Journal d’Analyse Mathématique

, Volume 99, Issue 1, pp 355–396

Global solutions of inhomogeneous Hamilton-Jacobi equations

  • Philippe Souplet
  • Qi S. Zhang

DOI: 10.1007/BF02789452

Cite this article as:
Souplet, P. & Zhang, Q.S. J. Anal. Math. (2006) 99: 355. doi:10.1007/BF02789452


We consider the viscous Hamilton-Jacobi (VHJ) equationutu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas considerable effort has been devoted to study the large time behavior of solutions of the equationutu=g(x,u), whereamplitude blow-up may occur if for instanceg(x,u)up asu→∞ andp>1, relatively little is known in the case of (VHJ). The aim of this paper is to investigate this question. More precisely, we study the relations between
  1. (i)

    the existence of global classical solutions

  2. (ii)

    the existence of stationary solutions (with gradient possibly singular on the boundary);

and we obtain a precise description of the global dynamics for (VHJ). Namely, we show that (i) implies (ii) and that in this case, all global solutions converge uniformly to the (unique) stationary solution. In the radial case, we prove that, conversely, (ii) implies (i). Moreover, for certain (smooth) functionsh, we obtain the existence of global classical solutions with gradient blowing up in infinite time. For 1p-2 or for the Cauchy problem, all solutions are global, but we establish similar relations between the existence of bounded or locally bounded solutions and the existence of stationary solutions. Our proofs depend on some new gradient estimates of solutions, local and global in space, obtained by Bernstein type arguments. As another consequence of these estimates we prove a parabolic Liouville-type theorem for solutions ofut\t-Δu=│Δup in ℝNx(\t-\t8,0). Various other results are obtained, including universal bounds for global solutions.

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • Philippe Souplet
    • 1
  • Qi S. Zhang
    • 2
  1. 1.Laboratoire Analyse Géométrie et Applications UMR CNRS 7539, Institut GaliléeUniversité Paris-NordVilletaneuseFrance
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA