GLOBAL EXISTENCE FOR NONLINEAR PARABOLIC PROBLEMS WITH MEASURE DATA.APPLICATIONS TO NON-UNIQUENESS FOR PARABOLIC PROBLEMS WITH CRITICAL GRADIENT TERMS

Abstract

In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum:{(phi(v))(t) - Delta V-p = f(x, t)(1 + phi(v)) + mu in Omega x (0, + infinity),v(x, t) = 0 on partial derivative Omega x (0, + infinity),v(x, 0) = v(0)(x) in Omega,where 1 < p < N, Omega is a bounded open subset of (N >= 2), Delta(p)u = div(vertical bar Delta u vertical bar(p-2)del u)) is the so called p-Laplacian operator, phi(s) =[(1 + vertical bar s vertical bar/p-1)(p-1)-1] sign s., phi(v(0)) is an element of L-1(Omega), mu is a P-I finite Radon measure and f is an element of L-infinity(Omega x(0, T)) for every T > 0. Then we apply this existence result to show wild nonuniqueness for a connected nonlinear parabolic problem having a gradient term with natural growth.