Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model
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ORIGINAL PAPER
Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model Xiangsheng Xu1 Received: 14 February 2020 / Accepted: 6 July 2020 Ó Springer Nature Switzerland AG 2020
Abstract In this paper we study partial regularity of weak solutions to the initial boundary value problem for the system div½ðI þ m mÞrp ¼ SðxÞ; ot m D2 Dm E2 ðm rpÞrp þ jmj2ðc1Þ m ¼ 0, where S(x) is a given function and D; E; c are given numbers. This problem has been proposed as a PDE model for biological transportation networks. The mathematical difficulty is due to the fact that the system in the model features both a quadratic nonlinearity and a cubic nonlinearity. The regularity issue seems to have a connection to a conjecture by De Giorgi (Congetture sulla continuita´ delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995). We also investigate the life-span of classical solutions. Our results show that local existence of a classical solution can always be obtained and the life-span of such a solution can be extended as far away as one wishes as long as the term kmðx; 0Þk1;X þ kSðxÞk2N;X is 3 made suitably small, where N is the space dimension and k kq;X denotes the norm in Lq ðXÞ. Keywords Biological network formulation Cubic nonlinearity Life-span of smooth solutions Partial regularity of weak solutions Mathematics Subject Classification Primary 35A01 35A09 35M33 35Q99
1 Introduction Network formulation and transportation networks are fundamental processes in living systems [1]. The angiogenesis of blood vessels, leaf venation, and creation of neural pathways in nervous systems are some of the well known examples. Tremendous interest has been shown for these phenomena from different scientific communities such as biologists, engineers, physicists, and computer scientists. Of particular interest is their property This article is part of the section ‘‘Applications of PDEs’’ edited by Hyeonbae Kang. & Xiangsheng Xu [email protected] 1
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA SN Partial Differential Equations and Applications
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SN Partial Differ. Equ. Appl. (2020)1:18
of optimal transport of fluids and other materials. The development of mathematical models for transportation networks and network formulation is a growing field. We would like to refer the reader to [2] for a comprehensive review and analysis of existing models. In this paper we are interested in the mathematical analysis of a PDE model first proposed by Hu and Cai in [14] that describes the pressure field of a network using a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. More precisely, let X be the network region, a bounded domain in RN , and T a positive number. Set XT ¼ X ð0; TÞ. We study the behavior of solutions to the system div½ðI þ m mÞrp ¼ SðxÞ
in
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