Topological invariants and solutions with a high complexity for scalar semilinear PDE
This paper is concerned with scalar parabolic equations with coefficients explicitly and strongly depending on spatial variables. It is well understood that a strong enough spatial variation of diffusion (or nonlinearity) leads to the existence of many stable solutions and implies a complicated enough behavior of solutions as \(t\to\infty\), especially in the case of one spatial variable [see \textit{J. K. Hale}, Math. Surveys and Monographs 25, Am. Math. Soc. (1988; Zbl 0642.58013) and Can. Appl. Math. Q. 5, 209-305 (1997; Zbl 0924.35002)). In the present paper the author studies equilibria and attractors which describe dynamics for large times, but the goal is to study dynamics generated by the parabolic equation, in particular to describe explicitly sets of functions which are invariant under the dynamics for all \(t\geq 0\), starting from the initial moment \(t=0\). Complexity of dynamics is determined by the size of the domain and very robust characteristics of coefficients. In particular, it is shown that the number of stable invariant components depends exponentially on the volume of the domain. The main tool is construction of conserved integer-valued functionals related to homotopy classes of functions and their spatial patterns. Persistence of stable oscillatory spatial patterns stems from persistence of certain homotopy invariants.
persistence of stable oscillatory spatial patterns
scientific article; zbMATH DE number 1556552