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This paper considers a semilinear integro-differential equation of Volterra type which interpolates semilinear heat and wave equations. Global existence of solutions is showed in spaces of Besov type based in Morrey spaces, namely BesovMorrey spaces. Our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat-wave equation. Some symmetries, self-similarity and asymptotic behavior of solutions are also investigated in the framework of Besov-Morrey spaces.

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This paper concerns with the heat equation in the half-space R n + with nonlinearity and singular potential on the boundary ∂R n + . We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak L p -spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in R n + . Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis − − → Ox n .

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