A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up
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 c...
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| Published in | Communications in partial differential equations Vol. 42; no. 3; pp. 436 - 473 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Taylor & Francis
04.03.2017
Taylor & Francis Ltd |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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
under the initial condition
and no-flux boundary conditions in balls Ω⊂ℝ
n
, where χ>0 and
.
The main results assert the existence of a unique classical solution, extensible in time up to a maximal T
max
∈(0,∞] which has the property that
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 (⋆), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either
with
if χ>1 and m
c
: = ∞ if χ≤1.
That these conditions are essentially optimal will be shown in a forthcoming paper in which (⋆) 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
∞
(Ω). |
|---|---|
| AbstractList | 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
under the initial condition
and no-flux boundary conditions in balls Ω⊂ℝ
n
, where χ>0 and
.
The main results assert the existence of a unique classical solution, extensible in time up to a maximal T
max
∈(0,∞] which has the property that
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 (⋆), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either
with
if χ>1 and m
c
: = ∞ if χ≤1.
That these conditions are essentially optimal will be shown in a forthcoming paper in which (⋆) 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
∞
(Ω). 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 [Formula omitted.] under the initial condition [Formula omitted.] and no-flux boundary conditions in balls Ω⊂Rn, where χ>0 and [Formula omitted.] . The main results assert the existence of a unique classical solution, extensible in time up to a maximal Tmax[element of](0,∞] which has the property that The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient ur to L∞ bounds for u and to upper bounds for [Formula omitted.] ; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z+ in terms of bounds for u and |ur|, with suitably mild dependence on the latter. As a consequence of (*), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either [Formula omitted.] with [Formula omitted.] if χ>1 and mc: = ∞ if χ[less-than or equal to]1. That these conditions are essentially optimal will be shown in a forthcoming paper in which (*) 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∞(Ω). 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 [Image omitted.] under the initial condition [Image omitted.] and no-flux boundary conditions in balls Omega sub(R) super(n) where chi >0 and [Image omitted.]. The main results assert the existence of a unique classical solution, extensible in time up to a maximal T sub(max)[isin](0, infinity ] which has the property that The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient u sub(r) to L super( infinity ) bounds for u and to upper bounds for [Image omitted.]; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z sub(+) in terms of bounds for u and |u sub(r)|, with suitably mild dependence on the latter. As a consequence of ([sstarf]), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either [Image omitted.] with [Image omitted.] if chi >1 and m sub(c): = infinity if chi less than or equal to 1. That these conditions are essentially optimal will be shown in a forthcoming paper in which ([sstarf]) 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 super( infinity )( Omega ). |
| Author | Winkler, Michael Bellomo, Nicola |
| Author_xml | – sequence: 1 givenname: Nicola surname: Bellomo fullname: Bellomo, Nicola organization: Department of Mathematics, Faculty of Sciences, King Abdulaziz University – sequence: 2 givenname: Michael surname: Winkler fullname: Winkler, Michael email: michael.winkler@math.uni-paderborn.de organization: Institut für Mathematik, Universität Paderborn |
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| SubjectTerms | Blowing Chemotaxis degenerate diffusion Flux flux limitation Mathematical models Nonlinearity Norms Optimization Partial differential equations Primary: 35K65 Secondary: 35B45 Upper bounds |
| Title | A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up |
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