Self-similarity and Uniqueness of Solutions for Semilinear Reaction-Diffusion Systems

We study the well-posedness of the initial value problem for a coupled semilinear reaction-diffusion system in Marcinkiewicz spaces $L^{(p_{1},\infty)}(\Omega)\times L^{(p_{2},\infty)}(\Omega)$. The exponents $p_{1},p_{2}$ of the initial value space are chosen to allow the existence of self-similar solutions (when $\Omega=\mathbb{R}^{n}$). As a nontrivial consequence of our coupling-term estimates, we prove the uniqueness of solutions in the scaling invariant class $C([0,\infty);L^{p_{1}}(\Omega)\times L^{p_{2}}(\Omega))$ regardless of their size and sign. We also analyze theasymptotic stability of the solutions, show the existence of a basin of attraction for each self-similar solution and that solutions in $L^{p_{1}}\times L^{p_{2}}$ \ present a simple long time behavior.