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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.

##### MSC:
 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