Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

TLDR: In this paper, two simple conservative systems of parabolic-elliptic and parabolicdegenerate type arising in modeling chemotaxis and angiogenesis were considered, and it was shown that weak solutions (which are equi-integrable in L1) exist even for large initial data.

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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaCa % aaleqabaWaaSaaaeaacaWGKbaabaGaaGOmaaaaaaaaaa!38A1! $$L^{\frac{d} {2}} $$ 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 L p spaces with max % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaada % GacaqaaiaaigdacaGG7aWaaSaaaeaacaWGKbaabaGaaGOmaaaacqGH % sislcaaIXaaacaGL9baacqGHKjYOcaWGWbGaeyipaWJaeyOhIuQaai % OlaaGaay5Eaaaaaa!42CD! $$\left\{ {\left. {1;\frac{d} {2} - 1} \right\} \leq p < \infty .} \right.$$ 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.