On the Cauchy problem for semilinear thermoelastic plate systems in the <i>L<sup>q</sup></i> framework

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.