Boundedness and stability of a 2-D parabolic-elliptic system arising in biological transport networks

Abstract This paper is concerned with the Dirichlet initial-boundary value problem of a 2-D parabolic-elliptic system of the following form $$\begin{cases}m_{t}-\kappa\Delta m+|m|^{2(\gamma-1)}m=(m\cdot\nabla_{p})\nabla_{p}, \\-\nabla \cdot[(\mathbf{I}+m\otimes m)\nabla_{p}]=S, \end{cases}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi>κ</mml:mi> <mml:mi>Δ</mml:mi> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>(</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>⋅</mml:mo> <mml:msub> <mml:mi>∇</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>∇</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>⋅</mml:mo> <mml:mo>[</mml:mo> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo>⊗</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>∇</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo/> </mml:mrow> </mml:math> which was proposed to model the formation of biological transport networks. Even if global weak solutions for this system are known to exist, how to improve the regularity of weak solutions is a challenging problem due to the peculiar cubic nonlinearity and the possible elliptic singularity of the system. Global-in-time existence of classical solutions has recently been established, showing that finite time singularities cannot emerge in this problem. However, whether or not singularities in infinite time can be precluded was still pending. In this work, we show that classical solutions of the initial-boundary value problem are uniformly bounded in time as long as γ ⩾ 1 and κ is suitably large, closing this gap in the literature. Moreover, the uniqueness of classical solutions is also achieved based on the uniform-in-time bounds. Furthermore, it is shown that the corresponding stationary problem possesses a unique classical stationary solution which is semi-trivial, and that is globally exponentially stable, i.e., all solutions of the time-dependent problem converge exponentially fast to the semi-trivial steady state for κ large enough.