A well posedness result for nonlinear viscoelastic equations with memory

Abstract

We establish an existence, uniqueness and continuous dependence result for the weak solutions to the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain |∂tu|ρ∂ttu−Δ∂ttu+γ(−Δ)θ∂tu−αΔu+∫0∞μ(s)Δu(t−s)ds+f(u)=h with Dirichlet boundary conditions. In particular, the parameter ρ belongs to the interval [0,4], the value 4 being critical for the Sobolev embeddings, while f can reach the critical polynomial order 5.