Nonlocal complement value problem for a global in time parabolic equation. (English) Zbl 1501.35130
Summary: The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator 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. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.
MSC:
| 35D40 | Viscosity solutions to PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R09 | Integro-partial differential equations |
| 47G20 | Integro-differential operators |
Keywords:
double nonlocality in space and timeReferences:
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