Applications of the theory of rotation of vector fields to the study of wave solutions of parabolic equations. (English. Russian original) Zbl 0711.35064

Trans. Mosc. Math. Soc. 1990, 59-108 (1990); translation from Tr. Mosk. Mat. O.-va 52, 58-109 (1989).
The authors consider the parabolic system \[ \partial u/\partial t=a(\partial^ 2u/\partial x^ 2)+f(u),\quad x\in {\mathbb{R}},\quad t>0, \] where \(a\in M_{n,n}({\mathbb{R}})\) is a positive definite matrix, f: \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\) a given function. Solutions of the form \(u(x,t)=w(x-ct)\) are to be obtained (w is a travelling wave with unknown constant velocity c), for which \(\lim_{x\to -\infty}w(x)=w_ -\), \(\lim_{x\to +\infty}w(x)=w_+\), where \(w_ -\) and \(w_+\) are given vectors. We mention two important results of the paper.
1) The rotation of a class of operators strongly connected to the given problem is constructed, the properties of the rotation are studied (a complete theory).
2) Using the obtained theory it is proved that under certain conditions on f there exists a unique travelling wave for the given system with the limits \(w_ -\) and \(w_+\) at -\(\infty\) resp. \(+\infty\).
Reviewer: P.Szilagyi


35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs