This research investigates an abstract Cauchy problem associated with a coupled reaction–diffusion system of two equations, subject to homogeneous Dirichlet or Neumann boundary conditions. The primary objective is to establish the global existence of a weak solution for this nonlinear system.

The authors simplify the analysis by applying a linear transformation (v̄ = v − (c/(a − d))u) to the original system. This transformation effectively decouples the principal parabolic operators, resulting in a reformulated system where each equation features a single diffusion term with constant coefficients.

The central theoretical achievement (Theorem 2.1) is the proof of existence for a positive global weak solution (u, v̄) to the transformed system. The solution is constructed within the functional framework C([0, T]; L¹(Ω)) ∩ L¹(0, T; ) and is represented using variation-of-constants formulas involving two distinct contraction semigroups, {Sₐ(t)} and {}, generated by the operators aΔ and dΔ in L¹(Ω), respectively.

Supporting lemmas establish crucial a priori estimates. Lemma 2.1 confirms the positivity of the solution, while Lemma 2.2 provides a uniform L¹-bound on the sum u + v̄, dependent on initial data and domain parameters, which is vital for proving global existence.

The overall methodology hinges on the theory of compact semigroups and a priori estimates to overcome the nonlinearities represented by the reaction terms f̃ and g̃. This work contributes to the broader mathematical theory of nonlinear parabolic systems, with potential implications for modeling phenomena in physics, biology, and ecology where such coupled reaction–diffusion equations are prevalent.