@article{63362, abstract = {{The system \left\{\begin{matrix} u_{t} = \mathrm{\Delta }u−\chi \mathrm{∇} \cdot \left(\frac{u}{v}\mathrm{∇}v\right)−uv + B_{1}(x,t), \\ v_{t} = \mathrm{\Delta }v + uv−v + B_{2}(x,t), \\ \end{matrix}\right.\:\:( \star ) is considered in a disk \mathrm{\Omega } \subset \mathbb{R}^{2} , with a positive parameter χ and given nonnegative and suitably regular functions B_{1} and B_{2} defined on \mathrm{\Omega } \times (0,\infty ) . In the particular version obtained when \chi = 2 , ( \star ) was proposed in [31] as a model for crime propagation in urban regions. Within a suitable generalized framework, it is shown that under mild assumptions on the parameter functions and the initial data the no-flux initial-boundary value problem for ( \star ) possesses at least one global solution in the case when all model ingredients are radially symmetric with respect to the center of Ω . Moreover, under an additional hypothesis on stabilization of the given external source terms in both equations, these solutions are shown to approach the solution of an elliptic boundary value problem in an appropriate sense. The analysis is based on deriving a priori estimates for a family of approximate problems, in a first step achieving some spatially global but weak initial regularity information which in a series of spatially localized arguments is thereafter successively improved. To the best of our knowledge, this is the first result on global existence of solutions to the two-dimensional version of the full original system ( \star ) for arbitrarily large values of χ . }}, author = {{Winkler, Michael}}, issn = {{0294-1449}}, journal = {{Annales de l'Institut Henri Poincaré C, Analyse non linéaire}}, number = {{6}}, pages = {{1747--1790}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation}}}, doi = {{10.1016/j.anihpc.2019.02.004}}, volume = {{36}}, year = {{2019}}, } @article{63363, abstract = {{ This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model [Formula: see text] for the population densities [Formula: see text] and [Formula: see text] of foragers and exploiters, as well as the nutrient concentration [Formula: see text], is considered in smoothly bounded domains [Formula: see text], [Formula: see text]. It is first shown that under an explicit condition linking the sizes of the resource production rate [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that [Formula: see text] satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆). }}, author = {{Winkler, Michael}}, issn = {{0218-2025}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, number = {{03}}, pages = {{373--418}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions}}}, doi = {{10.1142/s021820251950012x}}, volume = {{29}}, year = {{2019}}, } @article{63366, author = {{Winkler, Michael}}, issn = {{0362-546X}}, journal = {{Nonlinear Analysis}}, pages = {{102--116}}, publisher = {{Elsevier BV}}, title = {{{Instantaneous regularization of distributions from(C0)⋆×L2in the one-dimensional parabolic Keller–Segel system}}}, doi = {{10.1016/j.na.2019.01.017}}, volume = {{183}}, year = {{2019}}, } @article{63359, author = {{Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}}, issn = {{0944-2669}}, journal = {{Calculus of Variations and Partial Differential Equations}}, number = {{6}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{The fast signal diffusion limit in Keller–Segel(-fluid) systems}}}, doi = {{10.1007/s00526-019-1656-3}}, volume = {{58}}, year = {{2019}}, } @article{63364, author = {{Winkler, Michael}}, issn = {{0373-3114}}, journal = {{Annali di Matematica Pura ed Applicata (1923 -)}}, number = {{5}}, pages = {{1615--1637}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{How strong singularities can be regularized by logistic degradation in the Keller–Segel system?}}}, doi = {{10.1007/s10231-019-00834-z}}, volume = {{198}}, year = {{2019}}, } @article{63367, author = {{Winkler, Michael}}, issn = {{1021-9722}}, journal = {{Nonlinear Differential Equations and Applications NoDEA}}, number = {{6}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Does repulsion-type directional preference in chemotactic migration continue to regularize Keller–Segel systems when coupled to the Navier–Stokes equations?