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Abstract. This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents q 1 , q 2 ∈ (0, +∞) with q 1 < q 2 . In other words, when q belongs to different intervals (0, q 1 ), (q 1 , q 2 ), (q 2 , +∞), the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval (0, q 2 ]. However, when q ∈ (q 2 , +∞), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval (q 1 , +∞), while for q ∈ (0, q 1 ), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = q 1 is concerned, the other parameter λ will play an important role. In other words, when λ belongs to different interval (0, λ 1 ) or (λ 1 , +∞), where λ 1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

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In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operatorwhere [u] s is the Gagliardo seminorm of u, Ω ⊂ R N is a bounded domain with Lipschitz boundary, 0 < s < min{1, N/2}, L K is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator ) is the initial function, and M : [0, +∞) → [0, +∞) is a continuous function and there exist two constants m 0 > 0 and θ > 1 such thatAs is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156-70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228-50), the main results of Xiang et al (2018 Nonlinearity 31 3228-50) are as follows: • The local existence of nontrivial, nonnegative weak solution for 1 < θ < 2 * s /2, where 2 * s := 2N/(N − 2s).