[{"type":"journal_article","publication":"International Mathematics Research Notices","abstract":[{"lang":"eng","text":"Abstract\r\n We consider the spatially 2D version of the model $$\\begin{equation*} \\qquad\\quad\\left\\{ \\begin{array}{@{}rcll} n_t + u\\cdot\\nabla n &=& \\Delta n - \\nabla \\cdot \\big(nS(x,n,c) \\cdot \\nabla c \\big), \\qquad &\\qquad x\\in \\Omega, \\ t&gt;0, \\\\ c_t + u\\cdot \\nabla c &=& \\Delta c - n f(c), \\qquad &\\qquad x\\in \\Omega, \\ t&gt;0, \\\\ u_t &=& \\Delta u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla\\cdot u=0, \\qquad &\\qquad x\\in \\Omega, \\ t&gt;0, \\end{array} \\right. \\qquad \\qquad (\\star) \\end{equation*}$$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 ($\\star$)."}],"status":"public","_id":"63325","user_id":"31496","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["1073-7928","1687-0247"]},"issue":"11","year":"2019","citation":{"mla":"Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?” International Mathematics Research Notices, vol. 2021, no. 11, Oxford University Press (OUP), 2019, pp. 8106–52, doi:10.1093/imrn/rnz056.","bibtex":"@article{Winkler_2019, title={Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?}, volume={2021}, DOI={10.1093/imrn/rnz056}, number={11}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Winkler, Michael}, year={2019}, pages={8106–8152} }","short":"M. Winkler, International Mathematics Research Notices 2021 (2019) 8106–8152.","apa":"Winkler, M. (2019). Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems? International Mathematics Research Notices, 2021(11), 8106–8152. https://doi.org/10.1093/imrn/rnz056","ieee":"M. Winkler, “Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?,” International Mathematics Research Notices, vol. 2021, no. 11, pp. 8106–8152, 2019, doi: 10.1093/imrn/rnz056.","chicago":"Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?” International Mathematics Research Notices 2021, no. 11 (2019): 8106–52. https://doi.org/10.1093/imrn/rnz056.","ama":"Winkler M. Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems? International Mathematics Research Notices. 2019;2021(11):8106-8152. doi:10.1093/imrn/rnz056"},"intvolume":" 2021","page":"8106-8152","publisher":"Oxford University Press (OUP)","date_updated":"2025-12-18T19:59:29Z","date_created":"2025-12-18T19:35:55Z","author":[{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"volume":2021,"title":"Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?","doi":"10.1093/imrn/rnz056"},{"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","publisher":"Walter de Gruyter GmbH","date_updated":"2025-12-18T20:01:54Z","date_created":"2025-12-18T19:45:09Z","author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"volume":9,"year":"2019","citation":{"apa":"Winkler, M. (2019). The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1. Advances in Nonlinear Analysis, 9(1), 526–566. https://doi.org/10.1515/anona-2020-0013","mla":"Winkler, Michael. “The Role of Superlinear Damping in the Construction of Solutions to Drift-Diffusion Problems with Initial Data in L1.” Advances in Nonlinear Analysis, vol. 9, no. 1, Walter de Gruyter GmbH, 2019, pp. 526–66, doi:10.1515/anona-2020-0013.","bibtex":"@article{Winkler_2019, title={The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1}, volume={9}, DOI={10.1515/anona-2020-0013}, number={1}, journal={Advances in Nonlinear Analysis}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2019}, pages={526–566} }","short":"M. Winkler, Advances in Nonlinear Analysis 9 (2019) 526–566.","ama":"Winkler M. The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1. Advances in Nonlinear Analysis. 2019;9(1):526-566. doi:10.1515/anona-2020-0013","ieee":"M. Winkler, “The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1,” Advances in Nonlinear Analysis, vol. 9, no. 1, pp. 526–566, 2019, doi: 10.1515/anona-2020-0013.","chicago":"Winkler, Michael. “The Role of Superlinear Damping in the Construction of Solutions to Drift-Diffusion Problems with Initial Data in L1.” Advances in Nonlinear Analysis 9, no. 1 (2019): 526–66. https://doi.org/10.1515/anona-2020-0013."