Blow-up results for a logarithmic pseudo-parabolic p(.)-Laplacian type equation

Abita Rahmoune, Umberto Biccari. Blow-up results for a logarithmic pseudo-parabolic $p(.)$-Laplacian equation. (2021)

Abstract. In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic p(.)-Laplacian type equation with logarithmic nonlinearity:
$u_t-\Delta u_t-\text{div}\left(|\nabla u|^{p(.)-2}\nabla u\right) = |u|^{q(.)-2}u\ln(|u|), \quad (x,t)\in\Omega\times (0,+\infty) \\$

where $\Omega \subset \mathbb{R}^n$ is a bounded and regular domain, and the variable exponents p(.) and q(.) 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 $H^1_0(\Omega) \text{-norm as } t \rightarrow +\infty$.

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Arxiv: 2106.11620