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 $1\leqslant m\leqslant q\leqslant +\infty$) 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.