Asymptotic behaviour of the solution of a semilinear parabolic equation

We study the asymptotic behaviour ast tends to + of the solution of (u/t)–Lu+(u)–0,u_|=0 whereL is a second order self-adjoint elliptic operator and a maximal monotone graph of . If |(r)|/|r|²L¹ (-1, 1) and ₁ is the first eigenvalue ofL we prove thatu(.,t) converges uniformly on to some element of Ker (L + ₁I) and that the limit is nonzero if |(r)|/|r| is nondecreasing. We give also some properties of the limit (monotonicity, continuity, range).