Exponential stability in mean-square of parabolic quasilinear stochastic delay evolution equations

We initiate a study on the exponential stability of a parabolic quasilinear stochastic evolution equation with a variable delay which may be regarded as a perturbed quasilinear differential system. We first obtain the existence and uniqueness of a strong solution. Next, we establish exponential stability of a strong solution in the following sense: if the unperturbed system is exponentially stable and the perturbation is small enough, then the perturbed equation remains exponentially stable.