Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

Xiangsheng Xu

doi:10.3934/dcds.2020280

Article Contents

2020, Volume 40, Issue 11: 6289-6307. Doi: 10.3934/dcds.2020280

This issue Previous Article The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition Next Article On morawetz estimates with time-dependent weights for the Klein-Gordon equation

Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

Xiangsheng Xu^,

Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA

^* Corresponding author: Xiangsheng Xu
^* Corresponding author: Xiangsheng Xu

Received: December 2019

Revised: May 2020

Published online: November 2020

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Abstract

In this paper we study the initial boundary value problem for the system $ -\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p $ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $ (\mathbf{m}, p) $ with both $ |\nabla p| $ and $ |\nabla\mathbf{m}| $ being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for $ v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p $, and then suitably applying the De Giorge iteration method to the equation.

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Mathematics Subject Classification: Primary: 35B45, 35B65, 35M33, 35Q92.

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