Singular initial conditions for the heat equation with a noise term

Abstract

We consider the equation

$$\begin{array}{r@{=}l}u_t \, =\, u_{xx}+ u^\gamma\dot{W}, \quad t>0, \; 0\leq x \leq J,\\[1ex]u(0, x) \, =\, u_0 (x),\\[1ex]u(t, 0) \, = \, u(t, J) =0,\end{array}$$

where $\dot{W} = \dot{W}(t,x)$ is two-parameter white noise. We show local existence and uniqueness for unbounded initial conditions satisfying certain conditions. Our results are motivated by earlier work, which showed that, for large $\gamma$, solutions of this equation can blow up. One would wish to show that solutions can be extended beyond blowup, and our results can be viewed as a step in that direction.