3$ with fractional damping of the form $(-\Delta_x+1)^\frac{1}{2}\partial_t u$. The work extends previously known results for bounded domains in finite energy case. Furthermore, well-posedness and existence of locally-compact smooth attractors for the critical quintic non-linearity are obtained under less restrictive assumptions on non-linearity, relaxing some artificial technical conditions used before. This is achieved by virtue of new type Lyapunov functional that allows to establish extra space-time regularity of solutions of Strichartz type.