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 $n\geq1$. 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.
Lucas C.F. Ferreira. Marcelo Fernandes de Almeida. "Self-similarity, symmetries and asymptotic behavior in Morrey spaces for a fractional wave equation." Differential Integral Equations 25 (9/10) 957 - 976, September/October 2012. https://doi.org/10.57262/die/1356012377