Mathematics and Computer Science, Volume 2

21Viscoelastic Equation of p-Laplacian Hyperbolic Type with Logarithmic Source Term

Nazlı Irkıl^⋆ and Erhan Pişkin^†

Department of Mathematics, Dicle University, Diyarbakır, Turkey

Abstract

This paper aims to address the problem with viscolelastic p-Laplacian type equations with logarithmic nonlinearity,

in which p ≥ 2, under a convenient hypotheses on g (t). Under suitable conditions, we discuss global existence and blow up the results.

Keywords: Existence, blow up, viscoelastic equation, p–Laplacian type equation, logarithmic nonlinearity

21.1 Introduction

Let Ω be a bounded domain in Rⁿ (for n ≥ 1) with smooth boundary Σ = ∂Ω. We deal with the following problem in the initial boundary problem for (x, t) ∈ Ω × R⁺:

The memory kernel g(t) is a real function which has typical properties. In (21.1), if we take ln |u| = 1 and replace the second order as the fourth order, we have

Equation (21.2) was named as a fourth order weak viscoelastic plate equation with a lower order perturbation of p-Laplacian type and it can be taken as a general form of the one-dimensional nonlinear equation of elastoplastic microstructure flows. This type of equation was first considered by Andrade et al ...