},"page":"526-566","intvolume":" 9","publication_status":"published","publication_identifier":{"issn":["2191-950X"]},"issue":"1","language":[{"iso":"eng"}],"_id":"63337","user_id":"31496","abstract":[{"text":"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.","lang":"eng"}],"status":"public","type":"journal_article","publication":"Advances in Nonlinear Analysis"},{"language":[{"iso":"eng"}],"_id":"63334","user_id":"31496","status":"public","type":"journal_article","publication":"Journal of Differential Equations","title":"Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy","doi":"10.1016/j.jde.2019.10.046","date_updated":"2025-12-18T20:01:29Z","publisher":"Elsevier BV","date_created":"2025-12-18T19:40:07Z","author":[{"first_name":"Youshan","last_name":"Tao","full_name":"Tao, Youshan"},{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"volume":268,"year":"2019","citation":{"ieee":"Y. Tao and M. Winkler, “Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy,” Journal of Differential Equations, vol. 268, no. 9, pp. 4973–4997, 2019, doi: 10.1016/j.jde.2019.10.046.","chicago":"Tao, Youshan, and Michael Winkler. “Global Classical Solutions to a Doubly Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy.” Journal of Differential Equations 268, no. 9 (2019): 4973–97. https://doi.org/10.1016/j.jde.2019.10.046.","ama":"Tao Y, Winkler M. Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. Journal of Differential Equations. 2019;268(9):4973-4997. doi:10.1016/j.jde.2019.10.046","short":"Y. Tao, M. Winkler, Journal of Differential Equations 268 (2019) 4973–4997.","mla":"Tao, Youshan, and Michael Winkler. “Global Classical Solutions to a Doubly Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy.” Journal of Differential Equations, vol. 268, no. 9, Elsevier BV, 2019, pp. 4973–97, doi:10.1016/j.jde.2019.10.046.","bibtex":"@article{Tao_Winkler_2019, title={Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy}, volume={268}, DOI={10.1016/j.jde.2019.10.046}, number={9}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={4973–4997} }","apa":"Tao, Y., & Winkler, M. (2019). Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. Journal of Differential Equations, 268(9), 4973–4997. https://doi.org/10.1016/j.jde.2019.10.046"},"page":"4973-4997","intvolume":" 268","publication_status":"published","publication_identifier":{"issn":["0022-0396"]},"issue":"9"},{"date_updated":"2025-12-19T10:47:44Z","publisher":"Society for Industrial & Applied Mathematics (SIAM)","author":[{"first_name":"Nicola","full_name":"Bellomo, Nicola","last_name":"Bellomo"},{"last_name":"Painter","full_name":"Painter, Kevin J.","first_name":"Kevin J."},{"first_name":"Youshan","last_name":"Tao","full_name":"Tao, Youshan"},{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-19T10:47:32Z","volume":79,"title":"Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection","doi":"10.1137/19m1250261","publication_status":"published","publication_identifier":{"issn":["0036-1399","1095-712X"]},"issue":"5","year":"2019","citation":{"ieee":"N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler, “Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection,” SIAM Journal on Applied Mathematics, vol. 79, no. 5, pp. 1990–2010, 2019, doi: 10.1137/19m1250261.","chicago":"Bellomo, Nicola, Kevin J. Painter, Youshan Tao, and Michael Winkler. “Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.” SIAM Journal on Applied Mathematics 79, no. 5 (2019): 1990–2010. https://doi.org/10.1137/19m1250261.","ama":"Bellomo N, Painter KJ, Tao Y, Winkler M. Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection. SIAM Journal on Applied Mathematics. 2019;79(5):1990-2010. doi:10.1137/19m1250261","short":"N. Bellomo, K.J. Painter, Y. Tao, M. Winkler, SIAM Journal on Applied Mathematics 79 (2019) 1990–2010.","bibtex":"@article{Bellomo_Painter_Tao_Winkler_2019, title={Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection}, volume={79}, DOI={10.1137/19m1250261}, number={5}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bellomo, Nicola and Painter, Kevin J. and Tao, Youshan and Winkler, Michael}, year={2019}, pages={1990–2010} }","mla":"Bellomo, Nicola, et al. “Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.” SIAM Journal on Applied Mathematics, vol. 79, no. 5, Society for Industrial & Applied Mathematics (SIAM), 2019, pp. 1990–2010, doi:10.1137/19m1250261.","apa":"Bellomo, N., Painter, K. J., Tao, Y., & Winkler, M. (2019). Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection. SIAM Journal on Applied Mathematics, 79(5), 1990–2010. https://doi.org/10.1137/19m1250261"},"intvolume":" 79","page":"1990-2010","_id":"63349","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"SIAM Journal on Applied Mathematics","status":"public"},{"publication_status":"published","publication_identifier":{"issn":["0001-8708"]},"year":"2019","citation":{"ama":"Fila M, Winkler M. A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation. Advances in Mathematics. 