Advances in Differential Equations

Spectra of critical exponents in nonlinear heat equations with absorption

V. A. Galaktionov and P. J. Harwin

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It has been known from the beginning of the 1980's that the global $L^1$ solutions of the classical porous medium equation with absorption $u_t = \Delta u^m - u^p$ in $\mathbb R^N \times \mathbb R_+$, with $m,p>1$ change their large-time behavior at the critical absorption exponent $p_0=m+2/N$. We show that, actually, there exists an infinite sequence $\{p_k, \, k \ge 0\}$ of critical exponents generating a countable subset of different non-self-similar asymptotic patterns. The results are extended to the fully nonlinear dual porous-medium equation with absorption.

Article information

Adv. Differential Equations, Volume 9, Number 3-4 (2004), 267-298.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B33: Critical exponents 35B40: Asymptotic behavior of solutions 47N20: Applications to differential and integral equations


Galaktionov, V. A.; Harwin, P. J. Spectra of critical exponents in nonlinear heat equations with absorption. Adv. Differential Equations 9 (2004), no. 3-4, 267--298.

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