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

Philippe Souplet¹ ; Slim Tayachi²

¹ Département de Mathématiques INSSET Université de Picardie 02109 St-Quentin, France and Laboratoire de Mathématiques Appliquées UMR CNRS 7641 Université de Versailles 45 avenue des Etats-Unis 78035 Versailles, France
² Département de Mathématiques Faculté des Sciences de Tunis Université Tunis II, Campus Universitaire 1060 Tunis, Tunisia

Colloquium Mathematicum, Tome 88 (2001) no. 1, pp. 135-154

Résumé

Consider the nonlinear heat equation (E): $u_t-{\mit \Delta } u=|u|^{p-1}u+b|\nabla u|^q$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C_1 (T-t)^{-1/(p-1)} \leq \| u(t)\| _\infty \leq C_2 (T-t)^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if $u$ only satisfies the nonlinear parabolic inequality $u_t-u_{xx}\geq u^p$. More general inequalities of the form $u_t-u_{xx}\geq f(u)$ with, for instance, $f(u)=(1+u)\mathop {\rm log}\nolimits ^p(1+u)$ 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 $\dot v\geq f(v)$.

Détail

DOI : 10.4064/cm88-1-10

Keywords: consider nonlinear heat equation t mit delta p nabla prove large class radial positive nonglobal solutions has blowup estimates t t p leq infty leq t t p application method obtain upper estimate only satisfies nonlinear parabolic inequality t u geq general inequalities form t u geq instance mathop log nolimits treated results solutions parabolic inequality has essentially estimates solutions ordinary differential inequality dot geq