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We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very little regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
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Ana Rodríguez, Arturo de Pablo, Fernando Quirós, Anisotropic nonlocal diffusion equations with singular forcing. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 5, pp. 1167–1183DOI 10.1016/J.ANIHPC.2020.04.001