2019;357. doi:10.1016/j.aim.2019.106823","ieee":"M. Fila and M. Winkler, “A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation,” Advances in Mathematics, vol. 357, Art. no. 106823, 2019, doi: 10.1016/j.aim.2019.106823.","chicago":"Fila, Marek, and Michael Winkler. “A Gagliardo-Nirenberg-Type Inequality and Its Applications to Decay Estimates for Solutions of a Degenerate Parabolic Equation.” Advances in Mathematics 357 (2019). https://doi.org/10.1016/j.aim.2019.106823.","apa":"Fila, M., & Winkler, M. (2019). A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation. Advances in Mathematics, 357, Article 106823. https://doi.org/10.1016/j.aim.2019.106823","bibtex":"@article{Fila_Winkler_2019, title={A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation}, volume={357}, DOI={10.1016/j.aim.2019.106823}, number={106823}, journal={Advances in Mathematics}, publisher={Elsevier BV}, author={Fila, Marek and Winkler, Michael}, year={2019} }","mla":"Fila, Marek, and Michael Winkler. “A Gagliardo-Nirenberg-Type Inequality and Its Applications to Decay Estimates for Solutions of a Degenerate Parabolic Equation.” Advances in Mathematics, vol. 357, 106823, Elsevier BV, 2019, doi:10.1016/j.aim.2019.106823.","short":"M. Fila, M. Winkler, Advances in Mathematics 357 (2019)."},"intvolume":" 357","date_updated":"2025-12-19T10:49:47Z","publisher":"Elsevier BV","date_created":"2025-12-19T10:49:35Z","author":[{"full_name":"Fila, Marek","last_name":"Fila","first_name":"Marek"},{"last_name":"Winkler","full_name":"Winkler, Michael","first_name":"Michael"}],"volume":357,"title":"A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation","doi":"10.1016/j.aim.2019.106823","type":"journal_article","publication":"Advances in Mathematics","status":"public","_id":"63350","user_id":"31496","article_number":"106823","language":[{"iso":"eng"}]},{"type":"journal_article","publication":"Proceedings of the London Mathematical Society","abstract":[{"lang":"eng","text":"AbstractThis work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by\r\n\r\nwith 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\r\n\r\nMoreover, 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 ."}],"status":"public","_id":"63355","user_id":"31496","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0024-6115","1460-244X"]},"issue":"6","year":"2019","citation":{"ieee":"Y. Tao and M. Winkler, “Boundedness and stabilization in a population model with cross‐diffusion for one species,” Proceedings of the London Mathematical Society, vol. 119, no. 6, pp. 1598–1632, 2019, doi: 10.1112/plms.12276.","chicago":"Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population Model with Cross‐diffusion for One Species.” Proceedings of the London Mathematical Society 119, no. 6 (2019): 1598–1632. https://doi.org/10.1112/plms.12276.","ama":"Tao Y, Winkler M. Boundedness and stabilization in a population model with cross‐diffusion for one species. Proceedings of the London Mathematical Society. 2019;119(6):1598-1632. doi:10.1112/plms.12276","short":"Y. Tao, M. Winkler, Proceedings of the London Mathematical Society 119 (2019) 1598–1632.","bibtex":"@article{Tao_Winkler_2019, title={Boundedness and stabilization in a population model with cross‐diffusion for one species}, volume={119}, DOI={10.1112/plms.12276}, number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={1598–1632} }","mla":"Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population Model with Cross‐diffusion for One Species.” Proceedings of the London Mathematical Society, vol. 119, no. 6, Wiley, 2019, pp. 1598–632, doi:10.1112/plms.12276.","apa":"Tao, Y., & Winkler, M. (2019). Boundedness and stabilization in a population model with cross‐diffusion for one species. Proceedings of the London Mathematical Society, 119(6), 1598–1632. https://doi.org/10.1112/plms.12276"},"intvolume":" 119","page":"1598-1632","publisher":"Wiley","date_updated":"2025-12-19T10:54:09Z","author":[{"first_name":"Youshan","full_name":"Tao, Youshan","last_name":"Tao"},{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-19T10:54:01Z","volume":119,"title":"Boundedness and stabilization in a population model with cross‐diffusion for one species","doi":"10.1112/plms.12276"},{"year":"2019","citation":{"short":"Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 29 (2019) 2151–2182.","mla":"Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter Model with Different Taxis Strategies for Two Groups in Search of Food.” Mathematical Models and Methods in Applied Sciences, vol. 29, no. 11, World Scientific Pub Co Pte Ltd, 2019, pp. 2151–82, doi:10.1142/s021820251950043x.","bibtex":"@article{Tao_Winkler_2019, title={Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food}, volume={29}, DOI={10.1142/s021820251950043x}, number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={2151–2182} }","apa":"Tao, Y., & Winkler, M. (2019). Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food. Mathematical Models and Methods in Applied Sciences, 29(11), 2151–2182. https://doi.org/10.1142/s021820251950043x","ieee":"Y. Tao and M. Winkler, “Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food,” Mathematical Models and Methods in Applied Sciences, vol. 29, no. 11, pp. 2151–2182, 2019, doi: 10.1142/s021820251950043x.","chicago":"Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter Model with Different Taxis Strategies for Two Groups in Search of Food.” Mathematical Models and Methods in Applied Sciences 29, no. 11 (2019): 2151–82. https://doi.org/10.1142/s021820251950043x.","ama":"Tao Y, Winkler M. Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food. Mathematical Models and Methods in Applied Sciences. 2019;29(11):2151-2182. doi:10.1142/s021820251950043x"},"page":"2151-2182","intvolume":" 29","publication_status":"published","publication_identifier":{"issn":["0218-2025","1793-6314"]},"issue":"11","title":"Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food","doi":"10.1142/s021820251950043x","publisher":"World Scientific Pub Co Pte Ltd","date_updated":"2025-12-19T10:54:44Z","date_created":"2025-12-19T10:54:36Z","author":[{"full_name":"Tao, Youshan","last_name":"Tao","first_name":"Youshan"},{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"}],"volume":29,"abstract":[{"lang":"eng","text":" 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. "}],"status":"public","type":"journal_article","publication":"Mathematical Models and Methods in Applied Sciences","language":[{"iso":"eng"}],"_id":"63356","user_id":"31496"},{"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","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-19T10:51:33Z","date_created":"2025-12-19T10:51:24Z","author":[{"first_name":"Johannes","full_name":"Lankeit, Johannes","last_name":"Lankeit"},{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"volume":233,"year":"2019","citation":{"apa":"Lankeit, J., & Winkler, M. (2019). Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory. Israel Journal of Mathematics, 233(1), 249–296. https://doi.org/10.1007/s11856-019-1900-8","mla":"Lankeit, Johannes, and Michael Winkler. “Counterintuitive Dependence of Temporal Asymptotics on Initial Decay in a Nonlocal Degenerate Parabolic Equation Arising in Game Theory.” Israel Journal of Mathematics, vol. 233, no. 1, Springer Science and Business Media LLC, 2019, pp. 249–96, doi:10.1007/s11856-019-1900-8.","short":"J. Lankeit, M. Winkler, Israel Journal of Mathematics 233 (2019) 249–296.","bibtex":"@article{Lankeit_Winkler_2019, title={Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory}, volume={233}, DOI={10.1007/s11856-019-1900-8}, number={1}, journal={Israel Journal of Mathematics}, publisher={Springer Science and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2019}, pages={249–296} }","chicago":"Lankeit, Johannes, and Michael Winkler. “Counterintuitive Dependence of Temporal Asymptotics on Initial Decay in a Nonlocal Degenerate Parabolic Equation Arising in Game Theory.” Israel Journal of Mathematics 233, no. 1 (2019): 249–96. https://doi.org/10.1007/s11856-019-1900-8.","ieee":"J. Lankeit and M. Winkler, “Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory,” Israel Journal of Mathematics, vol. 233, no. 1, pp. 249–296, 2019, doi: 10.1007/s11856-019-1900-8.","ama":"Lankeit J, Winkler M. Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory. Israel Journal of Mathematics. 2019;233(1):249-296. doi:10.1007/s11856-019-1900-8"},"page":"249-296","intvolume":" 233","publication_status":"published","publication_identifier":{"issn":["0021-2172","1565-8511"]},"issue":"1","language":[{"iso":"eng"}],"_id":"63352","user_id":"31496","status":"public","type":"journal_article","publication":"Israel Journal of Mathematics"},{"type":"journal_article","publication":"Communications on Pure & Applied Analysis","status":"public","user_id":"31496","_id":"63358","language":[{"iso":"eng"}],"issue":"4","publication_status":"published","publication_identifier":{"issn":["1553-5258"]},"citation":{"ama":"Tao Y, Winkler M. