Parabolic problems whose Fujita critical exponent is not given by scaling

Fino A.Z. Torebek B.T.
April 2026 Springer Science and Business Media Deutschland GmbH

Calculus of Variations and Partial Differential Equations
2026 #65 Issue 4

This paper investigates the (fractional) heat equation with a nonlocal nonlinearity involving a Riesz potential: (Formula presented.) where α∈(0,n), β∈(0,2], n≥1, p>1. We introduce the Fujita-type critical exponent pFuj(n,β,α)=1+(β+α)/(n-α), which characterizes the global behavior of solutions: global existence for small initial data when p>pFuj(n,β,α), and finite-time blow-up when p≤pFuj(n,β,α). It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields psc=1+(β+α)/n, but instead arises in an unconventional manner, similar to the results of Cazenave et al. [Nonlinear Analysis, 68 (2008), 862-874] for the heat equation with a nonlocal nonlinearity of the form ∫0t(t-s)-γ|u(s)|p-1u(s)ds,0≤γ<1. The result on global existence for p>pFuj(n,2,α), provides a positive answer to the hypothesis proposed by Mitidieri and Pohozaev in [Proc. Steklov Inst. Math., 248 (2005) 164???185]. 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∈Lloc1, 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.

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College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait
Institute of Mathematics and Mathematical Modeling, 28 Shevchenko str., Almaty, 050010, Kazakhstan

College of Engineering and Technology
Institute of Mathematics and Mathematical Modeling

10 лет помогаем публиковать статьи Международный издатель

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