Observations on blow up and dead cores for nonlinear parabolic equations
type: Item
instance of: scholarly article
title: Observations on blow up and dead cores for nonlinear parabolic equations
zbMATH Open document ID: 0661.35053
DOI: 10.1007/BF01215255
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174086
review text: Consider the following problem \((P_ 1):\) \[ (1)\quad u_ t-\Delta u=u^ p-a| \nabla u|^ q\quad t>0,\quad x\in \Omega, \] \[ u=0,\quad t>0,\quad x\in \partial \Omega;\quad u(0,x)=\Phi_ 1(x),\quad x\in \Omega. \] It has been known for quite a while that solutions of a simpler problem (without the gradient term) can blow up in finite time. Recently Chipot and Weissler investigated whether the blow up effect persists when one adds a dissipative term to the equation. They discussed the case \(1<p\) and \(1<q\leq 2p/(p+1).\) In the present paper the case \(q=2\) is studied in some detail. For positive a and \(1<p<2\) the gradient term is of no lower power than the source term \(u^ p\). Therefore the damping term does control the source term in the sense that there is no blow up. For any \(p>2\), however, one may still have blow up phenomena, so that the restriction on q of Chipot and Weissler appears to be technical. Blow up phenomena for other functions f(u), instead of \(u^ p\), are studied as well. An important step in the proofs is a transformation to so-called dead problems, such as \((P_ 2):\) \[ (2)\quad v_ t-\Delta v=-h(v)\quad t>0,\quad x\in \Omega, \] \[ v=1,\quad t>0,\quad x\in \partial \Omega;\quad v(0,x)=\Phi_ 2(x)\quad x\in \Omega. \] The function u blows up if and only if v tends to zero in finite time. Problem \((P_ 2)\) is analyzed by methods which are essentially due to Bandle and Stakgold. Finally the case \(a<0\) in \((P_ 1)\) is treated. This causes an effect which enhances blow up in the sense that sometimes blow up will occur on all of \(\Omega\) and not in a single point only. iii) If \(p>2\), there is single point blow up. The proof of this Theorem can be reduced to some known blow up results, once one finds the right transformation. If we set \(v(t,x)=e^{u(t,x)}- 1\), we obtain the transformed equation \[ v_ t-\Delta v=(1+v)(\log (1+v))^ p. \] Now the result follows immediately from the paper by \textit{A. A. Lacey} [Proc. R. Soc. Edinb., Sect. A…
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4080031
zbMATH Keywords: blow up
zbMATH Keywords: dissipative term
zbMATH Keywords: dead problems
author: Bernd Kawohl
author: Lambertus A. Peletier
MaRDI profile type: Publication
cites work: Remarks on quenching
cites work: Diffusion and reaction with monotone kinetics
cites work: The Formation of the Dead Core in Parabolic Reaction-Diffusion Problems
cites work: Complete blow-up after \(T_{\max}\) for the solution of a semilinear heat equation
cites work: A Mathematical Analysis of Blowup for Thermal Reactions—the Spatially Nonhomogeneous Case
cites work: Extinction properties of semilinear heat equations with strong absorption
cites work: Q3697477
cites work: Q4728729
cites work: Asymptotically self‐similar blow‐up of semilinear heat equations
cites work: Convexity of level curves for an initial value problem
cites work: Global blow-up of a nonlinear heat equation
cites work: The Quenching of Solutions of Some Nonlinear Parabolic Equations
cites work: On the Existence of Positive Solutions of Semilinear Elliptic Equations
cites work: Existence of solutions of quenching problems
cites work: Global behaviour of solution branches for some Neumann problems depending on one or several parameters.
cites work: A strong maximum principle for some quasilinear elliptic equations
cites work: Single point blow-up for a semilinear initial value problem
OpenAlex ID: W2049016679
description: scientific article; zbMATH DE number 4080031