Critical regularity of nonlinearities in semilinear effectively damped wave models

Abstract

In this paper we consider the Cauchy problem for the semilinear effectively damped wave equation $ \begin{equation*} u_{tt}-u_{xx}+b(t)u_{t} = |u|^{3}\mu(|u|), \, \, \, u(0, x) = u_{0}(x), \, \, \, u_{t}(0, x) = u_{1}(x). \end{equation*} $ Our goal is to propose sharp conditions on $ \mu $ to obtain a threshold between global (in time) existence of small data Sobolev solutions (stability of the zero solution) and blow-up behaviour even of small data Sobolev solutions.

Keywords

• damped wave equations
• time dependent dissipation
• global existence
• blow-up
• critical regularity