Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy-problem when 1 < p < 2

Abstract

From the introduction: "We continue here the investigation initiated earlier by us [Trans. Amer. Math. Soc. 314 (1989), no. 1, 187–224] concerning the solvability of the Cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation (1) ut − div(|Du| p−2 Du)=0 in ST ≡ RN ×(0,T), 0<T<∞, N≥1. We study the case 1<p<2, consider only nonnegative solutions and investigate the solvability of the Cauchy problem when (1) is associated with an initial datum (2) u0 ∈ L1 loc (RN), u0≥0. It turns out that the Cauchy problem is solvable whenever (2) holds, regardless of the growth of x→u0(x) as |x|→∞. The weak solutions are shown to be unique whenever the initial datum is taken in the sense of L1 loc(RN). We also prove that every nonnegative weak solution of (1) possesses, as initial trace, a σ-finite Borel measure μ≥0. The case 1<p<2 is noticeably different from the case p>2, both in terms of results and techniques. The main difference stems from the fact that, unlike solutions in the case p>2, solutions of (1) are not, in general, locally bounded.''