2023/11/10 by Sankar, S., Mohan, Manil T., Karthikeyan, S.
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2311.05926
In the present paper, we study the existence and blow-up behavior to the following stochastic non-local reaction-diffusion equation: \ \beginaligned du(t,x)amp;=[(Δ+γ) u(t,x)+∫Duq(t,y)dy -kup(t,x)+δum(t,x)∫Dun(t,y)dy ]dt amp; +ηu(t,x)dBH(t), u(t,x)amp;=0, tgt;0, x∈ ∂ D, u(0,x)amp;=f(x) ≥ 0, x∈ D, \endaligned . where D⊂ ℝd (d ≥ 1) is a bounded domain with smooth boundary ∂ D. Here, k>0, γ, δ, η≥ 0 and p,q,n>1, m≥ 0 with m+n ≥ q≥ p. The initial data f is a non-negative bounded measurable function in class C2 which is not identically zero. Here, \ BH(t) \t ≥ 0 is a one-dimensional fractional Brownian motion with Hurst parameter (1)/(2) ≤ H<1 defined on a filtered probability space ( Ω, F, (Ft)t ≥ 0, ℙ ). 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.