A New Blow-Up Criterion to a Singular Non-Newton Polytropic Filtration Equation

: In this paper, a singular non-Newton polytropic ﬁltration equation under the initial-boundary value condition is revisited. The ﬁnite time blow-up results were discussed when the initial energy E ( u 0 ) was subcritical ( E ( u 0 ) < d ), critical ( E ( u 0 ) = d ), and supercritical ( E ( u 0 ) > d ), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a ﬁnite time blow-up result if E ( u 0 ) is independent of d . Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.


Introduction
In this paper, we are concerned with the following initial-boundary value problem: |x| −s u t − div(|∇u m | p−2 ∇u m ) = u q in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u 0 (x) where the initial value u 0 (x) is a nonnegative and nontrivial function, T ∈ (0, ∞] is the maximal existence time of solutions, Ω ⊂ R N (N > p) is a bounded domain with smooth boundary ∂Ω, , and the parameters satisfy Problem (1) in fact has its physical background.To be more precise, the volumetric moisture content θ(x), the macroscopic velocity − → V , and the density of the fluid u, under the assumption that a compressible fluid flows in a homogeneous isotropic rigid porous medium, are governed by the following equation [1]: where f (u) is the source.For the non-Newtonian fluid, provided that the fluid investigated is the polytropic gas, one obtains where c > 0, m > 0, λ > 0, p ≥ 2. Let θ(x) = |x| −s and f (u) = u q in (3); then, (1) can be deduced.For m = 1 and s = 2, in 2004, Tan in [2] considered the existence and asymptotic estimates of global solutions and the finite time blow-up of local solution based on the classical Hardy inequality [3].Later on, Wang [4] extended the results obtained by Tan to 0 ≤ s ≤ 2, proved the existence of a global solution by the Hardy-Sobolev inequality [5], and found two sufficient conditions for blowing up in finite time by variational methods and classical concave methods.Zhou [6] discussed the global existence and finite time blow-up of solutions to problem (1) by the potential well method and the Hardy-Sobolev inequality when the initial energy is subcritical, i.e., E(u 0 ) < d.For E(u 0 ) ≥ d, Xu and Zhou [7] discussed the behaviors of the solution by using the potential well method and some differential inequality techniques.Their results in fact extended previous one obtained by Hao and Zhou [8], where some blow-up conditions with E(u 0 ) ≥ d were obtained for m = 1 and p = 2 in problem (1).
The results above derived are not independent of the potential depth d.Naturally, we aim to present a new blow-up criterion when the initial energy is independent of d.In this paper, with the help of the Hardy-Sobolev inequality, we give a new blow-up result.Moreover, the upper and lower bounds of the blow-up time are derived.Our results extend the previous works in [9] and complement the results in [6,7].

Preliminaries
Throughout this paper, we denote by • p and ∇(•) p the norm on L p (Ω) and W 1,p 0 (Ω), respectively.Additionally, (•, •) represents the inner product in L 2 (Ω).In order to present our main results, let us begin by introducing some definitions, notations, and lemmas obtained in [6,7].
It is well known that problem (1) is degenerate if p > 2 at points where ∇u m = 0, and therefore there is no classical solution in general.For this, we state the definition of the weak solution.
Definition 1. (Weak solution) A nonnegative function u := u(x, t) satisfying u(x, 0) = u 0 (x) is called a weak solution of problem (1) dt < +∞, and u satisfies problem (1) in the distribution sense, that is (Finite time blow-up) Let u be a weak solution of problem (1) on Ω × (0, T).We say that u blows up at some finite time T if u exists for all t ∈ [0, T) and Therefore, for any u ∈ Q, define the energy functional and Nehari functional by Define the potential depth by where Nehari manifold and M is the optimal constant of the Sobolev embedding W In fact, M depends only on Ω, m, N, p, and q such that for all u ∈ Q it holds Lemma 1.Let u be a weak solution of problem (1); then, the energy functional E(u(t)) is nonincreasing with respect to t.Moreover, The following lemma is a descendant of Levine's concavity method [10,11].Further, the technique has also been revisited and presented in the book of Quittner and Souplet [12].
Lemma 2 ([10,11]).Suppose a positive, twice-differentiable function ψ(t) satisfies the inequality In order to prove our main results, we need the following Hardy-Sobolev inequality.

Lemma 3 ([5]). (Hardy-Sobolev
Then, there exists a positive constant C depending on β, n, N, and k such that for any u ∈ W 1,n 0 (R N ), it holds

Main Results and Its Proof
As shown in [6,7], the finite time blow-up results were discussed when the initial energy where d is the potential depth.Our first theorem will show a finite time blow-up result if E(u 0 ) is independent of d and give a upper bound of the blow-up time.
Theorem 1.Let u be a weak solution of problem (1), and let (2) hold.If then u blows up at some finite time T in the sense of Definition 1.Moreover, the upper of the blow-up time is given by Proof.This proof follows some ideas in [9,13].Firstly, we prove that u blows up in finite time.Suppose, on the contrary, that u is global, i.e., T = +∞.For the sake of simplicity, define hereafter Then, for all t ∈ [0, ∞), Hölder's inequality and ( 6) imply where we apply 0 < E(u(t)) ≤ E(u 0 ) if u is a global solution.Here, we prove that 0 < E(u(t)) ≤ E(u 0 ).Otherwise, there exists t 0 ∈ [0, ∞) such that E(t 0 ) ≤ 0.Then, by Remark 1.7 in [6], we know that u blows up in finite time, which is a contradiction.Multiplying the first equation of problem (1) with u m and integrating over Ω, and then recalling the definitions on E(u) and H(u) in ( 4) and ( 5), it follows that On the other hand, applying the Hardy-Sobolev inequality in Lemma 3 yields Noticing that mN+m+1−ms < p.Therefore, let us combine (11) with ( 12) and ( 17) to obtain here then by Lemma 1 and ( 14) Therefore, by ( 15), one has which contradicts (10) for sufficiently large t.Thus, u blows up in finite time.Moreover, ( 14) and ( 16) imply that L(t) is strictly increasing for t ∈ [0, ∞).
Secondly, let us estimate the upper bound of T. For any T * ∈ (0, T), γ > 0, and σ > 0, define an auxiliary function By a direct computation, one has Further, recall ( 14) and ( 6); then, Here, we have assumed that L(t) is strictly increasing for t ∈ [0, ∞).
Applying the Cauchy-Schwarz inequality and Young's inequality, one has Therefore, Since the arbitrariness of T * < T, for any γ ∈ 0, mp−(m+q) mp m+1 C .
Next, we shall derive a lower bound for the blow-up time T by combining the interpolation inequality with the first order differential inequalities.

Conclusions
In this paper, we establish new results on the blow-up in finite time of weak solutions to problem (1).Previous blow-up results were obtained by the potential well method and some differential inequalities, where the potential depth d plays an important role.We give a finite time blow-up result if E(u 0 ) is independent of d and obtain the upper bound and lower bound of the blow-up time by the classical Levine's concavity method and an interpolation inequality.In fact, our results illustrate that the blow-up phenomenon will happen when the initial energy is arbitrarily high.Blow-up rate estimates are also of importance; therefore, considering blow-up rate estimates will be the focus of our next work.