Blow-up results for a logarithmic pseudo-parabolic $p(\mathrm{.})$-Laplacian type equation

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(\mathrm{.})$-Laplacian type equation with logarithmic nonlinearity: $$u_t-\mathrm{\Delta }{u}_t-div\left({\mid \mathrm{\nabla }u\mid }^{p(\mathrm{.})-2}\mathrm{\nabla }u\right)=\mathrm{\mid }u{\mathrm{\mid }}^{q(\mathrm{.})-2}u\ln (\mathrm{\mid }u\mathrm{\mid }),(x,t)\in \mathrm{\Omega }\times (0,+\mathrm{\infty }),$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded and regular domain, and the variable exponents $p(\mathrm{.})$ and $q(\mathrm{.})$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(\mathrm{\Omega })$-norm as $t\to +\mathrm{\infty }$. This decay result is illustrated by a two-dimensional numerical example.