Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Global existence and non-existence of weak solutions for non-local stochastic semilinear reaction-diffusion equations driven by a fractional noise (2311.05926v1)

Published 10 Nov 2023 in math.PR

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[(\Delta+\gamma) u(t,x)+\int_{D}u{q}(t,y)dy -ku{p}(t,x)+\delta u{m}(t,x)\int_{D}u{n}(t,y)dy \right]dt &\quad+\eta u(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 $D\subset \mathbb{R}{d}\ (d \geq 1)$ is a bounded domain with smooth boundary $\partial D$. Here, $k>0, \gamma, \delta, \eta \geq 0$ and $p,q,n>1,\ m\geq 0$ with $m+n \geq q\geq p$. 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 \geq 0} $ is a one-dimensional fractional Brownian motion with Hurst parameter $\frac{1}{2} \leq H<1$ defined on a filtered probability space $\left( \Omega, \mathcal{F}, (\mathcal{F}{t})_{t \geq 0}, \mathbb{P} \right)$. 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.

Summary

We haven't generated a summary for this paper yet.