Uniqueness and nonuniqueness of solutions  to parabolic problemswith singular coefficients

Maria Assunta Pozio; Fabio Punzo; Alberto Tesei

doi:10.3934/dcds.2011.30.891

Article Contents

2011, Volume 30, Issue 3: 891-916. Doi: 10.3934/dcds.2011.30.891

This issue Previous Article Typical points for one-parameter families of piecewise expanding maps of the interval Next Article Billiards in ideal hyperbolic polygons

Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients

1.
Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", P.le A. Moro 5, I-00185 Roma
2.
Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, I-00185 Roma

Received: March 31, 2010

Revised: November 30, 2010

Published: August 2011

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Abstract

Uniqueness and nonuniqueness of solutions to the first initial-boundary value problem for degenerate semilinear parabolic equations, with possibly unbounded coefficients, are studied. Sub- and supersolutions of suitable auxiliary problems, such as the first exit time problem, are used to determine on which part of the boundary Dirichlet data must be given. As an application of the general results, we study uniqueness and nonuniqueness of bounded solutions to a semilinear Cauchy problem in the hyperbolic space $\mathbb{H}^n$ using the Poincaré model.

Keywords:

Degenerate parabolic equations; unbounded coefficients,
well-posedness; sub - supersolutions.

Mathematics Subject Classification: Primary: 35K65, 35K20; Secondary: 35B30.

Citation:

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