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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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