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

Philippe Souplet ; Slim Tayachi

Colloquium Mathematicae, Tome 89 (2001), p. 135-154 / Harvested from The Polish Digital Mathematics Library

Résumé

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).

Détail

Publié le : 2001-01-01

Zbl 0984.35077

EUDML-ID : urn:eudml:doc:283625