Abstract
In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate apbq (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present.
Highlights
Introduction and motivationIn this paper, we consider an initial-boundary value problem arising as a generic model for a one-step autocatalytic reaction
We examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications
The initial-boundary value problem is of semi-linear parabolic type, and in dimensionless form is given by ut = uxx + f (u) ∀(x, t) ∈ R × R+
Summary
We consider an initial-boundary value problem arising as a generic model for a one-step autocatalytic reaction. Q ≥ 1, the nonlinearity f : R → R is Lipschitz continuous on every closed bounded interval In this case, the initial-boundary value problem (1.1)–(1.4) has been studied extensively. The qualitative features concerning the solution to (1.1)–(1.5) with non-trivial initial data, in this case, do not exhibit travelling wave structure, but uniform convergence over x ∈ R, to the equilibrium state u = 1, as t → ∞. This represents a significant bifurcation in the structure of the solution to (1.1)–(1.5) for p ≥ 1 and 0 < p < 1, respectively.
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