Abstract.
We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the \(L^{\frac{d} {2}} \) norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the Lp spaces with max \(\left\{ {\left. {1;\frac{d} {2} - 1} \right\} \leq p < \infty .} \right.\) This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L1) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.
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Lecture by B. Perthame held at the “Presentation of MJM”, Milano, October 18, 2002
Received: March, 2003
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Corrias, L., Perthame, B. & Zaag, H. Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions. Milan j. math. 72, 1–28 (2004). https://doi.org/10.1007/s00032-003-0026-x
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DOI: https://doi.org/10.1007/s00032-003-0026-x