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure &amp; Applied Analysis. 2019;18(4):2047-2067. doi:10.3934/cpaa.2019092","ieee":"Y. Tao and M. Winkler, “A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production,” Communications on Pure &amp; Applied Analysis, vol. 18, no. 4, pp. 2047–2067, 2019, doi: 10.3934/cpaa.2019092.","chicago":"Tao, Youshan, and Michael Winkler. “A Chemotaxis-Haptotaxis System with Haptoattractant Remodeling: Boundedness Enforced by Mild Saturation of Signal Production.” Communications on Pure &amp; Applied Analysis 18, no. 4 (2019): 2047–67. https://doi.org/10.3934/cpaa.2019092.","apa":"Tao, Y., & Winkler, M. (2019). A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure &amp; Applied Analysis, 18(4), 2047–2067. https://doi.org/10.3934/cpaa.2019092","mla":"Tao, Youshan, and Michael Winkler. “A Chemotaxis-Haptotaxis System with Haptoattractant Remodeling: Boundedness Enforced by Mild Saturation of Signal Production.” Communications on Pure &amp; Applied Analysis, vol. 18, no. 4, American Institute of Mathematical Sciences (AIMS), 2019, pp. 2047–67, doi:10.3934/cpaa.2019092.","short":"Y. Tao, M. Winkler, Communications on Pure &amp; Applied Analysis 18 (2019) 2047–2067.","bibtex":"@article{Tao_Winkler_2019, title={A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production}, volume={18}, DOI={10.3934/cpaa.2019092}, number={4}, journal={Communications on Pure &amp; Applied Analysis}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={2047–2067} }"},"intvolume":" 18","page":"2047-2067","year":"2019","author":[{"first_name":"Youshan","full_name":"Tao, Youshan","last_name":"Tao"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-19T10:56:26Z","volume":18,"publisher":"American Institute of Mathematical Sciences (AIMS)","date_updated":"2025-12-19T10:56:33Z","doi":"10.3934/cpaa.2019092","title":"A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production"},{"language":[{"iso":"eng"}],"_id":"63357","user_id":"31496","status":"public","type":"journal_article","publication":"Journal of Differential Equations","title":"Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension","doi":"10.1016/j.jde.2019.01.014","date_updated":"2025-12-19T10:56:01Z","publisher":"Elsevier BV","author":[{"first_name":"Youshan","full_name":"Tao, Youshan","last_name":"Tao"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-19T10:55:53Z","volume":267,"year":"2019","citation":{"short":"Y. Tao, M. Winkler, Journal of Differential Equations 267 (2019) 388–406.","bibtex":"@article{Tao_Winkler_2019, title={Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension}, volume={267}, DOI={10.1016/j.jde.2019.01.014}, number={1}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={388–406} }","mla":"Tao, Youshan, and Michael Winkler. “Global Smooth Solvability of a Parabolic–Elliptic Nutrient Taxis System in Domains of Arbitrary Dimension.” Journal of Differential Equations, vol. 267, no. 1, Elsevier BV, 2019, pp. 388–406, doi:10.1016/j.jde.2019.01.014.","apa":"Tao, Y., & Winkler, M. (2019). Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension. Journal of Differential Equations, 267(1), 388–406. https://doi.org/10.1016/j.jde.2019.01.014","ama":"Tao Y, Winkler M. Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension. Journal of Differential Equations. 2019;267(1):388-406. doi:10.1016/j.jde.2019.01.014","chicago":"Tao, Youshan, and Michael Winkler. “Global Smooth Solvability of a Parabolic–Elliptic Nutrient Taxis System in Domains of Arbitrary Dimension.” Journal of Differential Equations 267, no. 1 (2019): 388–406. https://doi.org/10.1016/j.jde.2019.01.014.","ieee":"Y. Tao and M. Winkler, “Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension,” Journal of Differential Equations, vol. 267, no. 1, pp. 388–406, 2019, doi: 10.1016/j.jde.2019.01.014."