11 марта 2026 года в 14:00 пройдет совместный семинар Института математики имени В.И. Романовского Академии наук Республики Узбекистан, Института прикладной математики и автоматизации КБНЦ Российской академии наук и Analysis & PDE Center of Ghent University (Belgium) «Современные проблемы математической физики».
Докладчик: Professor Berikbol Torebek, Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan.
Название доклада: Parabolic problems whose Fujita critical exponent is not given by scaling.
Краткая аннотация:This work investigates the heat equation with a nonlocal nonlinearity involving a Riesz potential: ut −∆u =Iα(|u|p), x ∈Rn, t>0, where α ∈ (0,n), p > 1. We introduce the Fujita-type critical exponent pFuj(n, α) = 1 + 2+α n−α , which characterizes the global behavior of solutions: global existence for small initial data when p > pFuj(n,α), and finite-time blow-up when p ≤ pFuj(n,α). It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields psc = 1+(2+α)/n, but instead arises in an unconventional manner, similar to the results of Cazenave et al. [1] for the heat equation with a nonlocal nonlinearity of the form t 0 (t − s)−γ|u(s)|p−1u(s)ds, 0 ≤ γ < 1. The result on global existence for p > pFuj(n,α), provides a positive answer to the hypothesis proposed by Mitidieri and Pohozaev in [2]. We further establish global nonexistence results for the above heat equation, where the Riesz potential term Iα(|u|p) is replaced by a more general convolution operator (K ∗ |u|p), K ∈ L1 loc, thereby extending the Mitidieri–Pohozaev’s results established in the aforementioned work. Proofs of the blow-up results are obtained using a nonlinear capacity method specifically adapted to the structure of the problem, while global existence is established via a fixed point argument combined with the Hardy–Littlewood–Sobolev inequality. More detailed evidence can be found in the published full-text version [3].
References
- T. Cazenave, F. Dickstein, F. D. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal., 68 (2008), 862–874.
- E. Mitidieri, S. I. Pohozaev, Liouville theorems for some classes of nonlinear nonlocal problems, Proc. Steklov Inst. Math., 248 (2005) 164–185.
- A. Z. Fino, B. T. Torebek, Parabolic problems whose Fujita critical exponent is not given by scaling, Calc. Var. Partial Differential Equations, 65 (2026), 116.
Подключение к конференции Zoom:
https://us06web.zoom.us/j/88273567781?pwd=6TMH3UOjALVQmLsLlm6vTMPo5YpCYr.1
Идентификатор конференции: 882 7356 7781
Код доступа: 151325
Руководители семинара: академик Ш. А. Алимов, профессор А. В. Псху, профессор Р. Р. Ашуров, профессор Э. Каримов.
Секретарь семинара: Р. Т. Зуннунов.