Well-posedness and maximum principles for lattice reaction-diffusion equations

Antonín Slavík,Petr Stehlík,Jonáš Volek

doi:10.1515/anona-2016-0116

Antonín Slavík, Petr Stehlík + Show 1 more

Open Access

https://doi.org/10.1515/anona-2016-0116

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Abstract

Abstract Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity.

Highlights

The classical reaction-diffusion equation ∂t u = k∂xx u + f(u) is a nonlinear partial differential equation frequently used to describe the evolution of numerous natural quantities
Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. We study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time
This section is devoted to the study of continuous dependence of solutions to abstract dynamic equations with respect to the choice of the time scale

Summary

Introduction

Our focus lies on the existence, uniqueness, continuous dependence (both on the initial condition as well as on the underlying time structure/numerical discretization), and a priori bounds in the form of weak and strong maximum principles. Note that both continuous dependence and maximum principles are key assumptions in the proofs of the existence of traveling waves [21, 35]. These a priori bounds, as usual, depend strongly on the time structure. In the linear case f ≡ 0, the weak maximum principle was already proved in [29, Theorem 4.7], but the strong maximum principle is new even for linear equations

Local existence and uniqueness of solutions

Continuous dependence results

Weak maximum principle and global existence

Strong maximum principle