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

Philippe Souplet, and Slim Tayachi. "Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities." Colloquium Mathematicae 88.1 (2001): 135-154. <http://eudml.org/doc/283625>.

@article{PhilippeSouplet2001,
abstract = {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).},
author = {Philippe Souplet, Slim Tayachi},
journal = {Colloquium Mathematicae},
keywords = {positive radial solutions; semilinear parabolic equation; gradient nonlinearity; parabolic inequality},
language = {eng},
number = {1},
pages = {135-154},
title = {Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities},
url = {http://eudml.org/doc/283625},
volume = {88},
year = {2001},
}

TY - JOUR
AU - Philippe Souplet
AU - Slim Tayachi
TI - Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities
JO - Colloquium Mathematicae
PY - 2001
VL - 88
IS - 1
SP - 135
EP - 154
AB - 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).
LA - eng
KW - positive radial solutions; semilinear parabolic equation; gradient nonlinearity; parabolic inequality
UR - http://eudml.org/doc/283625
ER -