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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: October 10, 2012
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Abstract: We consider the Cauchy problem for the scalar conservation law

$\displaystyle \partial _t u+\partial _x f(u) =\displaystyle \frac {1}{g(u)},\qquad t>0,\ x\in \mathbb{R},$    

with $ g\in C^1(\mathbb{R})$, $ g(0)=0$, $ g(u)>0$ for $ u>0$, and assume that the initial datum $ u_0$ 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 $ u$ 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: http://dx.doi.org/10.1090/S0002-9939-2012-11694-6
PII: S 0002-9939(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