An existence theorem for a nonlinear evolution equation in viscoelasticity

We establish the existence of weak global solutions of initial-boundary value problems for a partial differential equation
which occours as the equation of motion in nonlinear Kelvin solids with nonlinear stress components. Each functionai 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 constantsK0,K1, some non negative functionK2∈L1(0,T) and for allx∈Ω, t∈[0, T], ξ and ν∈R.