Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis

Уравнение Агрегации С Анизотропной Диффузией

“…In fact, based on scaling arguments it is easy to argue that for m > 2 − 2/d, the diffusion term dominates when density becomes large, leading to global existence of solutions for all masses. This result was shown in [63] together with the global uniform bound of solutions for all times.…”

Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

“…) In the following section, we shall prepare several lemmas which will be used in the sequent sections. In Section 3, we introduce the results obtained in [30][31][32] concerning the existence of a time global strong solution of the approximated problem of (KS). In Section 4, we organize the proof of the decay of a solution (u, v).…”

“…This exponent q = m + 2 N is called the Fujita exponent [11]. For (KS) with N ≥ 1, m > 1, q ≥ 2, in [30][31][32]] the Fujita's exponent was found. Specifically, in [32] it was shown that (i) when q < m + 2 N , the problem (KS) is globally solvable without any restriction on the size of the initial data; and (ii) when m > 1 and…”

“…In [30][31][32], the following proposition concerning the existence of the strong solution was proved:…”

Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems

“…The existence of strong solution of (KS) ε was proved in [20]- [22]. The mass conservation law of u ε was established in [21, Proposition 7.1].…”