Monotonicity in time of large solutions to a nonlinear heat equation

V. A. Galaktionov, A. A. Lacey

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We consider the Cauchy problem for a two-dimensional semi-linear heat equation with radial symmetry ut := urr+ur/r + eu in R+ × (0, T); with smooth, bounded initial data u0(r). We prove that the solution u(r, t) becomes strictly monotone in time, ut > 0, at any point where u is large enough. The proof is based on intersection comparison of u(r, t) with the set {w(·)} of stationary solutions satisfying w? + w' /r + ew = 0 for r > 0. The above monotonicity result is shown to depend essentially on the global structure of the set {w}. The same result is found to hold for positive solutions u to the equation with power nonlinearity ut = ?u + up, 1 < p < (N + 2)/(N - 2)+. Several generalizations to boundary value problems and quasi-linear equations are given.

Original languageEnglish
Pages (from-to)1279-1301
Number of pages23
JournalRocky Mountain Journal of Mathematics
Volume28
Issue number4
Publication statusPublished - Dec 1998

Keywords

  • Intersection comparison
  • Maximum principle
  • Semilinear heat equation
  • Stationary solutions

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