Stochastic porous media equations in ℝ<sup>d</sup>
Existence and uniqueness of solutions to the stochastic porous media equation dX - Delta psi(X)dt = XdW in R-d are studied. Here, W is a Wiener process, psi is a maximal monotone graph in R x R such that psi(r) <= C vertical bar r vertical bar(m), for all r is an element of R. In this general case, the dimension is restricted to d >= 3, the main reason being the absence of a convenient multiplier result in the space H = {phi is an element of S' (R-d); vertical bar xi vertical bar (F phi)(xi) is an element of L-2(R-d)}, for d <= 2. When psi is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space H-1(R-d). If psi(r)r >= rho vertical bar r vertical bar(m+1) and m = d-2/d+2, we prove the finite time extinction with strictly positive probability. (C) 2014 Elsevier Masson SAS. All rights reserved.
103
4
1024-1052
1024-1052
Elsevier