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Reaction-Diffusion Equations on a Square

  • Zhen Mei
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 28)

Abstract

We study in this chapter bifurcations of the reaction-diffusion equation
$$ \frac{{\partial u}}{{\partial t}} = G\left( {u,\lambda ,d} \right) $$
(10.1a)
with
$$ G\left( {u,\lambda ,d} \right): = \left( \begin{array}{l}\Delta {u_1} + {f_1}\left( {{u_1},{u_2},\lambda } \right) \\d\Delta {u_2} + {f_2}\left( {{u_1},{u_2},\lambda } \right) \\\end{array} \right) $$
on the unit square Ω:— [0,1] x [0,1] with the homogeneous Dirichlet boundary conditions
$$ {u_1} = 0,\quad {u_2} = 0\;on\;\partial \Omega $$
(10.1b)
Here u :— (u 1 , u 2 ) T are state variables representing concentrations of immediate products; λ ∈ R p is a vector of control parameters and d ∈ R is the diffusion rate of the second substance. The functions f i : R 2+p R, i = 1,2, describe reactions among the substances. They are supposed to be sufficiently smooth and have a polynomial growth
$$ \left| {\partial _x^i{f_j}\left( {x,\lambda } \right)} \right| \le {c_1} + {c_2}{\left\| x \right\|^r}\quad for\;\left\| x \right\| \to \infty ,\quad i = 1,2,3,\;j = 1,2 $$
for some constants c 1, c 2, r ≥ 0. Furthermore, we assume
$$ {f_i}\left( {0,0,\lambda } \right) = 0\quad for\;all\;\lambda \in R,\;i = 1,2 $$
(10.2)

Keywords

Hopf Bifurcation Bifurcation Point Mode Interaction Fredholm Operator Reaction Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Zhen Mei
    • 1
  1. 1.Department of MathematicsUniversity of MarburgMarburgGermany

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