<abstract><p>In this paper, we study the double-diffusion perturbation equations when the flow is through a porous medium. If the initial conditions satisfy some constraint conditions, the Saint-Venant type spatial decay of solutions for double-diffusion perturbation equations is obtained. Based on the spatial decay bound, the structural stability for the double-diffusion perturbation equations is also established.</p></abstract>

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This article investigates the spatial behavior of the solutions of the Brinkman equations in a semi-infinite cylinder. We no longer require the solutions to satisfy any a priori assumptions at infinity. Using the energy estimation method and the differential inequality technology, the differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variables. In the case of decay, the structural stability of Brinkman fluid is also proved.