Existence and blow-up of solutions to the fractional stochastic heat equations

Abstract

In this work, we will show existence and blow-up of the solution to $$\frac{\partial }{\partial t} u_{t}(x) = \mathcal {L} u_{ t}(x) + \sigma (u_{t}(x))\eta $$ on a circle with space-time white noise $$\eta $$ . The operator $$\mathcal {L}$$ is taken to be the generator of a Levy process (superset of fractional derivatives) and $$\sigma $$ is a nonlinear function of form $$\sigma (x) \propto |x|^\gamma $$ , for $$\gamma >1$$ . We will develop a precise condition for existence or blow-up of the solution in terms of $$\gamma $$ and the Levy process corresponding to $$\mathcal {L}$$ .