Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms

Abstract

This study considers the blow-up in finite time and large-time behavioral properties of solutions to the initial-boundary value problem for generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms:v→t+(v→⋅∇)v→+∇P=div(ϰ|D(v→)|q−2D(v→)t+ν|D(v→)|p−2D(v→))+γ|v→|m−2v→,divv→(x,t)=0,(x,t)∈QT, where D(v→)=12(∇v→+∇v→T) is the rate of the strain tensor, v→(x,t) is the velocity field, P(x,t) is the pressure, ν is the viscosity kinematic coefficient, and ϰ is the viscosity relaxation coefficient. The coefficient γ and the exponents p, m are given constants.Two different cases are considered. In the case where γ<0 (with an absorption term), we prove that the solutions of the associated problem exhibit exponential and power decay, and, in the case where γ>0 (with nonlinear source term), the solutions blow up in a finite time under suitable assumptions on the exponents p, q, m and the coefficients ν, ϰ, γ, with certain initial data.