zbMATH — the first resource for mathematics

A critical exponent in a degenerate parabolic equation. (English) Zbl 1007.35043
The global existence of positive solutions of the equation \(u_t=u^p \Delta u+u^q\) in \({\mathbb R}^n\) is studied for \(p,q\geq 1\). It is shown that all positive solutions are global but unbounded if \(1\leq q<p+1\) (\(1\leq q<3/2\) if \(p=1\)) and the initial function \(u_0\) decays to zero rapidly enough as \(|x|\to\infty\). On the other hand, for \(q=p+1\) all positive solutions blow up in finite time. Both global and nonglobal solutions exist if \(q>p+1\).

35K65 Degenerate parabolic equations
35B33 Critical exponents in context of PDEs
Full Text: DOI
[1] Fujita, Journal of the Faculty of Science University Tokyo. Section A Mathematics 16 pp 105– (1966)
[2] Weissler, Israel Journal of Mathematics 38 pp 29– (1981)
[3] Levine, SIAM Review 32 pp 262– (1990)
[4] Blow-up in Quasilinear Parabolic Equations. De Gruyter Expositions in Mathematics Springer: Berlin, 1995.
[5] Luckhaus, Journal of Differential Equations 69 pp 1– (1987)
[6] Some results on degenerate parabolic equations not in divergence form. Ph.D. Thesis, www.math1.rwth-aachen.de/Forschung-Research/d_emath1.html, Aachen, 2000.
[7] Wiegner, Nonlinear Analysis Theory, Methods and Applications 28 pp 1977– (1997) · Zbl 0874.35061
[8] Friedman, Archive for Rational Mechanics and Analysis 96 pp 55– (1987)
[9] Wiegner, Differential and Integral Equations 7 pp 1641– · Zbl 0728.35084
[10] Winkler, Zeitschrift f?r Analysis und ihr Anwendungen 20 pp 677– (2001) · Zbl 0987.35089
[11] Functional Analytic Methods for Partial Differential Equations. Dekker: New York, 1997. · Zbl 0867.35003
[12] Linear and Quasi-linear Equations of Parabolic Type. 177-190 American Mathematical Society: Providence, RI, 1968.
[13] Deng, Journal of Mathematical Analysis and Applications 243 pp 85– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.