Classic solutions to the equation of local fluctuations of the Riesz gravitational fields and their properties - Litovchenko , V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2022)
UDC 517.937, 519.21We consider a pseudodifferential equation involving the Riesz operator of fractional differentiation, which is a natural generalization of the well-known equation of fractal diffusion. Its fundamental solution to the Cauchy problem is the density of probability distribution of local interaction forces for 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 an integrable singularity. In addition, the form of the classical solution of this problem is found and its smoothness properties and behavior at infinity are investigated. Moreover, under certain conditions on the fluctuation coefficient, we obtain an analogue of the maximum principle and use it to prove the uniqueness of the solution to the Cauchy problem.