Qualitative behavior of conservation laws with reaction term and nonconvex flux

Abstract

The aim of the paper is to study qualitative behavior of solutions to the equation \[ ∂ u ∂ t + ∂ f ( u ) ∂ x = g ( u ) , \frac {{\partial u}}{{\partial t}} + \frac {{\partial f\left ( u \right )}}{{\partial x}} = g\left ( u \right ) , \] where ( x , t ) ∈ R × R + , u = u ( x , t ) ∈ R \left ( x, t \right ) \in \mathbb {R} \times {\mathbb {R}_ + }, u = u\left ( x, t \right ) \in \mathbb {R} . The main new feature with respect to previous works is that the flux function f f may have finitely many inflection points, intervals in which it is affine, and corner points. The function g g is supposed to be zero at 0 and 1, and positive in between.