Global existence and non-existence of weak solutions for non-local stochastic semilinear reaction-diffusion equations driven by a fractional noise

Abstract

In the present paper, we study the existence and blow-up behavior to the following stochastic non-local reaction-diffusion equation: \begin{equation*} \left\{ \begin{aligned} du(t,x)&=\left[(Δ+γ) u(t,x)+\int_{D}u^{q}(t,y)dy -ku^{p}(t,x)+δu^{m}(t,x)\int_{D}u^{n}(t,y)dy \right]dt &\quad+ηu(t,x)dB^{H}(t), u(t,x)&=0, \ \ t>0, \ \ x\in \partial D, u(0,x)&=f(x) \geq 0, \ \ x\in D, \end{aligned} \right. \end{equation*} where $D\subset {\mathbb{R}}^d(d\ge 1)$ is a bounded domain with smooth boundary $\partial D$. Here, $k>0,\gamma ,\delta ,\eta \ge 0$ and $p,q,n>1,m\ge 0$ with $m+n\ge q\ge p$. The initial data $f$ is a non-negative bounded measurable function in class $C^2$ which is not identically zero. Here, ${\left\{B^H(t)\right\}}_{t\ge 0}$ is a one-dimensional fractional Brownian motion with Hurst parameter $\frac{1}{2}\le H<1$ defined on a filtered probability space $\left(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},\mathbb{P}\right)$. First, we estimate a lower bound for the finite-time blow-up and by choosing a suitable initial data, we obtain the upper bound for the finite-time blow-up of the above equation. Next, we provide a sufficient condition for the global existence of a weak solution of the above equation. Further, we obtain the bounds for the probability of blow-up solution.