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Spectrum of elliptic operators and stability of travelling waves. (English) Zbl 0952.35081
The authors consider an initial-boundary value problem for a class of semilinear parabolic systems. The spatial domain is an infinite cylinder \(\Omega\subset {\mathbb{R}}^n\) with the axis in the \(x_1-\)direction, the section of the cylinder being a bounded domain. The main results concern stability of the travelling wave solutions, \(u(x,t)=w(x_1-ct,x_2,\dots,x_n)\), where the constant \(c\) is the wave velocity. First, the authors study the location of the spectrum of the corresponding linear elliptic operator in \(\Omega\), which is obtained by a linearization about the travelling wave \(w(x)\). Then they prove the local and global stability of monotone travelling wave solutions. Finally, a minimax representation for the wave velocity is given.

35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35K55 Nonlinear parabolic equations