Convergence to periodic fronts in a class of semiliner parabolic equations (Q1376809)

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 \(\lambda\).