A well posedness result for nonlinear viscoelastic equations with memory

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 ρρ belongs to the interval [0,4][0,4], the value 4 being critical for the Sobolev embeddings, while ff can reach the critical polynomial order 5.