Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources

Abstract

In the paper, we consider a system of thermoelasticity of type I with viscoelastic damping and nonlinear sources. By using the Galerkin method and the Banach fixed point theorem, we first prove the local existence and uniqueness of weak solution. Secondly, by extending the potential well method, we prove that the local solution exists globally if its initial position starts inside a family of “potential wells.” In particular, we also establish an explicit and optimal decay rate of energy driven by the decay rate of the relaxation function which includes exponential, algebraic, and logarithmic decay rates. Finally, by using the continuation theorem and the concavity arguments due to Levine (Trans Am Math Soc 192:1–21, 1974), we show that the local solution blows up at finite time in the sense of Ball (Q J Math Oxf 28(4): 473–486, 1977) if its initial position starts outside the “potential wells.” Further, an upper bound for the blow-up time is also given explicitly. Notice that our results imply a sharp result on the global existence and blow-up of the local weak solution and they also allow a relatively large class of relaxation functions that generalize the existing results in the literature.