Mathematical results concerning a class of incompressible viscoelastic solids of differential type

In this paper we investigate several mathematical aspects concerning a class of incompressible viscoelastic solids of the differential type. The model that we consider can be viewed as a generalization of the Kelvin—Voigt viscoelastic solid. We obtain a uniqueness result and show that when the shear modulus of the viscoelastic solid is positive the solutions decay exponentially. We also show that if the shear modulus is negative, a physically unacceptable situation, we have exponential growth of the solutions, which is in keeping with physical expectations. The impossibility of localization of the solutions in finite time is also proved. The last section is devoted to the development of spatial decay estimates in the quasi-static case.