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摘要
This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller-Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic-elliptic system @@@ {ut = del center dot (u del u/root u(2) + vertical bar del u vertical bar(2) - chi(del) center dot u del u/root 1 + vertical bar del v vertical bar(2)), @@@ under theinitialcondition u vertical bar(t=0) = u(0) > 0 and no-flux boundary conditions in balls Omega subset of R-n. Rn, where chi > 0 and mu := 1/vertical bar Omega vertical bar f(Omega) u(0). @@@ The main results assert the existence of a unique classical solution, extensible in time up to a maximal T-max is an element of(0,infinity] which has the property @@@ if T-max < infinity then lim sup(t NE arrow Tmax) parallel to u(.,t)parallel to(L infinity)(Omega) = infinity. (star) @@@ The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient u(r) to L bounds for u and to upper bounds for ; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z(+) in terms of bounds for u and |u(r)|, with suitably mild dependence on the latter. @@@ As a consequence of (star), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either @@@ n >= 2 and chi < 1, or n = 1, chi > 0 and m < m(c), @@@ with m(c) := 1/root chi(2)-1 if chi > 1 and m(c):= if chi <= 1. @@@ That these conditions are essentially optimal will be shown in a forth-coming paper in which (star) will be used to derive complementary results on the occurrence of solutions blowing up in finite time with respect to the norm of u in L-infinity(Omega).
- 出版日期2017
