@article{63330,
author = {{Li, Genglin and Tao, Youshan and Winkler, Michael}},
issn = {{1531-3492}},
journal = {{Discrete and Continuous Dynamical Systems - B}},
number = {{11}},
pages = {{4383--4396}},
publisher = {{American Institute of Mathematical Sciences (AIMS)}},
title = {{{Large time behavior in a predator-prey system with indirect pursuit-evasion interaction}}},
doi = {{10.3934/dcdsb.2020102}},
volume = {{25}},
year = {{2020}},
}
@article{63327,
author = {{Winkler, Michael}},
issn = {{1468-1218}},
journal = {{Nonlinear Analysis: Real World Applications}},
publisher = {{Elsevier BV}},
title = {{{Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with gradient-dependent flux limitation}}},
doi = {{10.1016/j.nonrwa.2020.103257}},
volume = {{59}},
year = {{2020}},
}
@article{63333,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0362-546X}},
journal = {{Nonlinear Analysis}},
publisher = {{Elsevier BV}},
title = {{{A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy}}},
doi = {{10.1016/j.na.2020.111870}},
volume = {{198}},
year = {{2020}},
}
@article{63328,
author = {{Winkler, Michael}},
issn = {{0893-9659}},
journal = {{Applied Mathematics Letters}},
publisher = {{Elsevier BV}},
title = {{{Boundedness in a three-dimensional Keller–Segel–Stokes system with subcritical sensitivity}}},
doi = {{10.1016/j.aml.2020.106785}},
volume = {{112}},
year = {{2020}},
}
@article{63320,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{1553-5231}},
journal = {{Discrete & Continuous Dynamical Systems - A}},
number = {{1}},
pages = {{439--454}},
publisher = {{American Institute of Mathematical Sciences (AIMS)}},
title = {{{Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction}}},
doi = {{10.3934/dcds.2020216}},
volume = {{41}},
year = {{2020}},
}
@article{63335,
author = {{Winkler, Michael}},
issn = {{0036-1410}},
journal = {{SIAM Journal on Mathematical Analysis}},
number = {{2}},
pages = {{2041--2080}},
publisher = {{Society for Industrial & Applied Mathematics (SIAM)}},
title = {{{Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to the Navier--Stokes Equations}}},
doi = {{10.1137/19m1264199}},
volume = {{52}},
year = {{2020}},
}
@article{63318,
abstract = {{In a planar smoothly bounded domain$\Omega$, we consider the model for oncolytic virotherapy given by$$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$with positive parameters$ D_w $,$ D_z $and$\beta$. It is firstly shown that whenever$\beta \lt 1$, for any choice of$M \gt 0$, one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of$\beta \gt 0$, satisfies$$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$If$\beta \gt 1$, however, then for arbitrary initial data the corresponding is seen to have the property that$$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$This may be interpreted as indicating that$\beta$plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by$\beta = 1$.}},
author = {{TAO, YOUSHAN and Winkler, Michael}},
issn = {{0956-7925}},
journal = {{European Journal of Applied Mathematics}},
number = {{2}},
pages = {{301--316}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{A critical virus production rate for efficiency of oncolytic virotherapy}}},
doi = {{10.1017/s0956792520000133}},
volume = {{32}},
year = {{2020}},
}
@article{63314,
abstract = {{We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.}},
author = {{SURULESCU, CHRISTINA and Winkler, Michael}},
issn = {{0956-7925}},
journal = {{European Journal of Applied Mathematics}},
number = {{4}},
pages = {{618--651}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)}}},
doi = {{10.1017/s0956792520000236}},
volume = {{32}},
year = {{2020}},
}
@article{63265,
abstract = {{AbstractThe Cauchy problem in $${\mathbb {R}}^n$$
R
n
, $$n\ge 1$$
n
≥
1
, for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$
u
t
=
u
p
Δ
u
(
⋆
)
is considered in the strongly degenerate regime $$p\ge 1$$
p
≥
1
. The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies $$\begin{aligned} t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \rightarrow \infty \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
→
∞
as
t
→
∞
.