},"intvolume":" 267","page":"388-406","publication_status":"published","publication_identifier":{"issn":["0022-0396"]},"issue":"1"},{"volume":122,"date_created":"2025-12-19T10:52:04Z","author":[{"first_name":"Johannes","last_name":"Lankeit","full_name":"Lankeit, Johannes"},{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"}],"publisher":"Springer Fachmedien Wiesbaden GmbH","date_updated":"2025-12-19T10:52:11Z","doi":"10.1365/s13291-019-00210-z","title":"Facing Low Regularity in Chemotaxis Systems","issue":"1","publication_identifier":{"issn":["0012-0456","1869-7135"]},"publication_status":"published","page":"35-64","intvolume":" 122","citation":{"ama":"Lankeit J, Winkler M. Facing Low Regularity in Chemotaxis Systems. Jahresbericht der Deutschen Mathematiker-Vereinigung. 2019;122(1):35-64. doi:10.1365/s13291-019-00210-z","ieee":"J. Lankeit and M. Winkler, “Facing Low Regularity in Chemotaxis Systems,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 122, no. 1, pp. 35–64, 2019, doi: 10.1365/s13291-019-00210-z.","chicago":"Lankeit, Johannes, and Michael Winkler. “Facing Low Regularity in Chemotaxis Systems.” Jahresbericht Der Deutschen Mathematiker-Vereinigung 122, no. 1 (2019): 35–64. https://doi.org/10.1365/s13291-019-00210-z.","apa":"Lankeit, J., & Winkler, M. (2019). Facing Low Regularity in Chemotaxis Systems. Jahresbericht Der Deutschen Mathematiker-Vereinigung, 122(1), 35–64. https://doi.org/10.1365/s13291-019-00210-z","short":"J. Lankeit, M. Winkler, Jahresbericht Der Deutschen Mathematiker-Vereinigung 122 (2019) 35–64.","bibtex":"@article{Lankeit_Winkler_2019, title={Facing Low Regularity in Chemotaxis Systems}, volume={122}, DOI={10.1365/s13291-019-00210-z}, number={1}, journal={Jahresbericht der Deutschen Mathematiker-Vereinigung}, publisher={Springer Fachmedien Wiesbaden GmbH}, author={Lankeit, Johannes and Winkler, Michael}, year={2019}, pages={35–64} }","mla":"Lankeit, Johannes, and Michael Winkler. “Facing Low Regularity in Chemotaxis Systems.” Jahresbericht Der Deutschen Mathematiker-Vereinigung, vol. 122, no. 1, Springer Fachmedien Wiesbaden GmbH, 2019, pp. 35–64, doi:10.1365/s13291-019-00210-z."},"year":"2019","user_id":"31496","_id":"63353","language":[{"iso":"eng"}],"publication":"Jahresbericht der Deutschen Mathematiker-Vereinigung","type":"journal_article","status":"public"},{"_id":"63351","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"Nonlinear Analysis: Real World Applications","status":"public","publisher":"Elsevier BV","date_updated":"2025-12-19T10:50:59Z","author":[{"first_name":"Piotr","full_name":"Krzyżanowski, Piotr","last_name":"Krzyżanowski"},{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"},{"first_name":"Dariusz","full_name":"Wrzosek, Dariusz","last_name":"Wrzosek"}],"date_created":"2025-12-19T10:50:49Z","volume":48,"title":"Migration-driven benefit in a two-species nutrient taxis system","doi":"10.1016/j.nonrwa.2019.01.006","publication_status":"published","publication_identifier":{"issn":["1468-1218"]},"year":"2019","citation":{"short":"P. Krzyżanowski, M. Winkler, D. Wrzosek, Nonlinear Analysis: Real World Applications 48 (2019) 94–116.","mla":"Krzyżanowski, Piotr, et al. “Migration-Driven Benefit in a Two-Species Nutrient Taxis System.” Nonlinear Analysis: Real World Applications, vol. 48, Elsevier BV, 2019, pp. 94–116, doi:10.1016/j.nonrwa.2019.01.006.","bibtex":"@article{Krzyżanowski_Winkler_Wrzosek_2019, title={Migration-driven benefit in a two-species nutrient taxis system}, volume={48}, DOI={10.1016/j.nonrwa.2019.01.006}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier BV}, author={Krzyżanowski, Piotr and Winkler, Michael and Wrzosek, Dariusz}, year={2019}, pages={94–116} }","apa":"Krzyżanowski, P., Winkler, M., & Wrzosek, D. (2019). Migration-driven benefit in a two-species nutrient taxis system. Nonlinear Analysis: Real World Applications, 48, 94–116. https://doi.org/10.1016/j.nonrwa.2019.01.006","ama":"Krzyżanowski P, Winkler M, Wrzosek D. Migration-driven benefit in a two-species nutrient taxis system. Nonlinear Analysis: Real World Applications. 2019;48:94-116. doi:10.1016/j.nonrwa.2019.01.006","ieee":"P. Krzyżanowski, M. Winkler, and D. Wrzosek, “Migration-driven benefit in a two-species nutrient taxis system,” Nonlinear Analysis: Real World Applications, vol. 48, pp. 94–116, 2019, doi: 10.1016/j.nonrwa.2019.01.006.","chicago":"Krzyżanowski, Piotr, Michael Winkler, and Dariusz Wrzosek. “Migration-Driven Benefit in a Two-Species Nutrient Taxis System.” Nonlinear Analysis: Real World Applications 48 (2019): 94–116. https://doi.org/10.1016/j.nonrwa.2019.01.006."