On global existence and blowup of solutions of stochastic Keller-Segel type equation. (English) Zbl 1479.35146

Summary: In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.


35B44 Blow-up in context of PDEs
35K15 Initial value problems for second-order parabolic equations
35K59 Quasilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
92C17 Cell movement (chemotaxis, etc.)
Full Text: DOI arXiv


[1] Biler, P., Mathematical challenges in the theory of chemotaxis, Ann. Math. Sil., 32, 1, 43-63 (2018) · Zbl 1403.35300
[2] Biler, P.; Karch, G., Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10, 2, 247-262 (2010) · Zbl 1239.35177
[3] Blanchet, A.; Carrillo, JA; Masmoudi, N., Infinite time aggregation for the critical Patlak-Keller-Segel model in \(R^2\), Commun. Pure Appl. Math., 61, 10, 1449-1481 (2008) · Zbl 1155.35100
[4] Blanchet, A.; Dolbeault, J.; Perthame, B., Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differ. Equ., 44, 1-3 (2006) · Zbl 1112.35023
[5] Carrillo, J.; Choi, Y-P; Hauray, M., The derivation of swarming models: mean-field limit and Wasserstein distances. Collective dynamics from bacteria to crowds: an excursion through modeling, analysis and simulation series, CISM Int. Centre Mech. Sci., 553, 1-46 (2014)
[6] Coghi, M.; Flandoli, F., Propagation of chaos for interacting particles subject to environmental noise, Ann. Appl. Probab., 26, 3, 1407-1442 (2016) · Zbl 1345.60113
[7] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0761.60052
[8] Da Prato, G.; Zabczyk, J., Ergodicity for Infinite-Dimensional Systems (1996), Cambridge: Cambridge University Press, Cambridge · Zbl 0849.60052
[9] Debussche, A.; Glatt-Holtz, N.; Temam, R., Local martingale and pathwise solutions for an abstract fluids model, Physica D, 240, 14-15, 1123-1144 (2011) · Zbl 1230.60065
[10] Flandoli, F.; Galeati, L.; Luo, D., Delayed blow-up by transport noise, Commun. Partial. Differ. Equ., 46, 9, 1757-1788 (2021) · Zbl 1477.60096
[11] Hieber, M.; Misiats, O.; Stanzhytskyi, O., On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst., 40, 11, 6159-6177 (2020) · Zbl 1446.35273
[12] Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1-2), 183-217 (2009). doi:10.1007/s00285-008-0201-3 · Zbl 1161.92003
[13] Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105, 3, 103-165 (2003) · Zbl 1071.35001
[14] Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106, 2, 51-69 (2004) · Zbl 1072.35007
[15] Huang, H.; Qiu, J., The microscopic derivation and well-posedness of the stochastic Keller-Segel equation, J. Nonlinear Sci., 31, 1, 6-31 (2021) · Zbl 1464.35363
[16] Ikeda, N.; Watanabe, S., A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14, 619-633 (1977) · Zbl 0376.60065
[17] Jabin, P.-E., Wang, Z.: Mean field limit for stochastic particle systems. In: Active particles, vol. 1. Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., pp. 379-402. Springer, Cham (2017)
[18] Jäger, W.; Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329, 2, 819-824 (1992) · Zbl 0746.35002
[19] Keller, E.; Segel, L., Initiation of slide mold aggregation viewed as an instability, J. Theoret. Biol., 26, 399-415 (1970) · Zbl 1170.92306
[20] Krylov, N.: An Analytic Approach to Spdes. Stochastic Partial Differential Equations: Six Perspectives, vol. 64. AMS Mathematical Surveys and Monographs (1999) · Zbl 0933.60073
[21] Krylov, N.: Ito’s formula for the lp-norm of stochastic \(w_1^p\) -valued processes. Probab. Theory Related Fields 147(3), 583-605 (2010) · Zbl 1232.60050
[22] Li, D.; Rodrigo, JL; Zhang, X., Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoam., 26, 1, 295-332 (2010) · Zbl 1195.35182
[23] Misiats, O.; Stanzhytskyi, O.; Yip, N., Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theor. Probab., 29, 3, 996-1026 (2016) · Zbl 1360.60126
[24] Misiats, O.; Stanzhytskyi, O.; Yip, NK, Asymptotic analysis and homogenization of invariant measures, Stoch. Dyn., 19, 2, 1950015 (2019) · Zbl 1415.60053
[25] Misiats, O.; Stanzhytskyi, O.; Yip, NK, Invariant measures for stochastic reaction-diffusion equations with weakly dissipative nonlinearities, Stochastics, 92, 8, 1197-1222 (2020)
[26] Rosenzweig, M., Staffilani, G.: Global solutions of aggregation equations and other flows with random diffusion. arXiv preprint. arXiv:2109.09892 (2021)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.