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

2021/06/22 by Razik, Belhaoues, Umberto Biccari, Biccari, Umberto +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2106.11620

openalex publication_date 2021/06/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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-Δu_t-div(\vert ∇ u\vert^p(.)-2∇ u) =|u|^q(.)-2uln(|u|), (x,t)∈Ω×(0,+∞), where Ω⊂ℝⁿ 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 H₀¹(Ω)-norm as t→+∞. This decay result is illustrated by a two-dimensional numerical example.

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