Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations

Abstract

We analyze the well-posedness of the initial-value problem for the semilinear equation in Marcinkiewicz spaces $L^{(p,\infty)}$. Mild solutions are obtained in spaces with the right homogeneity to allow the existence of self-similar solutions. As a consequence of our results we prove that the class $C([0,T);L^{p}(\Omega)),\ 0 < T\leq\infty, \ p={\frac{n(\rho-1)}{2\gamma }},$ $\Omega=R^{n},$ has uniqueness of solutions (including large solutions) obtained in [19], [20] and [8]. The asymptotic stability of solutions is obtained, and as a consequence, a criterion for self-similarity persistence at large times is obtained.