A note on positive solutions for conservation laws with singular source

Authors:
D. Amadori and G. M. Coclite

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1613-1625

MSC (2010):
Primary 35B25, 35B09, 35L65

DOI:
https://doi.org/10.1090/S0002-9939-2012-11694-6

Published electronically:
October 10, 2012

MathSciNet review:
3020849

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Cauchy problem for the scalar conservation law

with , , for , and assume that the initial datum is nonnegative.

We show the existence of entropy solutions that are positive a.e. by means of an approximation of the equation that preserves positive solutions and by passing to the limit using a monotonicity argument. The difficulty lies in handling the singularity of the right-hand side (the source term) as possibly vanishes at the initial time. The source term is shown to be locally integrable.

Moreover, we prove a uniqueness and stability result for the above equation.

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Additional Information

**D. Amadori**

Affiliation:
Department of Pure & Applied Mathematics, University of L’Aquila, Via Vetoio 1, 67010 Coppito (L’Aquila), Italy

Email:
amadori@univaq.it

**G. M. Coclite**

Affiliation:
Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy

Email:
coclitegm@dm.uniba.it

DOI:
https://doi.org/10.1090/S0002-9939-2012-11694-6

Keywords:
Singular nonlinear problems,
positive solutions,
conservation laws

Received by editor(s):
August 26, 2011

Published electronically:
October 10, 2012

Communicated by:
Walter Craig

Article copyright:
© Copyright 2012
American Mathematical Society