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

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 $1⩽m⩽q⩽+\infty$) 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.

It updates the new version with general L^q estimates and nonlinearities