The first result of this study complements this by asserting that given any positive $$f\in C^0([0,\infty ))$$
f
∈
C
0
(
[
0
,
∞
)
)
fulfilling $$f(t)\rightarrow +\infty $$
f
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
one can find a positive nondecreasing function $$\phi \in C^0([0,\infty ))$$
ϕ
∈
C
0
(
[
0
,
∞
)
)
such that whenever $$u_0\in C^0({\mathbb {R}}^n)$$
u
0
∈
C
0
(
R
n
)
is radially symmetric with $$0< u_0 < \phi (|\cdot |)$$
0
<
u
0
<
ϕ
(
|
·
|
)
, the corresponding minimal solution u satisfies $$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
f
(
t
)
→
0
as
t
→
∞
.
Secondly, ($$\star $$
⋆
) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$
u
0
. It is shown that if the connected components of $$\{u_0>0\}$$
{
u
0
>
0
}
comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ($$\star $$
⋆
) satisfies $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$
0
<
lim inf
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
≤
lim sup
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
<
∞
.
Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$
{
u
0
>
0
}
contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.}},
author = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
number = {{S1}},
pages = {{3--23}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}}},
doi = {{10.1007/s10884-020-09892-x}},
volume = {{36}},
year = {{2020}},
}
@article{63325,
abstract = {{Abstract
We consider the spatially 2D version of the model $$\begin{equation*} \qquad\quad\left\{ \begin{array}{@{}rcll} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \big(nS(x,n,c) \cdot \nabla c \big), \qquad &\qquad x\in \Omega, \ t>0, \\ c_t + u\cdot \nabla c &=& \Delta c - n f(c), \qquad &\qquad x\in \Omega, \ t>0, \\ u_t &=& \Delta u + \nabla P + n\nabla\phi, \qquad \nabla\cdot u=0, \qquad &\qquad x\in \Omega, \ t>0, \end{array} \right. \qquad \qquad (\star) \end{equation*}$$for nutrient taxis processes, possibly interacting with liquid environments. Here the particular focus is on the situation when the chemotactic sensitivity $S$ is not a scalar function but rather attains general values in ${\mathbb{R}}^{2\times 2}$, thus accounting for rotational flux components in accordance with experimental findings and recent modeling approaches. Reflecting significant new challenges that mainly stem from apparent loss of energy-like structures, especially for initial data with large size, the knowledge on ($\star$) so far seems essentially restricted to results on global existence of certain generalized solutions with possibly quite poor boundedness and regularity properties; widely unaddressed seem aspects related to possible effects of such non-diagonal taxis mechanisms on the qualitative solution behavior, especially with regard to the fundamental question whether spatial structures may thereby be supported. The present work answers the latter in the negative in the following sense: under the assumptions that the initial data $(n_0,c_0,u_0)$ and the parameter functions $S$, $f$, and $\phi$ are sufficiently smooth, and that $S$ is bounded and $f$ is positive on $(0,\infty )$ with $f(0)=0$, it is shown that any nontrivial of these solutions eventually becomes smooth and satisfies $$\begin{equation*} n(\cdot,t)\to - \int_\Omega n_0, \quad c(\cdot,t)\to 0 \quad \text{and} \quad u(\cdot,t)\to 0 \qquad \text{as} \ t\to\infty, \end{equation*}$$uniformly with respect to $x\in \Omega$. By not requiring any smallness condition on the initial data, the latter seems new even in the corresponding fluid-free version obtained on letting $u\equiv 0$ in ($\star$).}},
author = {{Winkler, Michael}},
issn = {{1073-7928}},
journal = {{International Mathematics Research Notices}},
number = {{11}},
pages = {{8106--8152}},
publisher = {{Oxford University Press (OUP)}},
title = {{{Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?}}},
doi = {{10.1093/imrn/rnz056}},
volume = {{2021}},
year = {{2019}},
}
@article{63337,
