On the Cauchy problem for semilinear thermoelastic plate systems in the $L^q$ framework

TL;DR

This paper analyzes the semilinear thermoelastic plate systems in the entire space, deriving sharp estimates for solutions and establishing conditions for global existence or blow-up of solutions based on nonlinearities.

Contribution

It provides new sharp $(L^q o L^q)$ estimates for solutions and characterizes critical exponents for global existence and blow-up in the $L^q$ framework.

Findings

Derived sharp $(L^q o L^q)$ estimates for solutions.

Established conditions for global existence of small data solutions.

Identified critical exponents for blow-up of solutions.

Abstract

We mainly consider semilinear thermoelastic plate systems with general power nonlinearities in the whole space $\mathbb{R}^n$. By applying the Fourier analysis, some sharp $(L^q\cap L^m)-L^q$ estimates of solutions (with any $1\leqslant m\leqslant q\leqslant +\infty$) to the classical thermoelastic plate system are derived, which cover all known results in $\mathbb{R}^n$. Then, we investigate global in time existence of small data $L^q$ solutions (with any $q\in[1,+\infty]$) and blow-up of weak solutions for the semilinear thermoelastic plate systems under suitable conditions on the power exponents, which justify critical exponents for several classes of nonlinearities.