--- _id: '63672' article_number: '24' author: - first_name: Tobias full_name: Black, Tobias id: '23686' last_name: Black orcid: 0000-0001-9963-0800 - first_name: Shohei full_name: Kohatsu, Shohei last_name: Kohatsu - first_name: Duan full_name: Wu, Duan last_name: Wu citation: ama: Black T, Kohatsu S, Wu D. Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption. Journal of Evolution Equations. 2026;26(1). doi:10.1007/s00028-025-01163-w apa: Black, T., Kohatsu, S., & Wu, D. (2026). Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption. Journal of Evolution Equations, 26(1), Article 24. https://doi.org/10.1007/s00028-025-01163-w bibtex: '@article{Black_Kohatsu_Wu_2026, title={Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption}, volume={26}, DOI={10.1007/s00028-025-01163-w}, number={124}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Kohatsu, Shohei and Wu, Duan}, year={2026} }' chicago: Black, Tobias, Shohei Kohatsu, and Duan Wu. “Global Solvability and Large-Time Behavior in a Doubly Degenerate Migration Model Involving Saturated Signal Consumption.” Journal of Evolution Equations 26, no. 1 (2026). https://doi.org/10.1007/s00028-025-01163-w. ieee: 'T. Black, S. Kohatsu, and D. Wu, “Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption,” Journal of Evolution Equations, vol. 26, no. 1, Art. no. 24, 2026, doi: 10.1007/s00028-025-01163-w.' mla: Black, Tobias, et al. “Global Solvability and Large-Time Behavior in a Doubly Degenerate Migration Model Involving Saturated Signal Consumption.” Journal of Evolution Equations, vol. 26, no. 1, 24, Springer Science and Business Media LLC, 2026, doi:10.1007/s00028-025-01163-w. short: T. Black, S. Kohatsu, D. Wu, Journal of Evolution Equations 26 (2026). date_created: 2026-01-20T14:13:53Z date_updated: 2026-01-20T14:14:50Z department: - _id: '34' - _id: '10' - _id: '90' doi: 10.1007/s00028-025-01163-w intvolume: ' 26' issue: '1' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Global solvability and large-time behavior in a doubly degenerate migration model involving saturated signal consumption type: journal_article user_id: '23686' volume: 26 year: '2026' ... --- _id: '63249' abstract: - lang: eng text: "Abstract\r\n \r\n The model\r\n \r\n \r\n \ $$\\begin{aligned} \\left\\{ \\begin{array}{l}u_{tt} = \\big (\\gamma (\\Theta ) u_{xt}\\big )_x + au_{xx} - \\big (f(\\Theta )\\big )_x, \\\\[1mm] \\Theta _t = \\Theta _{xx} + \\gamma (\\Theta ) u_{xt}^2 - f(\\Theta ) u_{xt}, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ u\r\n \r\n \ tt\r\n \r\n \ \r\n =\r\n \ \r\n (\r\n \ \r\n γ\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ u\r\n \r\n \ xt\r\n \r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n x\r\n \ \r\n +\r\n \ a\r\n \r\n \ u\r\n \r\n \ xx\r\n \r\n \ \r\n -\r\n \ \r\n (\r\n \ \r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n x\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ [\r\n 1\r\n \ m\r\n m\r\n \ ]\r\n \r\n \ \r\n Θ\r\n \ t\r\n \r\n \ =\r\n \r\n \ Θ\r\n \r\n \ xx\r\n \r\n \ \r\n +\r\n \ γ\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n u\r\n \ \r\n xt\r\n \ \r\n 2\r\n \ \r\n -\r\n \ f\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n u\r\n \ \r\n xt\r\n \ \r\n \r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ for thermoviscoelastic evolution in one-dimensional Kelvin–Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if\r\n \r\n \ \r\n $$\\gamma _0>0$$\r\n \r\n \ \r\n \r\n \ γ\r\n 0\r\n \ \r\n >\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n is fixed, then there exists\r\n \r\n \r\n \ $$\\delta =\\delta (\\gamma _0)>0$$\r\n \ \r\n \ \r\n δ\r\n \ =\r\n δ\r\n \ (\r\n \r\n \ γ\r\n 0\r\n \ \r\n )\r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ with the property that for suitably regular initial data of arbitrary size an associated initial boundary value problem posed in an open bounded interval admits a global classical solution whenever\r\n \r\n \ \r\n $$\\gamma \\in C^2([0,\\infty ))$$\r\n \r\n \r\n \ γ\r\n ∈\r\n \ \r\n C\r\n \ 2\r\n \r\n \ \r\n (\r\n \ \r\n [\r\n \ 0\r\n ,\r\n \ ∞\r\n )\r\n \ \r\n )\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$f\\in C^2([0,\\infty ))$$\r\n \r\n \ \r\n f\r\n \ ∈\r\n \r\n \ C\r\n 2\r\n \ \r\n \r\n \ (\r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ are such that\r\n \r\n \ \r\n $$f(0)=0$$\r\n \ \r\n \ \r\n f\r\n \ (\r\n 0\r\n \ )\r\n =\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$|f(\\xi )| \\le K_f \\cdot (\\xi +1)^\\alpha $$\r\n \r\n \r\n \ \r\n |\r\n \ f\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ |\r\n \r\n \ ≤\r\n \r\n \ K\r\n f\r\n \ \r\n ·\r\n \ \r\n \r\n \ (\r\n ξ\r\n \ +\r\n 1\r\n \ )\r\n \r\n \ α\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ for all\r\n \r\n \r\n \ $$\\xi \\ge 0$$\r\n \r\n \r\n \ ξ\r\n ≥\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n and some\r\n \ \r\n \r\n \ $$K_f>0$$\r\n \r\n \r\n \ \r\n K\r\n \ f\r\n \r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ and\r\n \r\n \r\n \ $$\\alpha <\\frac{3}{2}$$\r\n \ \r\n \ \r\n α\r\n \ <\r\n \r\n \ 3\r\n 2\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n , and that\r\n \ \r\n \r\n \ $$\\begin{aligned} \\gamma _0 \\le \\gamma (\\xi ) \\le \\gamma _0 + \\delta \\qquad \\hbox {for all } \\xi \\ge 0. \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ γ\r\n 0\r\n \ \r\n ≤\r\n \ γ\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ ≤\r\n \r\n \ γ\r\n 0\r\n \ \r\n +\r\n \ δ\r\n \r\n \ for all\r\n \r\n \ ξ\r\n ≥\r\n \ 0\r\n .\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.\r\n " article_number: '108' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities. Journal of Evolution Equations. 2025;25(4). doi:10.1007/s00028-025-01144-z apa: Winkler, M. (2025). Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities. Journal of Evolution Equations, 25(4), Article 108. https://doi.org/10.1007/s00028-025-01144-z bibtex: '@article{Winkler_2025, title={Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities}, volume={25}, DOI={10.1007/s00028-025-01144-z}, number={4108}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Large-Data Regular Solutions in a One-Dimensional Thermoviscoelastic Evolution Problem Involving Temperature-Dependent Viscosities.” Journal of Evolution Equations 25, no. 4 (2025). https://doi.org/10.1007/s00028-025-01144-z. ieee: 'M. Winkler, “Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities,” Journal of Evolution Equations, vol. 25, no. 4, Art. no. 108, 2025, doi: 10.1007/s00028-025-01144-z.' mla: Winkler, Michael. “Large-Data Regular Solutions in a One-Dimensional Thermoviscoelastic Evolution Problem Involving Temperature-Dependent Viscosities.” Journal of Evolution Equations, vol. 25, no. 4, 108, Springer Science and Business Media LLC, 2025, doi:10.1007/s00028-025-01144-z. short: M. Winkler, Journal of Evolution Equations 25 (2025). date_created: 2025-12-18T19:02:51Z date_updated: 2026-04-23T12:19:51Z doi: 10.1007/s00028-025-01144-z intvolume: ' 25' issue: '4' language: - iso: eng project: - _id: '245' name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken für Leistungsschallanwendungen (NEPTUN)' publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities type: journal_article user_id: '31496' volume: 25 year: '2025' ... --- _id: '53316' abstract: - lang: eng text: "AbstractThe quasilinear Keller–Segel system

