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 $$T_{\alpha ,t}^{a}u(x,t)=\lambda \sigma (u(x,t)){\dot{W}}_t,x\in \mathbb{R},t\in [a,\infty ),0<\alpha <1,$$ with a non-random initial condition $u(x,0)=u_0(x),x\in \mathbb{R}$ assumed to be non-negative and bounded, $T_{\alpha ,t}^a$ is a conformable time - fractional derivative, $\sigma :\mathbb{R}\to \mathbb{R}$ is globally Lipschitz continuous, ${\dot{W}}_t$ a generalized derivative of Wiener process and $\lambda >0$ 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 $\lambda$. We also show that when the non-linear term $\sigma$ grows faster than linear, the energy of the solution blows-up at finite time for all $\alpha \in (0,1)$.

15 pages, submitted for publication, some crutial corrections to the previous version is made