Coupled reaction-diffusion equations
Consider the system of equations of the form: \[ \partial u^{\epsilon}_ k(t,x)/\partial t=\epsilon L_ ku^{\epsilon}_ k(t,x)+\epsilon^{-1}f_ k(x,u^{\epsilon}_ k)+\sum^{n}_{j=1}d_{kj}(u^{\epsilon}_ j-u^{\epsilon}_ k), \] (*) \[ x\in {\mathbb{R}}^ r,\quad t>0,\quad u^{\epsilon}_ k(0,x)=g_ k(x)\geq 0,\quad k=1,2,...,n, \] where \(L_ k=\sum^{r}_{i,j=1}a_ k^{ij}(x) \partial^ 2/\partial x_ i\partial x_ j\) are elliptic operators with smooth coefficients, \(d_{kj}>0\), \(f_ k(x,\cdot)\in {\mathcal F}=\{f\in C^ 1({\mathbb{R}}):\) \(f(u)>0\), \(0<u<1\), \(f(0)=f(1)=0\), \(f(u)<0\) for \(u\in [0,1]^ c\) and \(f'(0)=\sup_{u>0}f(u)/u\}\), for \(x\in {\mathbb{R}}^ r\) and the initial data \(g_ k\) are bounded and continuous except possibly on the boundaries of their supports. If \(\epsilon =1\), \(k=1\) and \(d_{1j}=0\), one has the usual reaction diffusion equation, and the above equations (*) are generalizations of the original Kolmogorov-Petrovskij-Piskunov equation. The asymptotic behavior of the solution of (*) uses probabilistic methods, especially the large deviation principle. This may be stated in the author's words as: ``We describe the behavior of \(\lim_{\epsilon \downarrow 0}u^{\epsilon}_ k(t,x)\) for the solutions of (*). In the case of systems, in general, the motion of the wave front will not be continuous, [and] can have jumps. Evolution of the wave front, in general, will not be Markovian. If \(\partial f_ k(x,0)/\partial u=c\) is independent of x and k, the motion of the front will be continuous, Markovian and can be described by the Huygens principle. The corresponding velocity field is homogeneous and isotropic not in a Riemannian metric, but in a Finsler metric. The main machinery is the Feynman-Kac formula and the large deviation principle.'' Some generalizations are also discussed. The material is technical and the reader should consult the paper for precise details which cannot be outlined here.
elliptic operators with smooth coefficients
Kolmogorov-Petrovskij- Piskunov equation
scientific article; zbMATH DE number 4194888