ABSTRACT
In this short chapter we consider real, parabolic, stochastic initial-Neumann boundary value problems of the form
du(x, t) = (u(x, t) + g(u(x, t)))dt + h(u(x, t))W (x, dt), (x, t) ∈ D ×R+,
u(x, 0) = ϕ(x), x ∈ D, ∂u(x, t) ∂n(x)
= 0, (x, t) ∈ ∂D ×R+ (18.1)
on a bounded domain D ⊂ Rd with a smooth boundary ∂D and satisfying the cone property, where d ∈ N+ and W (., t) is an L2(D)-valued Wiener process to be described below. Our aim is to investigate the long-time behavior of solutions to (18.1), assuming that the nonlinearities g and h are real-valued, twice continuously differentiable functions on a finite interval [u0, u1], vanish at the boundary points and are strictly positive on (u0, u1); moreover, we also assume that g is concave. Our main result is that for any d ∈ N+, the solutions to (18.1) converge to an asymptotic random variable which takes values in the set {u0, u1} provided the ratio hg remains bounded; moreover, if d = 1 or d = 2, and if the derivative g′(u0) is small enough, we show that the asymptotic state is equal to u1 almost surely. Also, if we replace g by −g in (18.1) and if |g′(u1)| is small enough, the asymptotic state is equal to u0 almost surely. In the second part of the article we consider the case of a one-dimensional (1D) heat equation driven by a space-time white noise. Under similar conditions as those outlined above, we also prove the convergence of solutions to a random variable which takes values in {u0, u1}, but this time a different method of proof is necessary.
