A degenerate parabolic equation modelling the spread of an epidemic (Q1821955)

This paper concerns the diffusive approximation of a model for the spread of an epidemic in a closed population without remotion: \[ s_ t=s s_{xx}-s(1-s)\quad in\quad {\mathbb{R}}\times (0,T),\quad 0\leq s\leq 1. \] A counterexample for uniqueness of solutions in a natural Lipschitz class is given. The authors formulate a more appropriate definition of solution and prove uniqueness and existence is results. The solution is constructed using \(C^{\infty}\) approximations of initial data. It is also shown that supp s\(=\sup p s_ 0\), \(0\leq t\leq T\), where \(s_ 0\) is the initial data. The main results of this paper hold for equations, \(u_ t=f(u)u_{xx}+g(u)\), under suitable assumptions on f and g.