@article{63253,
abstract = {{Abstract
The Neumann problem for the Keller-Segel system
{
u
t
=
∇
⋅
(
D
(
u
)
∇
u
)
−
∇
⋅
(
S
(
u
)
∇
v
)
,
0
=
Δ
v
−
μ
+
u
,
μ
=
−
∫
Ω
u
d
x
,
is considered in n-dimensional balls Ω with
n
⩾
2
, with suitably regular and radially symmetric, radially nonincreasing initial data u
0. The functions D and S are only assumed to belong to
C
2
(
[
0
,
∞
)
)
and to satisfy D > 0 and
S
⩾
0
on
[
0
,
∞
)
as well as
S
(
0
)
=
0
; in particular, diffusivities with arbitrarily fast decay are included.
In this general context, it is shown that it is merely the asymptotic behavior as
ξ
→
∞
of the expression
I
(
ξ
)
:=
S
(
ξ
)
ξ
2
n
D
(
ξ
)
,
ξ
>
0
,
which decides about the occurrence of blow-up: Namely, it is seen that
•
if
lim
ξ
→
∞
I
(
ξ
)
=
0
, then any such solution is global and bounded, that
•
if
lim sup
ξ
→
∞
I
(
ξ
)
<
∞
and
∫
Ω
u
0
is suitably small, then the corresponding solution is global and bounded, and that
•
if
lim inf
ξ
→
∞
I
(
ξ
)
>
0
, then at each appropriately large mass level m, there exist radial initial data u
0 such that
∫
Ω
u
0
=
m
, and that the associated solution blows up either in finite or in infinite time.
This especially reveals the presence of critical mass phenomena whenever
lim
ξ
→
∞
I
(
ξ
)
∈
(
0
,
∞
)
exists.}},
author = {{Ding, Mengyao and Winkler, Michael}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{12}},
publisher = {{IOP Publishing}},
title = {{{Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture}}},
doi = {{10.1088/1361-6544/ad871a}},
volume = {{37}},
year = {{2024}},
}
@article{53345,
abstract = {{AbstractA no-flux initial-boundary value problem forut=Δ(uϕ(v)),vt=Δv−uv,(⋆)is considered in smoothly bounded subdomains ofRnwithn⩾1and suitably regular initial data, whereφis assumed to reflect algebraic type cross-degeneracies by sharing essential features with0⩽ξ↦ξαfor someα⩾1. Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profileu∞and some null setN⋆⊂(0,∞), a corresponding global generalized solution, known to exist according to recent literature, satisfiesρ(u(⋅,t))⇀⋆ρ(u∞)in L∞(Ω) and v(⋅,t)→0in Lp(Ω)for all p⩾1as(0,∞)∖N⋆∋t→∞, whereρ(ξ):=ξ2(ξ+1)2,ξ⩾0. In the particular case when eithern⩽2andα⩾1is arbitrary, orn⩾1andα∈[1,2], additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states(u⋆,0)of (⋆) is stable with respect to a suitably chosen norm topology.}},
author = {{Winkler, Michael}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}},
number = {{8}},
pages = {{4438--4469}},
publisher = {{IOP Publishing}},
title = {{{Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction}}},
doi = {{10.1088/1361-6544/ace22e}},
volume = {{36}},
year = {{2023}},
}
@article{63281,
abstract = {{AbstractA no-flux initial-boundary value problem forut=Δ(uϕ(v)),vt=Δv−uv,(⋆)is considered in smoothly bounded subdomains ofRnwithn⩾1and suitably regular initial data, whereφis assumed to reflect algebraic type cross-degeneracies by sharing essential features with0⩽ξ↦ξαfor someα⩾1. Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profileu∞and some null setN⋆⊂(0,∞), a corresponding global generalized solution, known to exist according to recent literature, satisfiesρ(u(⋅,t))⇀⋆ρ(u∞)in L∞(Ω) and v(⋅,t)→0in Lp(Ω)for all p⩾1as(0,∞)∖N⋆∋t→∞, whereρ(ξ):=ξ2(ξ+1)2,ξ⩾0. In the particular case when eithern⩽2andα⩾1is arbitrary, orn⩾1andα∈[1,2], additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states(u⋆,0)of (⋆) is stable with respect to a suitably chosen norm topology.}},
author = {{Winkler, Michael}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{8}},
pages = {{4438--4469}},
publisher = {{IOP Publishing}},
title = {{{Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction}}},
doi = {{10.1088/1361-6544/ace22e}},
volume = {{36}},
year = {{2023}},
}
@article{33264,
abstract = {{We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.}},
