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On a strongly damped wave equation for the flame front. (English) Zbl 1221.35252
This article is concerned with a nonlinear one-dimensional wave equation (second order in time, fourth order in space) including a damping operator. The equation studied is shown to arise heuristically from the two-dimensional model of near equidiffusive flames and describes the evolution of the perturbation of the planar front of a moving flame. Solutions are sought for periodic boundary conditions.
The authors investigate the stability of the null solution with respect to the relevant model parameter \(\alpha\) related to the Lewis number. This analysis is based on the analyticity of the underlying linear semigroup of the linearized equation. Next it is shown that for \(\alpha\) close to \(1\), solutions of the nonlinear wave equation are close to solutions of the Kuramoto-Sivashinsky equation for the same initial data up to renormalization. This result holds true over a fixed time scale. Finally, numerical experiments illustrate and confirm the behavior of solutions in the asymptotic limit. An additional appendix gives a systematic derivation of the linear part of the governing equation.

MSC:
35L76 Higher-order semilinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B20 Perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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