Asymptotic estimates for solutions of inhomogeneous non-divergence diffusion equations with drifts

TL;DR

This paper analyzes the long-time behavior of solutions to inhomogeneous non-divergence diffusion equations with drifts, providing asymptotic estimates and convergence results for both linear and nonlinear cases.

Contribution

It introduces new asymptotic estimates for solutions of inhomogeneous non-divergence diffusion equations with drifts, including an inhomogeneous Landis-type Growth Lemma and convergence analysis.

Findings

Asymptotic estimates depend on the behavior of forcing and boundary data.

Solutions converge under conditions balancing nonlinear growth and data decay.

The inhomogeneous Growth Lemma is proven and applied to time-interval analysis.

Abstract

We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition. Starting with the reduced linear problem, we obtain the asymptotic estimates for the solutions, as time $t\to\infty$, depending on the asymptotic behavior of the forcing term and boundary data. These are established in both cases when the drifts are uniformly bounded, and unbounded as $t\to\infty$. For the nonlinear problem, we prove the convergence of the solutions under suitable conditions that balance the growth of the nonlinear term with the decay of the data. To take advantage of the diffusion in the non-divergence form, we prove an inhomogeneous version of the Landis-typed Growth Lemma and apply it to successive time-intervals. At each time step, the center for the barrier function is selected carefully to optimize the contracting factor. Our rigorous results show the robustness of the model.