Consider the stochastic partial differential equation
¶ $u_t=u_{xx}+u^\gammaW$
¶ where $x\in\mathbf{I}\equiv[0,J], W=W(t,x)$ is 2-parameter white
noise, and we assume that the initial function $u(0,x)$ is nonnegative and not
identically 0. We impose Dirichlet boundary conditions on u in the
interval I. We say that u blows up in finite time, with positive
probability, if there is a random time $T < \infty$ such that
$$ P(\lim_{t \uparrow T} \sup_{x} u(t, x) = \infty)
> 0. $$
It was known that if $\gamma<3/2$, then with
probability 1, $u$ does not blowup in finite time. It was also known that there
is a positive probability of finite time blowup for $\gamma$ sufficiently
large.
¶ We show that if $\gamma>3/2$, then there is a positive
probability that u blows up in finite time.