A degenerate parabolic equation modelling the spread of an epidemic. (English) Zbl 0617.35066

Ughi, Maura

Ann. Mat. Pura Appl., IV. Ser. 143, 385-400 (1986).

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.

Reviewer: S.M.Lenhart

MSC:

35K65	Degenerate parabolic equations
35K55	Nonlinear parabolic equations
92D25	Population dynamics (general)
35D05	Existence of generalized solutions of PDE (MSC2000)
35K15	Initial value problems for second-order parabolic equations

Keywords:

diffusive approximation; spread of an epidemic; closed population without remotion; uniqueness; existence; approximations of initial data

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