Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations

Abstract: This paper is concerned with the problem of regularization by noise of systems of reaction-diffusion equations with mass control. It is known that $\textit{strong}$ solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for sufficiently noise intensity and spectrum, the blow-up of strong solutions is delayed and an enhanced diffusion effect is also established. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the $L^p(L^q)$-approach to stochastic PDEs, highlighting new connections between the two areas.