Diffusive relaxation to equilibria for an extended reaction-diffusion system on the real line

We study the long-time behavior of the solutions of a two-component reaction-diffusion system on the real line, which describes the basic chemical reaction $A <=> 2 B$. Assuming that the initial densities of the species $A, B$ are bounded and nonnegative, we prove that the solution converges uniformly on compact sets to the manifold $E$ of all spatially homogeneous chemical equilibria. The result holds even if the species diffuse at very different rates, but the proof is substantially simpler for equal diffusivities. In the spirit of our previous work on extended dissipative systems [18], our approach relies on localized energy estimates, and provides an explicit bound for the time needed to reach a neighborhood of the manifold $E$ starting from arbitrary initial data. The solutions we consider typically do not converge to a single equilibrium as $t \to +\infty$, but they are always quasiconvergent in the sense that their omega-limit sets consist of chemical equilibria.