}}}, doi = {{10.1007/s00030-019-0600-8}}, volume = {{26}}, year = {{2019}}, } @article{63354, author = {{Souplet, Philippe and Winkler, Michael}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, number = {{2}}, pages = {{665--681}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Blow-up Profiles for the Parabolic–Elliptic Keller–Segel System in Dimensions $${n\geq 3}$$ n ≥ 3}}}, doi = {{10.1007/s00220-018-3238-1}}, volume = {{367}}, year = {{2018}}, } @article{63361, author = {{Winkler, Michael}}, issn = {{0025-5831}}, journal = {{Mathematische Annalen}}, number = {{3-4}}, pages = {{1237--1282}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases}}}, doi = {{10.1007/s00208-018-1722-8}}, volume = {{373}}, year = {{2018}}, } @article{63365, author = {{Winkler, Michael}}, issn = {{0022-0396}}, journal = {{Journal of Differential Equations}}, number = {{12}}, pages = {{8034--8066}}, publisher = {{Elsevier BV}}, title = {{{Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities}}}, doi = {{10.1016/j.jde.2018.12.019}}, volume = {{266}}, year = {{2018}}, } @article{63360, author = {{Winkler, Michael}}, issn = {{0022-1236}}, journal = {{Journal of Functional Analysis}}, number = {{5}}, pages = {{1339--1401}}, publisher = {{Elsevier BV}}, title = {{{A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization}}}, doi = {{10.1016/j.jfa.2018.12.009}}, volume = {{276}}, year = {{2018}}, } @article{63369, abstract = {{The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.}}, author = {{Cao, Xinru and Winkler, Michael}}, issn = {{0308-2105}}, journal = {{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}}, number = {{5}}, pages = {{939--955}}, publisher = {{Cambridge University Press (CUP)}}, title = {{{Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains}}}, doi = {{10.1017/s0308210518000057}}, volume = {{148}}, year = {{2018}}, } @article{63368, author = {{Winkler, Michael}}, issn = {{0167-8019}}, journal = {{Acta Applicandae Mathematicae}}, number = {{1}}, pages = {{1--17}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics with Mildly Saturated Chemotactic Sensitivity}}}, doi = {{10.1007/s10440-018-0211-0}}, volume = {{163}}, 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{63377, author = {{Winkler, Michael}}, issn = {{0044-2275}}, journal = {{Zeitschrift für angewandte Mathematik und Physik}}, number = {{2}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation}}}, doi = {{10.1007/s00033-018-0935-8}}, volume = {{69}}, year = {{2018}}, } @article{63375, author = {{Winkler, Michael}}, issn = {{1422-6928}}, journal = {{Journal of Mathematical Fluid Mechanics}}, number = {{4}}, pages = {{1889--1909}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?}}}, doi = {{10.1007/s00021-018-0395-0}}, volume = {{20}}, year = {{2018}}, } @article{63381, author = {{Winkler, Michael}}, issn = {{1424-3199}}, journal = {{Journal of Evolution Equations}}, number = {{3}}, pages = {{1267--1289}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components}}}, doi = {{10.1007/s00028-018-0440-8}}, volume = {{18}}, year = {{2018}}, } @article{63380, author = {{Winkler, Michael}}, issn = {{0022-0396}}, journal = {{Journal of Differential Equations}}, number = {{10}}, pages = {{6109--6151}}, publisher = {{Elsevier BV}}, title = {{{Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement}}}, doi = {{10.1016/j.jde.2018.01.027}}, volume = {{264}}, 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{63382, author = {{Winkler, Michael and Yokota, Tomomi}}, issn = {{0362-546X}}, journal = {{Nonlinear Analysis}}, pages = {{123--141}}, publisher = {{Elsevier BV}}, title = {{{Stabilization in the logarithmic Keller–Segel system}}}, doi = {{10.1016/j.na.2018.01.002}}, volume = {{170}}, year = {{2018}}, } @article{63371, abstract = {{Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.}}, author = {{HILLEN, T. and PAINTER, K. J. and Winkler, Michael}}, issn = {{0956-7925}}, journal = {{European Journal of Applied Mathematics}}, number = {{4}}, pages = {{645--684}}, publisher = {{Cambridge University Press (CUP)}}, title = {{{Global solvability and explicit bounds for non-local adhesion models}}}, doi = {{10.1017/s0956792517000328}}, volume = {{29}}, year = {{2017}}, }