@article{63256,
author = {{Nikolić, Vanja and Winkler, Michael}},
issn = {{0362-546X}},
journal = {{Nonlinear Analysis}},
publisher = {{Elsevier BV}},
title = {{{L∞ blow-up in the Jordan–Moore–Gibson–Thompson equation}}},
doi = {{10.1016/j.na.2024.113600}},
volume = {{247}},
year = {{2024}},
}
@article{63260,
abstract = {{AbstractA no‐flux initial‐boundary value problem for
is considered in a ball , where and .Under the assumption that , it is shown that for each , there exist and a positive with the property that whenever is nonnegative with , the global solutions to () emanating from the initial data have the property that
}},
author = {{Wang, Yulan and Winkler, Michael}},
issn = {{0025-584X}},
journal = {{Mathematische Nachrichten}},
number = {{6}},
pages = {{2353--2364}},
publisher = {{Wiley}},
title = {{{A singular growth phenomenon in a Keller–Segel–type parabolic system involving density‐suppressed motilities}}},
doi = {{10.1002/mana.202300361}},
volume = {{297}},
year = {{2024}},
}
@article{63262,
abstract = {{AbstractRadially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$
{
u
t
=
∇
⋅
(
D
(
u
)
∇
u
)
−
∇
⋅
(
u
S
(
u
)
∇
v
)
,
0
=
Δ
v
−
μ
+
u
,
μ
=
1
|
Ω
|
∫
Ω
u
,
are considered in a ball Ω = BR(0) ⊂ ℝn, where n ≥ 3 and R > 0.Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)m−1 and S(ξ) = (ξ + 1)−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling $$m + \lambda < 1 - {2 \over n}$$
m
+
λ
<
1
−
2
n
, a considerably large set of initial data u0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u0, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying $${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$
1
C
⋅
(
t
+
1
)
1
λ
≤
|
|
u
(
⋅
,
t
)
|
|
L
∞
(
Ω
)
≤
C
⋅
(
t
+
1
)
1
λ
f
o
r
a
l
l
t
>
0
as well as $$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$
|
|
u
(
⋅
,
t
)
|
|
L
1
(
Ω
\
B
r
0
(
0
)
)
→
0
as
t
→
∞
for all
r
0
∈
(
0
,
R
)
with some C > 0.}},
author = {{Winkler, Michael}},
issn = {{0021-2172}},
journal = {{Israel Journal of Mathematics}},
number = {{1}},
pages = {{93--127}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Complete infinite-time mass aggregation in a quasilinear Keller–Segel system}}},
doi = {{10.1007/s11856-024-2618-9}},
volume = {{263}},
year = {{2024}},
}
@article{63263,
author = {{Winkler, Michael}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
pages = {{423--456}},
publisher = {{Elsevier BV}},
title = {{{L∞ bounds in a two-dimensional doubly degenerate nutrient taxis system with general cross-diffusive flux}}},
doi = {{10.1016/j.jde.2024.04.028}},
volume = {{400}},
year = {{2024}},
}
@article{57820,
author = {{Nikolić, Vanja and Winkler, Michael}},
issn = {{0362-546X}},
journal = {{Nonlinear Analysis}},
publisher = {{Elsevier BV}},
title = {{{L∞ blow-up in the Jordan-Moore-Gibson-Thompson equation}}},
doi = {{10.1016/j.na.2024.113600}},
volume = {{247}},
year = {{2024}},
}
@article{63285,
author = {{Winkler, Michael}},
issn = {{1079-9389}},
journal = {{Advances in Differential Equations}},
number = {{11/12}},
publisher = {{Khayyam Publishing, Inc}},
title = {{{Absence of collapse into persistent Dirac-type singularities in a Keller-Segel-Navier-Stokes system involving local sensing}}},
doi = {{10.57262/ade028-1112-921}},
volume = {{28}},
year = {{2023}},
}
@article{63288,
abstract = {{Abstract
The Cauchy problem in
R
n
{{\mathbb{R}}}^{n}
,
n
≥
2
n\ge 2
, for
u
t
=
Δ
u
−
∇
⋅
(
u
S
⋅
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
\begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array}
is considered for general matrices
S
∈
R
n
×
n
S\in {{\mathbb{R}}}^{n\times n}
. A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to
BUC
(
R
n
)
∩
L
p
(
R
n
)
{\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n})
with some
p
∈
[
1
,
n
)
p\in \left[1,n)
, there exist
T
max
∈
(
0
,
∞
]
{T}_{\max }\in \left(0,\infty ]
and a uniquely determined
u
∈
C
0
(
[
0
,
T
max
)
;
BUC
(
R
n
)
)
∩
C
0
(
[
0
,
T
max
)
;
L
p
(
R
n
)
)
∩
C
∞
(
R
n
×
(
0
,
T
max
)
)
u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }))
such that with
v
≔
Γ
⋆
u
v:= \Gamma \star u
, and with
Γ
\Gamma
denoting the Newtonian kernel on
R
n
{{\mathbb{R}}}^{n}
, the pair
(
u
,
v
)
\left(u,v)
forms a classical solution of (
⋆
\star
) in
R
n
×
(
0
,
T
max
)
{{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })
, which has the property that
if
T
max
<
∞
,
then both
limsup
t
↗
T
max
‖
u
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
and
limsup
t
↗
T
max
‖
∇
v
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
.
