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

We analyze the well-posedness of the initial-value problem for the semilinear equation in Marcinkiewicz spaces L(p,∞). 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);Lᵖ(Ω)),\ 0 < T≤∞, \ p=n(ρ-1)/2γ , Ω=Rⁿ, 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.