Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the behaviour of stochastic heat equations on bounded domains   Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial
_{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under
Dirichlet boundary condition, we show that in the long run, the second moment
of the solution grows exponentially fast if $\lambda$ is large enough. But if
$\lambda$ is small, then the second moment eventually decays exponentially. If
we replace the Dirichlet boundary condition by the Neumann one, then the second
moment grows exponentially fast no matter what $\lambda$ is. We also provide
various extensions.

29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph