Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

• Autorzy: Philippe Souplet , Slim Tayachi
• Języki publikacji: EN

Abstrakty

EN

Consider the nonlinear heat equation (E): $u_{t} - Δu = |u|^{p-1}u + b|∇u|^{q}$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C₁(T-t)^{-1/(p-1)} ≤ ||u(t)||_{∞} ≤ C₂(T-t)^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{xx} ≥ u^{p}$. More general inequalities of the form $u_{t} - u_{xx} ≥ f(u)$ with, for instance, $f(u) = (1+u)log^{p}(1+u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary differential inequality v̇ ≥ f(v).

Kategorie tematyczne

• 35B60: Continuation and prolongation of solutions
• 35B35: Stability
• 35K60: Nonlinear initial value problems for linear parabolic equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 88
• Numer: 1
• Strony: 135-154

Daty

wydano

2001

Twórcy

autor

Philippe Souplet

• Département de Mathématiques INSSET, Université de Picardie, 02109 St-Quentin, France
• Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, France

autor

Slim Tayachi

• Département de Mathématiques, Faculté des Sciences de Tunis, Université Tunis II, Campus Universitaire, 1060 Tunis, Tunisia

Identyfikatory

DOI

10.4064/cm88-1-10