Volterra and Functional Differential Equations

FORMATION OF SINGULARITIES FOR A CONSERVATION LAW WITH DAMPING TERM

Reza Malek-Madani*

Department of Mathematics University of Wisconsin Madison, Wisconsin

* Current affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia.

DOI: 10.1201/9781003420026-20

1. INTRODUCTION

In this paper, we consider the behavior of smooth solutions of two first order hyperbolic equations which have lower order dissipation. The equations are

$$\begin{array}{c}{\text{u}}_{\text{t}}+\varphi {(\text{u)}}_{\text{x}}=\text{a(x)u}\\ \text{u(x,0)}={\text{u}}_{\text{0}}\end{array}$$ (1.1)

and

$$\begin{array}{c}{\text{u}}_{\text{t}}+\varphi {\text{(u)}}_{\text{x}}+{\int }_{\text{0}}^{\text{t}}{\text{a}}^{\prime }\text{(t}-\text{τ})\text{ψ}{(\text{u)}}_{\text{x}}\text{dτ}=0\\ \text{u(x,0)}={\text{u}}_{\text{0}}\text{(x) }\text{.}\end{array}$$ (1.2)

Problem (1.1) ...