},"intvolume":" 48","page":"94-116"},{"publication":"Annales de l'Institut Henri Poincaré C, Analyse non linéaire","type":"journal_article","status":"public","abstract":[{"text":"The system\r\n \r\n \r\n \\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 )\r\n \r\n \r\n \r\n is considered in a disk \r\n \r\n \\mathrm{\\Omega } \\subset \\mathbb{R}^{2}\r\n \r\n , with a positive parameter \r\n \r\n χ\r\n \r\n and given nonnegative and suitably regular functions \r\n \r\n B_{1}\r\n \r\n and \r\n \r\n B_{2}\r\n \r\n defined on \r\n \r\n \\mathrm{\\Omega } \\times (0,\\infty )\r\n \r\n . In the particular version obtained when \r\n \r\n \\chi = 2\r\n \r\n , (\r\n \r\n \\star\r\n \r\n ) was proposed in [31] as a model for crime propagation in urban regions.\r\n \r\n \r\n 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 (\r\n \r\n \\star\r\n \r\n ) possesses at least one global solution in the case when all model ingredients are radially symmetric with respect to the center of \r\n \r\n Ω\r\n \r\n . 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.\r\n \r\n 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.\r\n \r\n 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 (\r\n \r\n \\star\r\n \r\n ) for arbitrarily large values of \r\n \r\n χ\r\n \r\n .\r\n ","lang":"eng"}],"user_id":"31496","_id":"63362","language":[{"iso":"eng"}],"issue":"6","publication_identifier":{"issn":["0294-1449","1873-1430"]},"publication_status":"published","page":"1747-1790","intvolume":" 36","citation":{"apa":"Winkler, M. (2019). Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 36(6), 1747–1790. https://doi.org/10.1016/j.anihpc.2019.02.004","bibtex":"@article{Winkler_2019, title={Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation}, volume={36}, DOI={10.1016/j.anihpc.2019.02.004}, number={6}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael}, year={2019}, pages={1747–1790} }","short":"M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 36 (2019) 1747–1790.","mla":"Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional Cross-Diffusion System Modeling Urban Crime Propagation.” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, vol. 36, no. 6, European Mathematical Society - EMS - Publishing House GmbH, 2019, pp. 1747–90, doi:10.1016/j.anihpc.2019.02.004.","chicago":"Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional Cross-Diffusion System Modeling Urban Crime Propagation.” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 36, no. 6 (2019): 1747–90. https://doi.org/10.1016/j.anihpc.2019.02.004.","ieee":"M. Winkler, “Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation,” Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 36, no. 6, pp. 1747–1790, 2019, doi: 10.1016/j.anihpc.2019.02.004.","ama":"Winkler M. Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation. Annales de l’Institut Henri Poincaré C, Analyse non linéaire. 2019;36(6):1747-1790. doi:10.1016/j.anihpc.2019.02.004"},"year":"2019","volume":36,"author":[{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-19T10:58:29Z","date_updated":"2025-12-19T10:58:37Z","publisher":"European Mathematical Society - EMS - Publishing House GmbH","doi":"10.1016/j.anihpc.2019.02.004","title":"Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation"},{"doi":"10.1142/s021820251950012x","title":"Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions","date_created":"2025-12-19T10:59:03Z","author":[{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"volume":29,"publisher":"World Scientific Pub Co Pte Ltd","date_updated":"2025-12-19T10:59:10Z","citation":{"apa":"Winkler, M. (2019). Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions. Mathematical Models and Methods in Applied Sciences, 29(03), 373–418. https://doi.org/10.1142/s021820251950012x","mla":"Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional Doubly Tactic Resource Consumption Model Accounting for Social Interactions.” Mathematical Models and Methods in Applied Sciences, vol. 29, no. 03, World Scientific Pub Co Pte Ltd, 2019, pp. 373–418, doi:10.1142/s021820251950012x.","bibtex":"@article{Winkler_2019, title={Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions}, volume={29}, DOI={10.1142/s021820251950012x}, number={03}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2019}, pages={373–418} }","short":"M. Winkler, Mathematical Models and Methods in Applied Sciences 29 (2019) 373–418.","ama":"Winkler M. Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions. Mathematical Models and Methods in Applied Sciences. 