Reaction-Diffusion Equations on a Square

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))