Existence and asymptotic stability for viscoelastic evolution problems on compact manifolds
One considers the nonlinear viscoelastic evolution equation u(tt) + Au + F(x,t,u,u(t)) - g * Au = 0 on Gamma x (0, infinity) where Gamma is a compact manifold. When F not equal 0 and g = 0 we prove existence of global solutions as well as uniform (exponential and algebraic) decay rates. Furthermore, if F = 0 and g not equal 0 we prove that the dissipation introduced by the memory effect is strong enough to allow us to derive an exponential( or polynornial) decay rate provided the resolvent kernel of the relaxation function decays exponentially (or polynomially).
asymptotic stability viscoelastic evolution problem