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

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)$, fractional time derivative is in sense of Caputo, $-(-\Delta)^{\frac{\alpha}{2}}$ Laplacian, $\sigma$ a positive parameter, $\dot{W}$ space-time white $u_0(x)$ assumed to be non-negative, continuous bounded. We study first equation on $[0,\,1]$ homogeneous Drichlet boundary condition show that solution blows up finite if only $b$ satisfies Osgood condition, \[ \int_{c}^{\infty} \frac{ds}{b(s)} <\infty \] for some constant $c>0$. then consider same whole line above satisfied whenever up.