Nonlocal Kolmogorov–Petrovskii–Piskunov and Fisher equation on a closed manifold and its solutions
摘要
We study the dynamics of a resource distributed on a closed smooth manifold, for example, on a two-dimensional sphere—the Earth’s surface. It is assumed that this dynamics is described by the nonlocal Kolmogorov–Petrovskii–Piskunov and Fisher equation, the nonlocality of which is expressed by the dependence of the reaction term of the equation on the integral of the product of the sought solution with some integral kernel over the manifold. For example, if this kernel is equal to one, we obtain the dependence of the reaction term on the total volume of the resource on the manifold. Under natural restrictions on the parameters of the equation, a uniqueness theorem for the Caushy problem is proved f on assumption that initial data is bounded and nonnegative, and the solution has a continuous