\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty .
An exemplary application of this provides a result on global classical solvability in cases when
∣
S
+
1
∣
| S+{\bf{1}}|
is sufficiently small, where
1
=
diag
(
1
,
…
,
1
)
{\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1)
.}},
author = {{Winkler, Michael}},
issn = {{2391-5455}},
journal = {{Open Mathematics}},
number = {{1}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type}}},
doi = {{10.1515/math-2022-0578}},
volume = {{21}},
year = {{2023}},
}
@article{63287,
abstract = {{AbstractThe Cauchy problem in $$\mathbb {R}^n$$
R
n
is considered for the Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
Δ
u
-
∇
·
(
u
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data $$u_0$$
u
0
with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$
u
0
is continuous and bounded, then ($$\star $$
⋆
) admits a local-in-time classical solution, whereas if $$u_0(x)\rightarrow +\infty $$
u
0
(
x
)
→
+
∞
as $$|x|\rightarrow \infty $$
|
x
|
→
∞
, then no such solution can be found. Furthermore, a collection of three sufficient criteria for either global existence or global nonexistence indicates that with respect to the occurrence of finite-time blow-up, spatial decay properties of an explicit singular steady state plays a critical role. In particular, this underlines that explosions in ($$\star $$
⋆
) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.}},
author = {{Winkler, Michael}},
issn = {{2296-9020}},
journal = {{Journal of Elliptic and Parabolic Equations}},
number = {{2}},
pages = {{919--959}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity}}},
doi = {{10.1007/s41808-023-00230-y}},
volume = {{9}},
year = {{2023}},
}
@article{63289,
author = {{Winkler, Michael and Yokota, Tomomi}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
pages = {{1--28}},
publisher = {{Elsevier BV}},
title = {{{Avoiding critical mass phenomena by arbitrarily mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes systems}}},
doi = {{10.1016/j.jde.2023.07.029}},
volume = {{374}},
year = {{2023}},
}
@article{63271,
abstract = {{ As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. For any given conveniently regular and biologically meaningful initial data, smallness conditions on the prescribed resource growth and on the initial nutrient signal concentration are identified which ensure the global existence of a global classical solution to the corresponding no-flux initial-boundary value problem. Moreover, under additional assumptions on the food production source these solutions are shown to be bounded, and to stabilize toward semi-trivial equilibria in the large time limit, respectively. }},
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0218-2025}},
journal = {{Mathematical Models and Methods in Applied Sciences}},
number = {{01}},
pages = {{103--138}},
publisher = {{World Scientific Pub Co Pte Ltd}},
title = {{{Small-signal solutions to a nonlocal cross-diffusion model for interaction of scroungers with rapidly diffusing foragers}}},
doi = {{10.1142/s0218202523500045}},
volume = {{33}},
year = {{2023}},
}
@article{63267,
author = {{Ahn, Jaewook and Winkler, Michael}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
number = {{6}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system}}},
doi = {{10.1007/s00526-023-02523-5}},
volume = {{62}},
year = {{2023}},
}
@article{63270,
author = {{Li, Genglin and Winkler, Michael}},
issn = {{1539-6746}},
journal = {{Communications in Mathematical Sciences}},
number = {{2}},
pages = {{299--322}},
publisher = {{International Press of Boston}},
title = {{{Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities}}},
doi = {{10.4310/cms.2023.v21.n2.a1}},
volume = {{21}},
year = {{2023}},
}
@article{63273,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{1468-1218}},
journal = {{Nonlinear Analysis: Real World Applications}},
publisher = {{Elsevier BV}},
title = {{{Analysis of a chemotaxis-SIS epidemic model with unbounded infection force}}},
doi = {{10.1016/j.nonrwa.2022.103820}},
volume = {{71}},
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{63276,
abstract = {{The chemotaxis‐Stokes system
is considered along with homogeneous boundary conditions of no‐flux type for
and
, and of Dirichlet type for
, in a smoothly bounded domain
. Under the assumption that
, that
is bounded on each of the intervals