2019;29(03):373-418. doi:10.1142/s021820251950012x","ieee":"M. Winkler, “Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions,” Mathematical Models and Methods in Applied Sciences, vol. 29, no. 03, pp. 373–418, 2019, doi: 10.1142/s021820251950012x.","chicago":"Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional Doubly Tactic Resource Consumption Model Accounting for Social Interactions.” Mathematical Models and Methods in Applied Sciences 29, no. 03 (2019): 373–418. https://doi.org/10.1142/s021820251950012x."},"page":"373-418","intvolume":" 29","year":"2019","issue":"03","publication_status":"published","publication_identifier":{"issn":["0218-2025","1793-6314"]},"language":[{"iso":"eng"}],"user_id":"31496","_id":"63363","status":"public","abstract":[{"text":" 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 (⋆). ","lang":"eng"}],"type":"journal_article","publication":"Mathematical Models and Methods in Applied Sciences"},{"user_id":"31496","_id":"63366","language":[{"iso":"eng"}],"type":"journal_article","publication":"Nonlinear Analysis","status":"public","date_created":"2025-12-19T11:01:12Z","author":[{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"volume":183,"date_updated":"2025-12-19T11:01:21Z","publisher":"Elsevier BV","doi":"10.1016/j.na.2019.01.017","title":"Instantaneous regularization of distributions from(C0)⋆×L2in the one-dimensional parabolic Keller–Segel system","publication_status":"published","publication_identifier":{"issn":["0362-546X"]},"citation":{"mla":"Winkler, Michael. “Instantaneous Regularization of Distributions From<mml:Math Xmlns:Mml=\"http://Www.W3.Org/1998/Math/MathML\" Display=\"inline\" Overflow=\"scroll\" Id=\"d1e19\" Altimg=\"si17.Gif\"><mml:Msup><mml:Mrow><mml:Mrow><mml:Mo>(</Mml:Mo><mml:Msup><mml:Mrow><mml:Mi>C</Mml:Mi></Mml:Mrow><mml:Mrow><mml:Mn>0</Mml:Mn></Mml:Mrow></Mml:Msup><mml:Mo>)</Mml:Mo></Mml:Mrow></Mml:Mrow><mml:Mrow><mml:Mo>⋆</Mml:Mo></Mml:Mrow></Mml:Msup><mml:Mo>×</Mml:Mo><mml:Msup><mml:Mrow><mml:Mi>L</Mml:Mi></Mml:Mrow><mml:Mrow><mml:Mn>2</Mml:Mn></Mml:Mrow></Mml:Msup></Mml:Math>in the One-Dimensional Parabolic Keller–Segel System.” Nonlinear Analysis, vol. 183, Elsevier BV, 2019, pp. 102–16, doi:10.1016/j.na.2019.01.017.","short":"M. Winkler, Nonlinear Analysis 183 (2019) 102–116.","bibtex":"@article{Winkler_2019, title={Instantaneous regularization of distributions from<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\" id=\"d1e19\" altimg=\"si17.gif\"><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>in the one-dimensional parabolic Keller–Segel system}, volume={183}, DOI={10.1016/j.na.2019.01.017}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Winkler, Michael}, year={2019}, pages={102–116} }","apa":"Winkler, M. (2019). Instantaneous regularization of distributions from<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\" id=\"d1e19\" altimg=\"si17.gif\"><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>in the one-dimensional parabolic Keller–Segel system. Nonlinear Analysis, 183, 102–116. https://doi.org/10.1016/j.na.2019.01.017","ama":"Winkler M. Instantaneous regularization of distributions from<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\" id=\"d1e19\" altimg=\"si17.gif\"><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>in the one-dimensional parabolic Keller–Segel system. Nonlinear Analysis. 2019;183:102-116. doi:10.1016/j.na.2019.01.017","ieee":"M. Winkler, “Instantaneous regularization of distributions from<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\" id=\"d1e19\" altimg=\"si17.gif\"><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>in the one-dimensional parabolic Keller–Segel system,” Nonlinear Analysis, vol. 183, pp. 102–116, 2019, doi: 10.1016/j.na.2019.01.017.","chicago":"Winkler, Michael. “Instantaneous Regularization of Distributions From<mml:Math Xmlns:Mml=\"http://Www.W3.Org/1998/Math/MathML\" Display=\"inline\" Overflow=\"scroll\" Id=\"d1e19\" Altimg=\"si17.Gif\"><mml:Msup><mml:Mrow><mml:Mrow><mml:Mo>(</Mml:Mo><mml:Msup><mml:Mrow><mml:Mi>C</Mml:Mi></Mml:Mrow><mml:Mrow><mml:Mn>0</Mml:Mn></Mml:Mrow></Mml:Msup><mml:Mo>)</Mml:Mo></Mml:Mrow></Mml:Mrow><mml:Mrow><mml:Mo>⋆</Mml:Mo></Mml:Mrow></Mml:Msup><mml:Mo>×</Mml:Mo><mml:Msup><mml:Mrow><mml:Mi>L</Mml:Mi></Mml:Mrow><mml:Mrow><mml:Mn>2</Mml:Mn></Mml:Mrow></Mml:Msup></Mml:Math>in the One-Dimensional Parabolic Keller–Segel System.” Nonlinear Analysis 183 (2019): 102–16. https://doi.org/10.1016/j.na.2019.01.017."},"page":"102-116","intvolume":" 183","year":"2019"},{"intvolume":" 58","citation":{"apa":"Wang, Y., Winkler, M., & Xiang, Z. 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