Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 78, pp. 1-13.

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

Author: Jonas Volek (Univ. of West Bohemia, Pilsen, Czech Republic)

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.