Positive effects of multiplicative noise on the explosion of nonlinear fractional stochastic differential equations

TL;DR

This paper demonstrates that multiplicative noise can delay the blow-up of solutions in nonlinear fractional stochastic PDEs, extending previous results and applying to fractional Keller-Segel and Fisher-KPP equations.

Contribution

It shows how noise influences blow-up times in fractional stochastic PDEs and extends existing results to fractional derivatives with new analytical methods.

Findings

Noise delays blow-up time in nonlinear fractional stochastic PDEs.

Existence and uniqueness of solutions proved using Galerkin approximation.

Validation of hypotheses in fractional Keller-Segel and Fisher-KPP equations.

Abstract

For the nonlinear stochastic partial differential equation which is driven by multiplicative noise of the form \[D_t^\beta u = \left[ { - {{\left( { - \Delta } \right)}^s}u + \zeta \left( u \right)} \right]dt + A\sum\limits_{m \in Z_0^d} {\sum\limits_{j = 1}^{d - 1} {{\theta _m}{\sigma _{m,j}}\left( x \right)} } \circ dW_t^{m,j},\;\; s \ge 1,\;\;\frac{1}{2} < \beta < 1,\] where $D_{t}^\beta $ denotes the Caputo derivative, $A>0$ is a constant depending on the noise intensity, $\circ$ represent the Stratonovich-type stochastic differential, we consider the blow-up time of its solutions. We find that the introduction of noise can effectively delay the blow-up time of the solution to the deterministic differential equation when $\zeta$ in the above equation satisfies some assumptions. A key element in our construction is using the Galerkin approximation and a priori estimates methods to prove the existence and uniqueness of the solutions to the above stochastic equations, which can be regarded as the fractional order extension of the conclusions in \cite{flandoli2021delayed}. We also verify the validation of hypotheses in the time fractional Keller-Segel and time fractional Fisher-KPP equations in 3D case.