$$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t=\\nabla \\cdot (D(u)\\nabla u) - \\nabla \\cdot (S(u)\\nabla v), \\\\ v_t=\\Delta v-v+u, \\end{array}\\right. \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ t\r\n \r\n \ =\r\n ∇\r\n \ ·\r\n \r\n \ (\r\n D\r\n \ \r\n (\r\n \ u\r\n )\r\n \ \r\n ∇\r\n \ u\r\n )\r\n \ \r\n -\r\n \ ∇\r\n ·\r\n \ \r\n (\r\n \ S\r\n \r\n \ (\r\n u\r\n \ )\r\n \r\n \ ∇\r\n v\r\n \ )\r\n \r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n v\r\n \ t\r\n \r\n \ =\r\n Δ\r\n \ v\r\n -\r\n \ v\r\n +\r\n \ u\r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ \r\n endowed with homogeneous Neumann boundary conditions is considered in a bounded domain

$$\\Omega \\subset {\\mathbb {R}}^n$$

\r\n \ \r\n Ω\r\n ⊂\r\n \ \r\n \r\n R\r\n \ \r\n n\r\n \ \r\n \r\n ,

$$n \\ge 3$$

\r\n \r\n \ n\r\n ≥\r\n \ 3\r\n \r\n , with smooth boundary for sufficiently regular functions D and S satisfying

$$D>0$$

\r\n \r\n \ D\r\n >\r\n \ 0\r\n \r\n on

$$[0,\\infty )$$

\r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n ,

$$S>0$$

\r\n \r\n \ S\r\n >\r\n \ 0\r\n \r\n on

$$(0,\\infty )$$

\r\n \r\n \ (\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n and

$$S(0)=0$$

\r\n \r\n \ S\r\n (\r\n \ 0\r\n )\r\n \ =\r\n 0\r\n \ \r\n . On the one hand, it is shown that if

$$\\frac{S}{D}$$

\r\n \r\n \ S\r\n D\r\n \ \r\n satisfies the subcritical growth condition

$$\\begin{aligned} \\frac{S(s)}{D(s)} \\le C s^\\alpha \\qquad \\text{ for } \\text{ all } s\\ge 1 \\qquad \\text{ with } \\text{ some } \\alpha < \\frac{2}{n} \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n S\r\n \ (\r\n s\r\n \ )\r\n \r\n \ \r\n D\r\n \ (\r\n s\r\n \ )\r\n \r\n \ \r\n ≤\r\n \ C\r\n \r\n \ s\r\n α\r\n \ \r\n \r\n \r\n for\r\n \ \r\n \r\n all\r\n \r\n s\r\n ≥\r\n \ 1\r\n \r\n \r\n with\r\n \ \r\n \r\n some\r\n \r\n α\r\n <\r\n \ \r\n 2\r\n \ n\r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ and

$$C>0$$

\r\n \r\n \ C\r\n >\r\n \ 0\r\n \r\n , then for any sufficiently regular initial data there exists a global weak energy solution such that

$${ \\mathrm{{ess}}} \\sup _{t>0} \\Vert u(t) \\Vert _{L^p(\\Omega )}<\\infty $$

\r\n \ \r\n ess\r\n \r\n \ sup\r\n \r\n \ t\r\n >\r\n \ 0\r\n \r\n \ \r\n \r\n \r\n \ ‖\r\n u\r\n \ \r\n (\r\n \ t\r\n )\r\n \ \r\n ‖\r\n \ \r\n \r\n \r\n \ L\r\n p\r\n \ \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n <\r\n \ ∞\r\n \r\n for some

$$p > \\frac{2n}{n+2}$$

\r\n \r\n \ p\r\n >\r\n \ \r\n \r\n 2\r\n \ n\r\n \r\n \ \r\n n\r\n \ +\r\n 2\r\n \ \r\n \r\n \r\n \ . On the other hand, if

$$\\frac{S}{D}$$

\r\n \r\n \ S\r\n D\r\n \ \r\n satisfies the supercritical growth condition

$$\\begin{aligned} \\frac{S(s)}{D(s)} \\ge c s^\\alpha \\qquad \\text{ for } \\text{ all } s\\ge 1 \\qquad \\text{ with } \\text{ some } \\alpha > \\frac{2}{n} \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n S\r\n \ (\r\n s\r\n \ )\r\n \r\n \ \r\n D\r\n \ (\r\n s\r\n \ )\r\n \r\n \ \r\n ≥\r\n \ c\r\n \r\n \ s\r\n α\r\n \ \r\n \r\n \r\n for\r\n \ \r\n \r\n all\r\n \r\n s\r\n ≥\r\n \ 1\r\n \r\n \r\n with\r\n \ \r\n \r\n some\r\n \r\n α\r\n >\r\n \ \r\n 2\r\n \ n\r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ and

$$c>0$$

\r\n \r\n \ c\r\n >\r\n \ 0\r\n \r\n , then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value

