Global Existence and Finite-Time Blow-Up for a Coupled Darcy-Forchheimer-Brinkman System with Quadratic Reaction Dynamics

TL;DR

This paper analyzes a complex coupled system modeling reactive transport in porous media, establishing conditions for global existence, decay, and finite-time blow-up of solutions, supported by numerical simulations.

Contribution

It provides new results on existence, uniqueness, decay, and blow-up phenomena for a nonlinear Darcy-Forchheimer-Brinkman system with quadratic reaction terms.

Findings

Global existence and uniqueness for initial data with concentration between 0 and 1.

Exponential decay of concentration to zero for certain initial conditions.

Finite-time blow-up occurs when initial concentration exceeds 1.

Abstract

We study a nonlinear system coupling the Darcy-Forchheimer-Brinkman equations with a convection-diffusion-reaction equation, arising in reactive transport through porous media. The model features a nonlinear viscosity coupling, Forchheimer inertial drag, convective transport, and a quadratic reaction term. We establish the existence of local-in-time weak solutions for general initial data. Under the physically relevant condition on initial data $0 \leq c_0 \leq 1$, a maximum principle for the concentration is proved, yielding global existence and uniqueness of weak solutions in two and three space dimensions. For higher regular initial data, we obtain the existence, uniqueness, and continuous dependence of strong solutions. In this regime, the concentration decays exponentially to zero in $L^p$-norm for all $1 \leq p \leq \infty$ with a uniform decay rate. In contrast, if $c_0 > 1$, we demonstrate the occurrence of finite-time blow-up of solutions and derive an explicit upper bound for the blow-up time. Finally, numerical simulations based on the finite element method are presented to illustrate both the decay behavior and finite-time blow-up predicted by the theory.