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 vertical bar partial derivative(t)u vertical bar(rho)partial derivative(tt)u - Delta partial derivative(tt)u + gamma(-Delta)(theta)partial derivative(t)u -alpha Delta u + integral(infinity)(0)mu(s)Delta u(t - s)ds + f (u) = h with Dirichlet boundary conditions. In particular, the parameter rho belongs to the interval [0, 4], the value 4 being critical for the Sobolev embeddings, while f can reach the critical (C) 2013 Elsevier Ltd. All rights reserved.