$$\\alpha = \\frac{2}{n}$$

\r\n \ \r\n α\r\n =\r\n \ \r\n 2\r\n \ n\r\n \r\n \ \r\n for

$$n \\ge 3$$

\r\n \r\n \ n\r\n ≥\r\n \ 3\r\n \r\n , without any additional assumption on the behavior of D(s) as

$$s \\rightarrow \\infty $$

\r\n \ \r\n s\r\n →\r\n \ ∞\r\n \r\n , in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type

$$Q(s) = \\exp (-s^\\beta )$$

\r\n \ \r\n Q\r\n \r\n \ (\r\n s\r\n \ )\r\n \r\n \ =\r\n exp\r\n \ \r\n (\r\n \ -\r\n \r\n \ s\r\n β\r\n \ \r\n )\r\n \ \r\n \r\n ,

$$s \\ge 0$$

\r\n \r\n \ s\r\n ≥\r\n \ 0\r\n \r\n , for global solvability the exponent

$$\\beta = \\frac{n-2}{n}$$

\r\n \ \r\n β\r\n =\r\n \ \r\n \r\n n\r\n \ -\r\n 2\r\n \ \r\n n\r\n \ \r\n \r\n is seen to be critical.\r\n" article_number: '26' author: - first_name: Christian full_name: Stinner, Christian last_name: Stinner - first_name: Michael full_name: Winkler, Michael last_name: Winkler citation: ama: Stinner C, Winkler M. A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations. 2024;24(2). doi:10.1007/s00028-024-00954-x apa: Stinner, C., & Winkler, M. (2024). A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations, 24(2), Article 26. https://doi.org/10.1007/s00028-024-00954-x bibtex: '@article{Stinner_Winkler_2024, title={A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects}, volume={24}, DOI={10.1007/s00028-024-00954-x}, number={226}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Stinner, Christian and Winkler, Michael}, year={2024} }' chicago: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations 24, no. 2 (2024). https://doi.org/10.1007/s00028-024-00954-x. ieee: 'C. Stinner and M. Winkler, “A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects,” Journal of Evolution Equations, vol. 24, no. 2, Art. no. 26, 2024, doi: 10.1007/s00028-024-00954-x.' mla: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations, vol. 24, no. 2, 26, Springer Science and Business Media LLC, 2024, doi:10.1007/s00028-024-00954-x. short: C. Stinner, M. Winkler, Journal of Evolution Equations 24 (2024). date_created: 2024-04-07T12:29:25Z date_updated: 2024-04-07T12:36:21Z doi: 10.1007/s00028-024-00954-x intvolume: ' 24' issue: '2' keyword: - Mathematics (miscellaneous) language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects type: journal_article user_id: '31496' volume: 24 year: '2024' ... --- _id: '53542' abstract: - lang: eng text: "AbstractThis work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for

$$L^1$$

\r\n \r\n \ L\r\n 1\r\n \ \r\n initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as

$$L^1$$

\r\n \r\n \ L\r\n 1\r\n \ \r\n asymptotic convergence without the assumption of bi-K-invariance." article_number: '34' author: - first_name: Efthymia full_name: Papageorgiou, Efthymia id: '100325' last_name: Papageorgiou citation: ama: Papageorgiou E. Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces. Journal of Evolution Equations. 2024;24(2). doi:10.1007/s00028-024-00959-6 apa: Papageorgiou, E. (2024). Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces. Journal of Evolution Equations, 24(2), Article 34. https://doi.org/10.1007/s00028-024-00959-6 bibtex: '@article{Papageorgiou_2024, title={Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces}, volume={24}, DOI={10.1007/s00028-024-00959-6}, number={234}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Papageorgiou, Efthymia}, year={2024} }' chicago: Papageorgiou, Efthymia. “Asymptotic Behavior of Solutions to the Extension Problem for the Fractional Laplacian on Noncompact Symmetric Spaces.” Journal of Evolution Equations 24, no. 2 (2024). https://doi.org/10.1007/s00028-024-00959-6. ieee: 'E. Papageorgiou, “Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces,” Journal of Evolution Equations, vol. 24, no. 2, Art. no. 34, 2024, doi: 10.1007/s00028-024-00959-6.' mla: Papageorgiou, Efthymia. “Asymptotic Behavior of Solutions to the Extension Problem for the Fractional Laplacian on Noncompact Symmetric Spaces.” Journal of Evolution Equations, vol. 24, no. 2, 34, Springer Science and Business Media LLC, 2024, doi:10.1007/s00028-024-00959-6. short: E. Papageorgiou, Journal of Evolution Equations 24 (2024). date_created: 2024-04-17T13:18:30Z date_updated: 2024-04-17T13:20:29Z department: - _id: '555' doi: 10.1007/s00028-024-00959-6 intvolume: ' 24' issue: '2' keyword: - Mathematics (miscellaneous) language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces type: journal_article user_id: '100325' volume: 24 year: '2024' ... --- _id: '63257' abstract: - lang: eng text: AbstractThe quasilinear Keller–Segel system

