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Convergence to periodic fronts in a class of semiliner parabolic equations. (English) Zbl 0887.35070
The following system of nonlinear parabolic equations, which has applications to propagation of a combustion front in a solid-state periodically inhomogeneous mixture, is considered: $u_t-\Delta u = f(x,u,\nabla u). \tag{1}$ Here the nonlinear kinetic function $$f$$ is periodic in $$x$$ and $$u$$. Using a technique based on the maximum principle, it is proved that (1) has a unique solution of the form $u=\frac{t}{T} + v(x,t), \tag{2}$ where $$v(x,t)$$ has the same periodicity in $$x$$ as the function $$f$$ in (1), and is periodic in $$t$$ with a unique period $$T$$. Moreover, it is proved that the solution (2) is a dynamical attractor in a class of general $$x$$-periodic solutions to (1). In the particular case when the function $$f$$ in (1) does not depend on $$u$$, the function $$v$$ in the solution (2) does not depend on $$t$$, the first term in the solution being $$\lambda t$$ with a unique constant $$\lambda$$.

##### MSC:
 35K55 Nonlinear parabolic equations 80A25 Combustion
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