abstract = {{AbstractIn boundedn-dimensional domainsΩ, the Neumann problem for the parabolic equation$$\begin{array}{} \displaystyle u_t = \nabla \cdot \Big( A(x,t)\cdot\nabla u\Big) + \nabla \cdot \Big(b(x,t)u\Big) - f(x,t,u)+g(x,t) \end{array}$$(*)is considered for sufficiently regular matrix-valuedA, vector-valuedband real valuedg, and withfrepresenting superlinear absorption in generalizing the prototypical choice given byf(⋅, ⋅,s) =sαwithα> 1. Problems of this form arise in a natural manner as sub-problems in several applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto type in mathematical biology, and accordingly a natural space for initial data appears to beL1(Ω).The main objective thus consists in examining how far solutions can be constructed for initial data merely assumed to be integrable, with major challenges potentially resulting from the interplay between nonlinear degradation on the one hand, and the possibly destabilizing drift-type action on the other in such contexts. Especially, the applicability of well-established methods such as techniques relying on entropy-like structures available in some particular cases, for instance, seems quite limited in the present setting, as these typically rely on higher initial regularity properties.The first of the main results shows that in the general framework of (*), nevertheless certain global very weak solutions can be constructed through a limit process involving smooth solutions to approximate variants thereof, provided that the ingredients of the latter satisfy appropriate assumptions with regard to their stabilization behavior.The second and seemingly most substantial part of the paper develops a method by which it can be shown, under suitably stregthened hypotheses on the integrability ofband the degradation parameterα, that the solutions obtained above in fact form genuine weak solutions in a naturally defined sense. This is achieved by properly exploiting a weak integral inequality, as satisfied by the very weak solution at hand, through a testing procedure that appears to be novel and of potentially independent interest.To underline the strength of this approach, both these general results are thereafter applied to two specific cross-diffusion systems. Inter alia, this leads to a statement on global solvability in a logistic Keller-Segel system under the assumptionα>$\begin{array}{} \frac{2n+4}{n+4} \end{array}$on the respective degradation rate which seems substantially milder than any previously found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto system some apparently first results on global solvability forL1initial data are derived.}},
author = {{Winkler, Michael}},
issn = {{2191-950X}},
journal = {{Advances in Nonlinear Analysis}},
number = {{1}},
pages = {{526--566}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1}}},
doi = {{10.1515/anona-2020-0013}},
volume = {{9}},
year = {{2019}},
}
@article{63334,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
number = {{9}},
pages = {{4973--4997}},
publisher = {{Elsevier BV}},
title = {{{Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy}}},
doi = {{10.1016/j.jde.2019.10.046}},
volume = {{268}},
year = {{2019}},
}
@article{63349,
author = {{Bellomo, Nicola and Painter, Kevin J. and Tao, Youshan and Winkler, Michael}},
issn = {{0036-1399}},
journal = {{SIAM Journal on Applied Mathematics}},
number = {{5}},
pages = {{1990--2010}},
publisher = {{Society for Industrial & Applied Mathematics (SIAM)}},
title = {{{Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection}}},
doi = {{10.1137/19m1250261}},
volume = {{79}},
year = {{2019}},
}
@article{63355,
abstract = {{AbstractThis work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by
with positive parameters and , and nonnegative constants and . Beyond some statements on global existence, the literature apparently provides only few results on qualitative behavior of solutions; in particular, questions related to boundedness as well as to large time asymptotics in seem unsolved so far.In the present paper it is inter alia shown that if and is a bounded convex domain with smooth boundary, then whenever and are nonnegative, the associated Neumann initial‐boundary value problem for possesses a global classical solution which in fact is bounded in the sense that
