Asymptotic behavior of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

Abstract

Consider the following class of conformable time-fractional stochastic equation $$T_{α,t}^a u(x,t)=λσ(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0<α<1,$$ with a non-random initial condition $u(x,0)=u_0(x),\,x\in\mathbb{R}$ assumed to be non-negative and bounded, $T_{α,t}^a$ is a conformable time - fractional derivative, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is globally Lipschitz continuous, $\dot{W}_t$ a generalized derivative of Wiener process and $λ>0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $λ$. We also show that when the non-linear term $σ$ grows faster than linear, the energy of the solution blows-up at finite time for all $α\in (0,1)$.