SOME ASPECTS OF SOLUTIONS OF SPACE-TIME FRACTIONAL STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH OSGOOD CONDITION

Abstract

In this paper we discuss the following problem with additive noise, \[\begin{cases} \frac{\partial^{\beta} u(t,x) }{\partial t}=-(-\triangle)^{\frac{\alpha}{2}} u(t,x)+b(u(t,x))+\sigma\dot{W}(t,x),~~t>0, \\u(0,x)=u_{0}(x),\end{cases},\] where $\alpha \in(0,2) $ and $ \beta \in (0,1)$, the fractional time derivative is in the sense of Caputo, $-(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian, $\sigma$ is a positive parameter, $\dot{W}$ is a space-time white noise, $u_0(x)$ is assumed to be non-negative, continuous and bounded. We study first the equation on $[0,\,1]$ with homogeneous Drichlet boundary condition and show that the solution of the equation blows up in finite time if and only if $b$ satisfies the Osgood condition, \[ \int_{c}^{\infty} \frac{ds}{b(s)} <\infty \] for some constant $c>0$. We then consider the same equation on the whole line and show that the above Osgood condition is satisfied whenever the solution of the equation blows up.