Moreover, the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness property is studied in the general situation when is arbitrary and no longer necessarily convex. If , then in both cases and , an explicit smallness condition on is identified as sufficient for stabilization of any nontrivial solutions toward a corresponding unique nontrivial spatially homogeneous steady state. If and , then without any further assumption all nonzero solutions are seen to approach the equilibrium (0,1). As a by‐product, this particularly improves previous knowledge on nonexistence of nonconstant equilibria of .}},
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0024-6115}},
journal = {{Proceedings of the London Mathematical Society}},
number = {{6}},
pages = {{1598--1632}},
publisher = {{Wiley}},
title = {{{Boundedness and stabilization in a population model with cross‐diffusion for one species}}},
doi = {{10.1112/plms.12276}},
volume = {{119}},
year = {{2019}},
}
@article{63356,
abstract = {{ This work deals with a taxis cascade model for food consumption in two populations, namely foragers directly orienting their movement upward the gradients of food concentration and exploiters taking a parasitic strategy in search of food via tracking higher forager densities. As a consequence, the dynamics of both populations are adapted to the space distribution of food which is dynamically modified in time and space by the two populations. This model extends the classical one-species chemotaxis-consumption systems by additionally accounting for a second taxis mechanism coupled to the first in a consecutive manner. It is rigorously proved that for all suitably regular initial data, an associated Neumann-type initial-boundary value problem for the spatially one-dimensional version of this model possesses a globally defined bounded classical solution. Moreover, it is asserted that the considered two populations will approach spatially homogeneous distributions in the large time limit, provided that either the total population number of foragers or that of exploiters is appropriately small. }},
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0218-2025}},
journal = {{Mathematical Models and Methods in Applied Sciences}},
number = {{11}},
pages = {{2151--2182}},
publisher = {{World Scientific Pub Co Pte Ltd}},
title = {{{Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food}}},
doi = {{10.1142/s021820251950043x}},
volume = {{29}},
year = {{2019}},
}
@article{63352,
author = {{Lankeit, Johannes and Winkler, Michael}},
issn = {{0021-2172}},
journal = {{Israel Journal of Mathematics}},
number = {{1}},
pages = {{249--296}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory}}},
doi = {{10.1007/s11856-019-1900-8}},
volume = {{233}},
year = {{2019}},
}
@article{63358,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{1553-5258}},
journal = {{Communications on Pure & Applied Analysis}},
number = {{4}},
pages = {{2047--2067}},
publisher = {{American Institute of Mathematical Sciences (AIMS)}},
title = {{{A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production}}},
doi = {{10.3934/cpaa.2019092}},
volume = {{18}},
year = {{2019}},
}
@article{63357,
author = {{Tao, Youshan and Winkler, Michael}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
number = {{1}},
pages = {{388--406}},
publisher = {{Elsevier BV}},
title = {{{Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension}}},
doi = {{10.1016/j.jde.2019.01.014}},
volume = {{267}},
year = {{2019}},
}
@article{63353,
author = {{Lankeit, Johannes and Winkler, Michael}},
issn = {{0012-0456}},
journal = {{Jahresbericht der Deutschen Mathematiker-Vereinigung}},
number = {{1}},
pages = {{35--64}},
publisher = {{Springer Fachmedien Wiesbaden GmbH}},
title = {{{Facing Low Regularity in Chemotaxis Systems}}},
doi = {{10.1365/s13291-019-00210-z}},
volume = {{122}},
year = {{2019}},
}
@article{63351,
author = {{Krzyżanowski, Piotr and Winkler, Michael and Wrzosek, Dariusz}},
issn = {{1468-1218}},
journal = {{Nonlinear Analysis: Real World Applications}},
pages = {{94--116}},
publisher = {{Elsevier BV}},
title = {{{Migration-driven benefit in a two-species nutrient taxis system}}},
doi = {{10.1016/j.nonrwa.2019.01.006}},
volume = {{48}},
year = {{2019}},
}