Nonlocal complement value problem for a global in time parabolic equation

Abstract

The overreaching goal of this paper is to investigate the existence and uniqueness of weak solutions to a semilinear parabolic equation involving symmetric integrodifferential operators of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. The existence and uniqueness of a weak solution of the nonlocal complement value problem is proven under fair conditions on the interaction potential.

Weak solvability of a boundary value problem for a parabolic equation with a global-in-time term that contains a weighted integral

“…The strong solvability of the same problem was proven in [7], where the semigroup approach was employed. In [8], the authors consider the problem with the Laplace operator replaced by a more complex non-local operator that can be thought as the fractional Laplacian. The uniqueness of the solution is proven for sufficiently small T in all these works.…”