author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}},
number = {{4}},
pages = {{2073--2120}},
publisher = {{IOP Publishing}},
title = {{{Amplified steady state bifurcations in feedforward networks}}},
doi = {{10.1088/1361-6544/ac5463}},
volume = {{35}},
year = {{2022}},
}
@article{19939,
author = {{Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{5}},
pages = {{2335--2363}},
title = {{{Detection of high codimensional bifurcations in variational PDEs}}},
doi = {{10.1088/1361-6544/ab7293}},
volume = {{33}},
year = {{2020}},
}
@article{19935,
abstract = {{A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples. }},
author = {{McLachlan, Robert I and Offen, Christian}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{2895--2927}},
title = {{{Bifurcation of solutions to Hamiltonian boundary value problems}}},
doi = {{10.1088/1361-6544/aab630}},
year = {{2018}},
}
@article{8755,
abstract = {{Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.}},
author = {{Klus, Stefan and Gelß, Patrick and Peitz, Sebastian and Schütte, Christof}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{7}},
pages = {{3359--3380}},
title = {{{Tensor-based dynamic mode decomposition}}},
doi = {{10.1088/1361-6544/aabc8f}},
volume = {{31}},
year = {{2018}},
}
@article{63370,
author = {{Espejo, Elio and Winkler, Michael}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{4}},
pages = {{1227--1259}},
publisher = {{IOP Publishing}},
title = {{{Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization}}},
doi = {{10.1088/1361-6544/aa9d5f}},
volume = {{31}},
year = {{2018}},
}
@article{63376,
author = {{Winkler, Michael}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
number = {{5}},
pages = {{2031--2056}},
publisher = {{IOP Publishing}},
title = {{{A critical blow-up exponent in a chemotaxis system with nonlinear signal production}}},
doi = {{10.1088/1361-6544/aaaa0e}},
volume = {{31}},
year = {{2018}},
}
@article{34659,
author = {{Black, Tobias}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}},
number = {{6}},
pages = {{1865--1886}},
publisher = {{IOP Publishing}},
title = {{{Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant}}},
doi = {{10.1088/0951-7715/29/6/1865}},
volume = {{29}},
year = {{2016}},
}
@article{26511,
author = {{Barkhofen, Sonja and Faure, F and Weich, T}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{1829--1858}},
title = {{{Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum}}},
doi = {{10.1088/0951-7715/27/8/1829}},
year = {{2014}},
}
@article{31296,
author = {{Barkhofen, Sonja and Faure, F and Weich, Tobias}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}},
number = {{8}},
pages = {{1829--1858}},
publisher = {{IOP Publishing}},
title = {{{Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum}}},
doi = {{10.1088/0951-7715/27/8/1829}},
volume = {{27}},
year = {{2014}},
}
@article{16554,
author = {{Dellnitz, Michael and Froyland, Gary and Sertl, Stefan}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{1171--1188}},
title = {{{On the isolated spectrum of the Perron-Frobenius operator}}},
doi = {{10.1088/0951-7715/13/4/310}},
year = {{2000}},
}
@article{16532,
author = {{Dellnitz, M and Heinrich, C}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{1039--1066}},
title = {{{Admissible symmetry increasing bifurcations}}},
doi = {{10.1088/0951-7715/8/6/009}},
year = {{1995}},
}
@article{16542,
author = {{Dellnitz, M and Melbourne, I}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{1067--1075}},
title = {{{A note on the shadowing lemma and symmetric periodic points}}},
doi = {{10.1088/0951-7715/8/6/010}},
year = {{1995}},
}
@article{16548,
author = {{Dellnitz, M and Melbourne, I and Marsden, J E}},
issn = {{0951-7715}},
journal = {{Nonlinearity}},
pages = {{979--996}},
title = {{{Generic bifurcation of Hamiltonian vector fields with symmetry}}},
doi = {{10.1088/0951-7715/5/4/008}},
year = {{1992}},
}