Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)This paper is concerned with a fractional PDE that interpolates semilinear heat and wave equations. We show results on global-in-time well-posedness for small initial data in the critical Morrey spaces and space dimension n >= 1. We also remark how to derive the local-in-time version of the results. Qualitative properties of solutions like self-similarity, antisymmetry and positivity are also investigated. Moreover, we analyze the asymptotic stability of the solutions and obtain a class of asymptotically self-similar solutions.2541921957976Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [00334-0/2011]