{"title":"A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects","abstract":[{"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","lang":"eng"}],"issue":"2","intvolume":" 24","date_updated":"2024-04-07T12:36:21Z","volume":24,"publication_status":"published","doi":"10.1007/s00028-024-00954-x","date_created":"2024-04-07T12:29:25Z","publication_identifier":{"issn":["1424-3199","1424-3202"]},"author":[{"first_name":"Christian","last_name":"Stinner","full_name":"Stinner, Christian"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"publication":"Journal of Evolution Equations","language":[{"iso":"eng"}],"citation":{"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","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.","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.","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.","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} }","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","short":"C. Stinner, M. Winkler, Journal of Evolution Equations 24 (2024)."},"article_number":"26","keyword":["Mathematics (miscellaneous)"],"type":"journal_article","publisher":"Springer Science and Business Media LLC","status":"public","_id":"53316","user_id":"31496","year":"2024"}