Volterra and Functional Differential Equations

A NONLINEAR CONSERVATION LAW WITH MEMORY

J. A. Nohel

Matheatics Research Center University of Wisconsin - Madison Madison, Wisconsin

DOI: 10.1201/9781003420026-8

1 INTRODUCTION

In this paper we study the model nonlinear volterra functional differential equation (with infinite memory)

$${\text{u}}_{\text{t}}+\varphi \left(\text{u}\right){\text{x}}_{\text{x}}+{{\displaystyle \int }}_{-\infty }^{\text{t}}{\text{a}}^{\prime }\left(\text{t}-\text{τ}\right)\text{ψ}\left(\text{u}\left(\text{τ},\text{x}\right)\right){\text{x}}^{\text{dτ}}=\text{f}\left(\text{t},\text{x}\right),-\infty <\text{t}<\infty ,0\le \text{x}\le 1,$$(1.1)

where ϕ,ψ: ℝ → ℝ are given smooth constitutive functions, a: (0,∞) → ℝ is a given memory kernel, and f: ℝ ×[0,1] → ℝ is a given function representing an external force; subseripts denote partial derivatives and ‘= d/dt. The motivation and the assumptions under which (1.1) is studied are provided by the more ...