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

TL;DR

This paper investigates the long-term behavior and non-existence of global solutions for a class of conformable time-fractional stochastic equations, highlighting conditions leading to solution blow-up and asymptotic properties.

Contribution

It provides new insights into the asymptotic behavior and blow-up phenomena of solutions to conformable time-fractional stochastic equations under specific conditions.

Findings

Solutions exhibit specific asymptotic behaviors depending on initial conditions.

Non-linear growth of the term causes finite-time blow-up of energy.

Conditions are identified under which solutions do not exist globally.

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, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ 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 $\alpha\in (0,1)$.