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

  • * Corresponding author: Xiangsheng Xu

    * Corresponding author: Xiangsheng Xu
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  • 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.

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


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