with arbitrary
, and that with some
and
, we have
It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution.}},
author = {{Tian, Yu and Winkler, Michael}},
issn = {{0170-4214}},
journal = {{Mathematical Methods in the Applied Sciences}},
number = {{14}},
pages = {{15667--15683}},
publisher = {{Wiley}},
title = {{{Keller–Segel–Stokes interaction involving signal‐dependent motilities}}},
doi = {{10.1002/mma.9419}},
volume = {{46}},
year = {{2023}},
}
@article{63275,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{2163-2480}},
journal = {{Evolution Equations and Control Theory}},
number = {{6}},
pages = {{1676--1687}},
publisher = {{American Institute of Mathematical Sciences (AIMS)}},
title = {{{Global smooth solutions in a three-dimensional cross-diffusive SIS epidemic model with saturated taxis at large densities}}},
doi = {{10.3934/eect.2023031}},
volume = {{12}},
year = {{2023}},
}
@article{63277,
author = {{Painter, Kevin J. and Winkler, Michael}},
issn = {{0036-1399}},
journal = {{SIAM Journal on Applied Mathematics}},
number = {{5}},
pages = {{2096--2117}},
publisher = {{Society for Industrial & Applied Mathematics (SIAM)}},
title = {{{Phenotype Switching in Chemotaxis Aggregation Models Controls the Spontaneous Emergence of Large Densities}}},
doi = {{10.1137/22m1539393}},
volume = {{83}},
year = {{2023}},
}
@article{63283,
abstract = {{AbstractThe parabolic problem $$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta \big (u\phi (v)\big ), \\ v_t=\Delta v-uv, \end{array} \right. \end{aligned}$$
u
t
=
Δ
(
u
ϕ
(
v
)
)
,
v
t
=
Δ
v
-
u
v
,
is considered in smoothly bounded subdomains of $${\mathbb {R}}^n$$
R
n
with arbitrary $$n\ge 1$$
n
≥
1
. Under the assumptions that $$\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))$$
ϕ
∈
C
0
(
[
0
,
∞
)
)
∩
C
3
(
(
0
,
∞
)
)
is positive on $$(0,\infty )$$
(
0
,
∞
)
and satisfies $$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^\alpha }>0 \quad {\text{ and }} \quad \limsup _{\xi \searrow 0} \big \{ \xi ^\beta |\phi '(\xi )| \big \}<\infty \end{aligned}$$
lim inf
ξ
↘
0
ϕ
(
ξ
)
ξ
α
>
0
and
lim sup
ξ
↘
0
{
ξ
β
|
ϕ
′
(
ξ
)
|
}
<
∞
with some $$\alpha >0$$
α
>
0
and $$\beta >0$$
β
>
0
, for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices $$\alpha =0$$
α
=
0
and $$\alpha \in (0,2)$$
α
∈
(
0
,
2
)
, or were restricted to low-dimensional settings by requiring that $$n\le 2$$
n
≤
2
.}},
author = {{Winkler, Michael}},
issn = {{0044-2275}},
journal = {{Zeitschrift für angewandte Mathematik und Physik}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration–consumption interaction}}},
doi = {{10.1007/s00033-022-01925-3}},
volume = {{74}},
year = {{2023}},
}
@article{63255,
author = {{Li, Genglin and Winkler, Michael}},
issn = {{0003-6811}},
journal = {{Applicable Analysis}},
number = {{1}},
pages = {{45--64}},
publisher = {{Informa UK Limited}},
title = {{{Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities}}},
doi = {{10.1080/00036811.2023.2173183}},
volume = {{103}},
year = {{2023}},
}
@article{63261,
abstract = {{
The taxis-type migration–consumption model accounting for signal-dependent motilities, as given by
u_{t} = \Delta (u\phi(v))
,
v_{t} = \Delta v-uv
, is considered for suitably smooth functions
\phi\colon[0,\infty)\to\R
which are such that
\phi>0
on
(0,\infty)
, but that in addition
\phi(0)=0
with
\phi'(0)>0
. In order to appropriately cope with the diffusion degeneracies thereby included, this study separately examines the Neumann problem for the linear equation
V_{t} = \Delta V + \nabla\cdot ( a(x,t)V) + b(x,t)V
and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of
a
and
b
. This is thereafter used as a key tool in the derivation of a result on global existence of solutions to the equation above, smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative effect genuinely due to degeneracy in diffusion, a criterion on initial smallness of the second component is identified as sufficient for this limit state to be spatially nonconstant.
}},
author = {{Winkler, Michael}},
issn = {{0294-1449}},
journal = {{Annales de l'Institut Henri Poincaré C, Analyse non linéaire}},
number = {{1}},
pages = {{95--127}},
publisher = {{European Mathematical Society - EMS - Publishing House GmbH}},
title = {{{A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion}}},
doi = {{10.4171/aihpc/73}},
volume = {{41}},
year = {{2023}},
}