$$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$

ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u,endowed with homogeneous Neumann boundary conditions is considered in a bounded domain

$$\Omega \subset {\mathbb {R}}^n$$

Ω⊂Rn,

$$n \ge 3$$

n≥3, with smooth boundary for sufficiently regular functionsDandSsatisfying

$$D>0$$

D>0on

$$[0,\infty )$$

[0,∞),

$$S>0$$

S>0on

$$(0,\infty )$$

(0,∞)and

$$S(0)=0$$

S(0)=0. On the one hand, it is shown that if

$$\frac{S}{D}$$

SDsatisfies the subcritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$

S(s)D(s)≤Csαforalls≥1withsomeα<2nand

$$C>0$$

C>0, then for any sufficiently regular initial data there exists a global weak energy solution such that

$${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$

esssupt>0‖u(t)‖Lp(Ω)<∞for some

$$p > \frac{2n}{n+2}$$

p>2nn+2. On the other hand, if

$$\frac{S}{D}$$

SDsatisfies the supercritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$

S(s)D(s)≥csαforalls≥1withsomeα>2nand

$$c>0$$

c>0, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value

$$\alpha = \frac{2}{n}$$

α=2nfor

$$n \ge 3$$

n≥3, without any additional assumption on the behavior ofD(s) as

$$s \rightarrow \infty $$

s→∞, in particular without requiring any algebraic lower bound forD. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type

