Strong solutions to a nonlinear stochastic aggregation-diffusion equation

Abstract

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to “regularize” the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on [Formula: see text]. If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model.