Advances in Differential Equations

Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity

Daniele Andreucci and Anatoli F. Tedeev

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We investigate the connection between the geometry of an unbounded domain $\Omega$ and the existence and qualitative behaviour of solutions to a degenerate doubly non linear parabolic equation posed in $\Omega$. The domain $\Omega$ is assumed to be ``narrowing'' at infinity in a suitable sense, so that it has infinite volume. On the boundary of $\Omega$ we prescribe a homogeneous Neumann condition. Among other results, we prove sharp estimates for the finite speed of propagation of the support of positive solutions originating from initial data with bounded support. This is done by means of a new approach, which is flexible enough to be applied to the geometry at hand, and to cover the case of initial data measures. We also show that even if the initial datum has finite mass, the solution need not be globally bounded over $\Omega$ for a fixed positive time. We provide sharp estimates for such solutions. Our main tool is a new embedding inequality connected with the geometry of $\Omega$.

Article information

Adv. Differential Equations, Volume 5, Number 7-9 (2000), 833-860.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35D05 35K55: Nonlinear parabolic equations


Andreucci, Daniele; Tedeev, Anatoli F. Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity. Adv. Differential Equations 5 (2000), no. 7-9, 833--860.

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