Classical Solutions of the Equation of Local Fluctuations of Riesz Gravitational Fields and their Properties

We consider a pseudodifferential equation involving the Riesz operator of fractional differentiation, which is a natural generalization of the well-known equation of fractal diffusion. The fundamental solution of the Cauchy problem for this equation is the probability distribution density for the forces of local interaction of moving objects in the corresponding Riesz gravitational field. For this equation, we establish the correct solvability of the Cauchy problem in the class of unbounded discontinuous initial functions with integrable singularities. In addition, we determine the form of the classical solution of this problem, analyze the properties of smoothness, and investigate its behavior at infinity. Moreover, under certain conditions imposed on the fluctuation coefficient, we establish an analog of the maximum principle and apply it to prove the unique solvability of the Cauchy problem.