Von Karman thermoelastic plates: Existence and nonexistence of global solutions

Abstract

This paper undertakes a study of behavior of solutions corresponding to von Karman thermoelastic plates involving a nonlinear source. It is well-known that the source term gives rise to finite-time blow-up of solutions and drives the system to possible instability. So, it is of interest to explore the mechanism of the source and the dissipation. In this paper, on the one hand, for low initial energy, we apply modified potential well method and differential inequality argument to prove the solution globally exists and decays exponentially to zero if the initial data starts from the stable set and fails to exist globally when the initial data starts from the unstable set. On the other hand, for high initial energy, with the aid of the monotonicity of the $ L^{2} $ norm of the solution with a positive perturbed constant and some differential inequality argument as well as energy inequalities, we are in a position to prove that the solution to the studied problem blows up in finite time. At last, an explicit lower bound for blow-up time is obtained.