{"citation":{"ama":"Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. <i>International Mathematics Research Notices</i>. 2022;2023(19):16336-16393. doi:<a href=\"https://doi.org/10.1093/imrn/rnac286\">10.1093/imrn/rnac286</a>","ieee":"M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System,” <i>International Mathematics Research Notices</i>, vol. 2023, no. 19, pp. 16336–16393, 2022, doi: <a href=\"https://doi.org/10.1093/imrn/rnac286\">10.1093/imrn/rnac286</a>.","bibtex":"@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={<a href=\"https://doi.org/10.1093/imrn/rnac286\">10.1093/imrn/rnac286</a>}, number={19}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393} }","short":"M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393.","apa":"Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. <i>International Mathematics Research Notices</i>, <i>2023</i>(19), 16336–16393. <a href=\"https://doi.org/10.1093/imrn/rnac286\">https://doi.org/10.1093/imrn/rnac286</a>","chicago":"Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” <i>International Mathematics Research Notices</i> 2023, no. 19 (2022): 16336–93. <a href=\"https://doi.org/10.1093/imrn/rnac286\">https://doi.org/10.1093/imrn/rnac286</a>.","mla":"Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” <i>International Mathematics Research Notices</i>, vol. 2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:<a href=\"https://doi.org/10.1093/imrn/rnac286\">10.1093/imrn/rnac286</a>."},"keyword":["General Mathematics"],"type":"journal_article","publisher":"Oxford University Press (OUP)","volume":2023,"publication_identifier":{"issn":["1073-7928","1687-0247"]},"user_id":"31496","author":[{"last_name":"Winkler","full_name":"Winkler, Michael","first_name":"Michael"}],"year":"2022","status":"public","_id":"53319","publication_status":"published","doi":"10.1093/imrn/rnac286","date_created":"2024-04-07T12:33:44Z","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n               <jats:p>The Neumann problem for (0.1)$$ \\begin{align}&amp; V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where $n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega \\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big| \\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}&amp; \\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}} L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau&amp;gt;0$, and $T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to \\infty } \\frac {L(s^{\\lambda _0})}{L(s)} &amp;lt;\\infty $ for some $\\lambda _0&amp;gt;1$. In the particular case when $p=n\\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$ whenever $b&amp;gt;0$ and $\\alpha&amp;gt;\\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) - \\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb {R}}$, $0&amp;lt;D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu&amp;gt;0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.</jats:p>"}],"publication":"International Mathematics Research Notices","title":"A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System","date_updated":"2024-04-07T12:36:06Z","page":"16336-16393","issue":"19","intvolume":"      2023","language":[{"iso":"eng"}]}