Electron. J. Diff. Equ., Vol. 2014 (2014), No. 78, pp. 1-13.

Maximum and minimum principles for nonlinear transport equations on discrete-space domains

Jonas Volek

Abstract:
We consider nonlinear scalar transport equations on the domain with discrete space and continuous time. As a motivation we derive a conservation law on these domains. In the main part of the paper we prove maximum and minimum principles that are later applied to obtain an a priori bound which is applied in the proof of existence of solution and its uniqueness. Further, we study several consequences of these principles such as boundedness of solutions, sign preservation, uniform stability and comparison theorem which deals with lower and upper solutions.

Submitted December 30, 2013. Published March 19, 2014.
Math Subject Classifications: 82C70, 34A33, 35B50.
Key Words: Nonlinear transport equation; discrete-space domain; maximum principles; existence; uniqueness; sign preservation; uniform stability; nonlinear comparison theorem.

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Jonas Volek
Department of Mathematics and NTIS
New Technologies for the Information Society - European Centre of Excellence
Faculty of Applied Sciences, University of West Bohemia in Pilsen
Univerzitni 8, 306 14 Pilsen, Czech Republic
email: volek1@kma.zcu.cz

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