Large-Time Behavior for Conservation Laws with Source in a Bounded Domain

Abstract

Here u=u(x, t), u # R. Precise assumptions will be given later. We are, in particular, interested in the asymptotic behavior of solutions to problem (1) (3). Equation of the form (1) is called reaction-convection equation or balance law. The correct setting to deal with for such equations is the one given by entropy solutions, in the sense of Kruz$ kov in [15]. In such class it is well known that the Cauchy problem has a unique solution, continuously depending on initial data in L-norm. In this article we consider boundary conditions and we analyze the influence of such conditions on the whole solution as time goes on. Such conditions for equations of the form (1) were considered in [1]. It turns out that, in order to have existence, uniqueness and continuous dependence, conditions (3) have to be interpreted in a non classical fashion. Following [1], we look for entropy solutions of (1), such that their boundary values belong to appropriate sets (details will be given later on). Article ID jdeq.1999.3669, available online at http: www.idealibrary.com on