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

Consider the following class of conformable time-fractional stochastic equation, for any x∈R fixed, Tα,tau(x,t)=λσ(u(x,t))Ẇt,t∈[a,∞),0<α<1, with a non-random initial condition u(x,0)=u0(x),x∈R assumed to be non-negative and bounded, Tα,ta is a conformable time-fractional derivative, σ:R→R is globally Lipschitz continuous, Ẇt 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∈(0,1).