Analysis of the weak formulation of a coupled nonlinear parabolic system modeling a heat exchanger

This paper establishes the existence, uniqueness and time-space regularity of the weak solution to a nonlinear coupled parabolic system modeling temperature evolution in a coaxial heat exchanger with source terms and spatially varying coefficients. The system is formulated in a weak sense and the analysis relies on a Faedo-Galerkin method tailored to handle the nonlinear coupling and heterogeneous domains. Under suitable assumptions on the initial data and source terms, enhanced regularity in both time and space is obtained. In contrast with classical scalar models, the study addresses a multi-component system with realistic boundary conditions and complex interfacial dynamics.