Evolution Equations With Temporary Memory: Nonlocal Problems, Analytical Frameworks, and High‐Order PDE Applications

ABSTRACT In this paper, we consider a family of first‐order evolution equations with temporary memory and boundary conditions. The domain is partitioned into subdomains; the values of the solution at interior points of these subdomains are treated as functional parameters. This allows us to reduce the original family of evolution equations with temporary memory to a family of initial‐interior problems on each subdomain for evolution equations with parameters. Using the solutions of these initial‐interior problems, we introduce general solutions in the sense of Dzhumabaev for the family of evolution equations with temporary memory and establish their fundamental properties. Based on these general solutions, together with the boundary condition and continuity conditions across the interior partition lines, we construct a system of functional equations with respect to the parameters. The coefficients and right‐hand sides of this system are determined by solving family initial‐interior problems for evolution equations on the subdomains. It is shown that the solvability of the problem is equivalent to the solvability of the constructed system. We propose methods for solving problems that rely on constructing and solving these parameter systems. The results extend the theory of evolution equations with temporary memory or discontinuities and provide a foundation for future numerical methods and applications in dynamical systems and neural networks.