Nonlinear integroparabolic equations on unbounded domains: existence of classical solutions with special properties

Classical solvability is established for a certain nonlinear integrodifferential parabolic equation, on unbounded domains in several dimensions. The model equation of the Fokker–Planck type represents a regularized version of an equation recently derived by J. A. Acebron and R. Spigler for the physical problem of describing the time evolution of large populations of nonlinearly globally coupled random oscillators. Precise estimates are obtained for the decay of convolutions with fundamental solutions of linear parabolic equations on unbounded domains in $R^n$. Existence of a classical solution with special properties is established.