Well-posedness of elliptic-parabolic differential problem with integral condition

Abstract

In this paper, we study a class of nonlocal boundary value problems for elliptic-parabolic equations subject to integral-type conditions. Such problems naturally emerge in various physical and engineering contexts, including diffusion processes in composite materials and systems with memory or nonlocal interactions. The model considered involves a mixed-type equation in which the elliptic and parabolic components are coupled through nonlocal boundary terms, while the boundary conditions incorporate integral constraints that generalize the traditional Dirichlet and Neumann formulations. To investigate the solvability of this problem, we employ analytical methods based on the theory of parabolic and elliptic operators in weighted H¨older spaces, which are particularly suitable for handling boundary singularities and ensuring regularity of solutions. We establish the existence, uniqueness, and continuous dependence of solutions on the input data, thereby proving the well-posedness of the problem. Furthermore, we derive coercivity inequalities for solutions of the associated mixed nonlocal boundary problems, which guarantee their stability and provide essential tools for studying related inverse and control problems. The findings extend several classical results and offer a unified approach to the analysis of nonlocal elliptic-parabolic models.