Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model

Xiangsheng Xu ()
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Xiangsheng Xu: Mississippi State University

Partial Differential Equations and Applications, 2020, vol. 1, issue 4, 1-31

Abstract: Abstract In this paper we study partial regularity of weak solutions to the initial boundary value problem for the system $$-\text {div}\left[ (I+\mathbf{m}\otimes \mathbf{m})\nabla p\right] =S(x),\ \ \partial _t\mathbf{m}-D^2\Delta \mathbf{m}-E^2(\mathbf{m}\cdot \nabla p)\nabla p+|\mathbf{m}|^{2(\gamma -1)}{} \mathbf{m}=0$$ - div ( I + m ⊗ m ) ∇ p = S ( x ) , ∂ t m - D 2 Δ m - E 2 ( m · ∇ p ) ∇ p + | m | 2 ( γ - 1 ) m = 0 , where S(x) is a given function and $$D, E, \gamma$$ D , E , γ 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(Congetture sulla continuitá delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995).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 $$\Vert \mathbf{m}(x,0)\Vert _{\infty , \Omega }+\Vert S(x)\Vert _{\frac{2N}{3}, \Omega }$$ ‖ m ( x , 0 ) ‖ ∞ , Ω + ‖ S ( x ) ‖ 2 N 3 , Ω is made suitably small, where N is the space dimension and $$\Vert \cdot \Vert _{q,\Omega }$$ ‖ · ‖ q , Ω denotes the norm in $$L^q(\Omega )$$ L q ( Ω ) .

Keywords: Biological network formulation; Cubic nonlinearity; Life-span of smooth solutions; Partial regularity of weak solutions; Primary 35A01; 35A09; 35M33; 35Q99 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00021-3

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