An existence theorem for a nonlinear evolution equation in viscoelasticity

Abstract

We establish the existence of weak global solutions of initial-boundary value problems for $$u_{tt} - \sum\limits_{i = 1}^N {\frac{d}{{dx_i }}a} ,(x,t,u_{x_i } ) - \Delta _N u_t = f$$ which occours as the equation of motion in nonlinear Kelvin solids with stress components σi=αi(x,t,uxi)+uxti=1,2...,N.. Each functiona i is required to be sufficiently smooth and must satisfy the following conditions:a)∣αi(x,t,η)∣⩽K0{∣η∣p-1+1}a)αi(x,t,η)≶K1∣η∣p-2η η≶0c)(∂/∂t)αi(x,t,η)≶K2(t){∣η∣p-2+1}η≶0d)[αi(x,t,η)—αi(x,t,ξ)].(η—ξ⩾0 for somep≥2, some positive constantsK 0,K 1, some non negative functionK 2∈L 1(0,T) and for allx∈Ω, t∈[0, T], ξ and ν∈R.