$$Q(s) = \exp (-s^\beta )$$

Q(s)=exp(-sβ),

$$s \ge 0$$

s≥0, for global solvability the exponent

$$\beta = \frac{n-2}{n}$$

β=n-2nis seen to be critical. article_number: '26' author: - first_name: Christian full_name: Stinner, Christian last_name: Stinner - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Stinner C, Winkler M. A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations. 2024;24(2). doi:10.1007/s00028-024-00954-x apa: Stinner, C., & Winkler, M. (2024). A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations, 24(2), Article 26. https://doi.org/10.1007/s00028-024-00954-x bibtex: '@article{Stinner_Winkler_2024, title={A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects}, volume={24}, DOI={10.1007/s00028-024-00954-x}, number={226}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Stinner, Christian and Winkler, Michael}, year={2024} }' chicago: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations 24, no. 2 (2024). https://doi.org/10.1007/s00028-024-00954-x. ieee: 'C. Stinner and M. Winkler, “A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects,” Journal of Evolution Equations, vol. 24, no. 2, Art. no. 26, 2024, doi: 10.1007/s00028-024-00954-x.' mla: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations, vol. 24, no. 2, 26, Springer Science and Business Media LLC, 2024, doi:10.1007/s00028-024-00954-x. short: C. Stinner, M. Winkler, Journal of Evolution Equations 24 (2024). date_created: 2025-12-18T19:06:36Z date_updated: 2025-12-18T20:14:21Z doi: 10.1007/s00028-024-00954-x intvolume: ' 24' issue: '2' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects type: journal_article user_id: '31496' volume: 24 year: '2024' ... --- _id: '63295' abstract: - lang: eng text: AbstractWe introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate. article_number: '14' author: - first_name: Johannes full_name: Lankeit, Johannes last_name: Lankeit - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Lankeit J, Winkler M. Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations. 2022;22(1). doi:10.1007/s00028-022-00768-9 apa: Lankeit, J., & Winkler, M. (2022). Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations, 22(1), Article 14. https://doi.org/10.1007/s00028-022-00768-9 bibtex: '@article{Lankeit_Winkler_2022, title={Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects}, volume={22}, DOI={10.1007/s00028-022-00768-9}, number={114}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2022} }' chicago: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations 22, no. 1 (2022). https://doi.org/10.1007/s00028-022-00768-9. ieee: 'J. Lankeit and M. Winkler, “Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects,” Journal of Evolution Equations, vol. 22, no. 1, Art. no. 14, 2022, doi: 10.1007/s00028-022-00768-9.' mla: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations, vol. 22, no. 1, 14, Springer Science and Business Media LLC, 2022, doi:10.1007/s00028-022-00768-9. short: J. Lankeit, M. Winkler, Journal of Evolution Equations 22 (2022). date_created: 2025-12-18T19:22:46Z date_updated: 2025-12-18T20:08:35Z doi: 10.1007/s00028-022-00768-9 intvolume: ' 22' issue: '1' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects type: journal_article user_id: '31496' volume: 22 year: '2022' ... --- _id: '63381' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components. Journal of Evolution Equations. 2018;18(3):1267-1289. doi:10.1007/s00028-018-0440-8 apa: Winkler, M. (2018). Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components. Journal of Evolution Equations, 18(3), 1267–1289. https://doi.org/10.1007/s00028-018-0440-8 bibtex: '@article{Winkler_2018, title={Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components}, volume={18}, DOI={10.1007/s00028-018-0440-8}, number={3}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1267–1289} }' chicago: 'Winkler, Michael. “Global Mass-Preserving Solutions in a Two-Dimensional Chemotaxis-Stokes System with Rotational Flux Components.” Journal of Evolution Equations 18, no. 3 (2018): 1267–89. https://doi.org/10.1007/s00028-018-0440-8.' ieee: 'M. Winkler, “Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components,” Journal of Evolution Equations, vol. 18, no. 3, pp. 1267–1289, 2018, doi: 10.1007/s00028-018-0440-8.' mla: Winkler, Michael. “Global Mass-Preserving Solutions in a Two-Dimensional Chemotaxis-Stokes System with Rotational Flux Components.” Journal of Evolution Equations, vol. 18, no. 3, Springer Science and Business Media LLC, 2018, pp. 1267–89, doi:10.1007/s00028-018-0440-8. short: M. Winkler, Journal of Evolution Equations 18 (2018) 1267–1289. date_created: 2025-12-19T11:08:43Z date_updated: 2025-12-19T11:08:50Z doi: 10.1007/s00028-018-0440-8 intvolume: ' 18' issue: '3' language: - iso: eng page: 1267-1289 publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components type: journal_article user_id: '31496' volume: 18 year: '2018' ... --- _id: '34665' author: - first_name: Tobias full_name: Black, Tobias id: '23686' last_name: Black orcid: 0000-0001-9963-0800 - first_name: Johannes full_name: Lankeit, Johannes last_name: Lankeit - first_name: Masaaki full_name: Mizukami, Masaaki last_name: Mizukami citation: ama: Black T, Lankeit J, Mizukami M. Singular sensitivity in a Keller–Segel-fluid system. Journal of Evolution Equations. 2017;18(2):561-581. doi:10.1007/s00028-017-0411-5 apa: Black, T., Lankeit, J., & Mizukami, M. (2017). Singular sensitivity in a Keller–Segel-fluid system. Journal of Evolution Equations, 18(2), 561–581. https://doi.org/10.1007/s00028-017-0411-5 bibtex: '@article{Black_Lankeit_Mizukami_2017, title={Singular sensitivity in a Keller–Segel-fluid system}, volume={18}, DOI={10.1007/s00028-017-0411-5}, number={2}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2017}, pages={561–581} }' chicago: 'Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “Singular Sensitivity in a Keller–Segel-Fluid System.” Journal of Evolution Equations 18, no. 2 (2017): 561–81. https://doi.org/10.1007/s00028-017-0411-5.' ieee: 'T. Black, J. Lankeit, and M. Mizukami, “Singular sensitivity in a Keller–Segel-fluid system,” Journal of Evolution Equations, vol. 18, no. 2, pp. 561–581, 2017, doi: 10.1007/s00028-017-0411-5.' mla: Black, Tobias, et al. “Singular Sensitivity in a Keller–Segel-Fluid System.” Journal of Evolution Equations, vol. 18, no. 2, Springer Science and Business Media LLC, 2017, pp. 561–81, doi:10.1007/s00028-017-0411-5. short: T. Black, J. Lankeit, M. Mizukami, Journal of Evolution Equations 18 (2017) 561–581. date_created: 2022-12-21T09:47:13Z date_updated: 2022-12-21T10:05:25Z department: - _id: '34' - _id: '10' - _id: '90' doi: 10.1007/s00028-017-0411-5 intvolume: ' 18' issue: '2' keyword: - Mathematics (miscellaneous) language: - iso: eng page: 561-581 publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Singular sensitivity in a Keller–Segel-fluid system type: journal_article user_id: '23686' volume: 18 year: '2017' ...