2022/11/06 by Sankar, S., Mohan, Manil T., Karthikeyan, S. · 1 citation
#35R60 #60H15 #74H35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2211.03111
In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction-diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by du1(t,x)amp;=amp;[ Δαu1(t,x)+γ1u1(t,x)+u^1+β12(t,x) ]dt amp; +k11u1(t,x)dBH1(t)+k12u1(t,x)dBH2(t), du2(t,x)amp;=amp;[ Δαu2(t,x)+γ2u2(t,x)+u^1+β21(t,x) ]dt amp; +k21u2(t,x)dBH1(t)+k22u2(t,x)dBH2(t), for x ∈ ℝd, t ≥ 0, along with ui(0,x)=fi(x), · amp;x ∈ ℝd, where Δα is the fractional power -(-Δ)^\fracα2 of the Laplacian, 0<α≤ 2 and βi>0, γi>0 and kij≥ 0, i,j=1,2 are constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that β1≥ β2>0 with Hurst index 1/2 ≤ H < 1, we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solutions to our considered system.