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Optimal bounds and blow up phenomena for parabolic problems in narrowing domains

Published online by Cambridge University Press:  14 November 2011

Daniele Andreucci  and
Anatoli F. Tedeev
Daniele Andreucci
Affiliation:
Dip. Metodi e Modelli, Università La Sapienza, via AScarpa 16, 00161 Rome, Italy
Anatoli F. Tedeev
Affiliation:
Institute of Applied Mathematics, Academy of Sciences, R. Luxemburg st. 74, 340114 Donetsk, Ukraine
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Extract

We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1163 - 1180
Copyright
Copyright © Royal Society of Edinburgh 1998

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