Mathematische Annalen

, Volume 320, Issue 1, pp 87–113

Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

  • Marek Fila
  • Philippe Souplet
  • Fred B. Weissler
Original articles

DOI: 10.1007/PL00004471

Cite this article as:
Fila, M., Souplet, P. & Weissler, F. Math Ann (2001) 320: 87. doi:10.1007/PL00004471

Abstract.

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces \(L^q_\delta\), where \(\delta\) is the distance to the boundary. In particular, we prove an optimal \(L^q_\delta-L^r_\delta\) estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for \(t\geq \tau>0\) by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data.

Mathematics Subject Classification (1991): 35B45, 35K15, 35K60, 46E30, 47D06

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Marek Fila
    • 1
  • Philippe Souplet
    • 2
  • Fred B. Weissler
    • 4
  1. 1.Institute of Applied Mathematics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia (e-mail: fila@fmph.uniba.sk) SK
  2. 2.Département de Mathématiques, INSSET, Université de Picardie, 02109 St-Quentin, France FR
  3. 3.Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45, avenue des Etats-Unis, 78035 Versailles, France (e-mail: souplet@math.uvsq.fr) FR
  4. 4.LAGA, UMR CNRS 7539, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France (e-mail: weissler@math.univ-paris13.fr) FR