Long-time vanishing properties of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equations of tho following type (PE) partial derivative (t)u - del .A (x, t, u, delu) + f (x, u) = 0 where f(x, u) approximate to b(x)\u\(q-1)u, b being a nonnegative bounded and measurable function and q a real number such that 0 less than or equal to q < 1. We give criteria which imply that any solution of the above equations vanishes in finite time and these criteria are associated to semi-classical limits of some Schrodinger operators. We also give a series of sufficient conditions on b(x) which imply that any supersolution with positive initial data does not to vanish identically for any positive t. (C) 2001 Editions scientifiques et medicales Elsevier SAS.