Unconditional Global Stability of Solutions to High-Dimensional Parabolic-Elliptic Coupled System: Initial-Boundary Value Problem

Abstract

Abstract. This paper demonstrates the unconditional global stability of solutions to the initial-boundary value problem for a parabolic-elliptic coupled system in the two-dimensional or three-dimensional half-space. It is shown that as time approaches infinity, the solution converges to a planar rarefaction wave at a specific decay rate. The concept of unconditional global stability means that these conclusions hold under arbitrarily large initial perturbation and wave strength. The proof relies on transforming the parabolic-elliptic coupled system into a nonlocal scalar equation in the half-space. By capitalizing on this transformation, the maximum principle is employed to deduce the boundedness and monotonicity of the solution, which are crucial for energy estimates under large perturbations and the construction of one-dimensional smooth rarefaction waves. The primary challenge lies in the higher-order energy estimates due to boundary layer. We first estimate the energy of tangential derivatives and then use iterative methods based on the governing equations to derive estimates of all derivatives.