Blow-up behaviour of one-dimensional semilinear parabolic equations

  • M.A. Herrero

    Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
  • J.J.L. Velázquez

    Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain

Abstract

Consider the Cauchy problem

where u0(x) is continuous, nonnegative and bounded, and F(u) = up with p > 1, or F(u) = eu. Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) → (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(−x) for x > 0, there holds

1) If with p > 1, then

uniformly on compact sets |ξ| ≦ R with R > 0,

2) If F(u) = eu, then

uniformly on compact sets |ξ| ≦ R with R > 0.

Résumé

On considère le problème de Cauchy

u0(x) est une fonction continue, non négative et bornée, et F(u) = up avec p > 1 ou F(u) = eu. Nous supposons que u explose au point x = 0 en temps T > 0. Dans ce travail, nous obtenons tous les comportements asymptotiques possibles de la solution u(x, t) quand (x, t) → (0, T).

Cite this article

M.A. Herrero, J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), no. 2, pp. 131–189

DOI 10.1016/S0294-1449(16)30217-7