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

Volume 88 / 2001

Philippe Souplet, Slim Tayachi Colloquium Mathematicum 88 (2001), 135-154 MSC: 35K60, 35B35, 35B60. DOI: 10.4064/cm88-1-10


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 $\dot v\geq f(v)$.


  • Philippe SoupletDépartement 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
  • Slim TayachiDépartement de Mathématiques
    Faculté des Sciences de Tunis
    Université Tunis II, Campus Universitaire
    1060 Tunis, Tunisia

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