Singular perturbation for a class of nonlinear reaction diffusion systems
Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\) and consider the singularly perturbed reaction-diffusion system \[ u_ t=(\epsilon L_ 1(t,x)+L_ 2(t,x))u+f(t,x,u,w,\epsilon),\quad w_ t=g(t,x,u,w,\epsilon), \] for \(0<t\leq T\) and \(x\in \Omega\) together with initial conditions for u, w and Dirichlet boundary conditions for u. Here, u and w are vector-valued functions of t and x, \(L_ 1\) is a uniformly elliptic operator of second order and diagonal form, \(L_ 2\) a differential operator of first order (both with respect to x), and \(\epsilon >0\) is a small parameter. The functions f and g are assumed to be strictly decreasing in u and w. Making use of comparison theorems, the author establishes an asymptotic expansion formula to arbitrary order for u and w with respect to \(\epsilon\). This expansion is uniformly valid throughout [0,T]\(\times {\bar \Omega}\).
scientific article; zbMATH DE number 4165480