General Decay and Blow-up Results for a Thermo-viscoelastic System with Logarithmic Nonlinearity

This paper deals with a nonlinear thermo -viscoelastic system with logarithmic nonlinearity of the form \(\begin{aligned}&u_{tt} - \Delta u + \int _0^t g(t-s)\Delta u(s)ds+ v= |u|^{p-2}u \ln \vert u\vert , \\&v_{t} -\Delta v = u_{t}, \end{aligned}\) $$\begin{array}{cc}& u_{\mathit{tt}}-\mathrm{\Delta }u+{\int }_0^{t}g(t-s)\mathrm{\Delta }u(s)ds+v={|u|}^{p-2}uln|u|,\hfill \\ \hfill & v_t-\mathrm{\Delta }v=u_t,\hfill \end{array}$$ on a bounded domain \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ . By using the modified potential-well method, we first show the global existence and the finite-time blow-up results of solutions. Further, we give explicit and general decay rates of the energy functional associated with the global solution under a general class of relaxation function \(g\) $g$ which includes exponential, logarithmic, and polynomial rates. Besides, the lower and upper bounds for the blow-up time are also studied.