Existence and non-existence of global solutions for a discrete semilinear heat equation
Keisuke Matsuya Tetsuji Tokihiro
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 209-220 doi: 10.3934/dcds.2011.31.209
Existence of global solutions to initial value problems for a discrete analogue of a $d$-dimensional semilinear heat equation is investigated. We prove that a parameter $\alpha$ in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for $0<\alpha \le 2/d$, and, for $\alpha > 2/d$, existence of non-trivial global solutions for sufficiently small initial data.
keywords: global solution. semilinear heat equation Discretization
Spatial pattern of discrete and ultradiscrete Gray-Scott model
Keisuke Matsuya Mikio Murata
Discrete & Continuous Dynamical Systems - B 2015, 20(1): 173-187 doi: 10.3934/dcdsb.2015.20.173
Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.
keywords: Gray-Scott model cellular automaton ultradiscretization pattern formation. discretization

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