The local theory for viscous hamilton-Jacobi equations in Lebesgue spaces

Matania Ben-Artzi, Philippe Souplet*, Fred B. Weissler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

99 Scopus citations

Abstract

We consider viscous Hamilton-Jacobi equations of the form where a ε ℝ, a ≠ 0 and p ≥ 1. We provide an extensive investigation of the local Cauchy problem for (VHJ) for irregular initial data u0, namely for u0 in Lebesgue spaces Lq = Lq (ℝN), 1 ≤ q < ∞. The case of initial data measures or in Sobolev spaces is also considered. ut - Δu = a ∇up, x ε ℝN, t > 0, u(x, 0) = u0(x), x ε ℝN, When p < 2, we prove well-posedness in Lq for q ≥ qc = N (p - 1)/(2 - p). This holds without sign restriction neither on a nor on u0. In the case a > 0 and u0 ≥ 0 (repulsive gradient term) we show that existence fails in all Lq spaces when p ≥ 2. When p < 2, we prove that both existence and uniqueness fail if 1 ≤ q < qc. Rather surprisingly, in the case a < 0 and u0 ≥ 0 (absorbing gradient term), we show that existence holds in L1 while it may fail in measures. More precisely, we obtain existence in Lq for any q ≥ 1 when p ≤ 2 (and also for p > 2 under some additional assumption on u0), whereas nonexistence occurs for a large class of measure initial data if p > (N+2)/(N+1). In particular, a critical exponent for existence and uniqueness in the scale of Lq spaces appears if the gradient term is repulsive, while none occurs if it is absorbing.

Original languageEnglish
Pages (from-to)343-378
Number of pages36
JournalJournal des Mathematiques Pures et Appliquees
Volume81
Issue number4
DOIs
StatePublished - 2002

Keywords

  • Critical exponents
  • Lebesgue spaces
  • Nonexistence
  • Nonlinear parabolic equations
  • Nonuniqueness
  • Viscous Hamilton-Jacobi equations
  • Well-posedness

Fingerprint

Dive into the research topics of 'The local theory for viscous hamilton-Jacobi equations in Lebesgue spaces'. Together they form a unique fingerprint.

Cite this