Existence of Global Positive Solutions to Order-[ m ] Tridiagonal Reaction-Diffusion Systems via Semigroup Methods

Semilinear parabolic tridiagonal reaction-diffusion systems of order [ m ] model coupled diffusion-reaction processes in physics, biology, and chemistry. These systems take the form\frac{\partial U}{\partial t}-D\Delta U=F\left( U\right) \text{ \ \ \ \ dans \ }\Omega \times \left( 0,+\infty \right) , with Neumann boundary conditions and positive initial data . Here, [ \Delta_m ] denotes the tridiagonal Laplacian matrix. Proving global existence, uniqueness, and positivity of solutions is vital for understanding long-term dynamics, yet remains challenging due to nonlinear reactions. This work establishes these properties for a broad class of reaction terms [ f_i ].

We employ compact semigroup theory generated by the tridiagonal diffusion operator [ A = \operatorname{diag}(d_1,\dots,d_m) \Delta_m ] on [ [C(\overline{\Omega})]^m ]. Key tools include positivity preservation through maximum principles and a priori bounds via comparison principles and fixed-point arguments in suitable Banach spaces, handling general nonlinearities with sublinear growth

Under assumptions of positive initial data and reaction terms satisfying [ f_i(t,x,\xi) \geq 0 ] for [ \xi \geq 0 ] with controlled growth, the system admits a unique global positive solution remaining bounded for all [ t > 0 ].

These results provide a robust framework for tridiagonal reaction-diffusion systems, applicable to multi-species models. The semigroup-[ L^1 ] approach extends to higher-order or non-local interactions, opening avenues for complex pattern formation studies.