Fujita - type theorems for a quasilinear parabolic di ﬀ erential inequality with weighted nonlocal source term

: This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic di ﬀ erential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic ﬁ ltration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the ﬁ rst and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and max - imum principles or assumptions on symmetry or behavior at in ﬁ nity of the solutions are required


Introduction
We consider a quasilinear parabolic differential inequality with weighted nonlocal source , , , and the non- The differential operator L A could be both the weighted polytropic filtration operator and the generalized mean curvature operator, which are related to filtration theory and differential geometry etc., see [24,26]. The positive weights ( ) a x and ( ) K x are measurable functions and satisfy which may be singular or degenerate at the origin. Note that the differential inequality (1.1) is called nonlocal differential inequality in the sense that it is not defined pointwise. Meanwhile, norm type nonlocal terms appear in population dynamics and theory of biological populations, see [7,22]. We are interested in finding sufficient conditions for the nonexistence of nonnegative nontrivial weak solutions for (1.1), especially the results related to critical Fujita exponents. The pioneering article in this subject was by Fujita [6], where he considered the Cauchy problem for semilinear parabolic equation − = u u u Δ t q and obtained the critical exponent = + / q N 1 2 F nonnegative nontrivial solutions. Precisely, nonexistence of solutions, that is blow-up, holds when < < + / q N 1 1 2 , while blow-up can occur when > + / q N 1 2 depending on the size of u 0 . Since then, there have been a number of extensions of Fujita results in many directions. Note that, Hayakawa [10] and Weissler [25] proved that the critical case = q q F belongs to the blow-up case. In fact, for the Cauchy problem of parabolic equations with pure power like nonlinearity, there have been extensive literature and results on the critical (Fujita) exponents for the existence and nonexistence of the solutions, see monographs [24,26] and survey papers [3,8,14].
Motivated by previous results of Mitidieri and Pohozaev in [17,18] and Kartsatos and Kurta in [12], we first study the Fujita-type nonexistence of nonnegative nontrivial weak solutions for a wide class of nonlocal quasilinear parabolic differential inequalities in the framework of the test function. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.
We assume throughout this article that L A is defined as follows: , , is a usual Carathéodory function in × × + N N satisfying weak ellipticity, namely,   [4,5] for studying the ellipticproblem, while the subcase of (1.5) with = = σ m 0 appears in [2], which is known as p-weak coercivity.
Operators satisfying the aforementioned assumptions can be given by Thus, our analysis includes weighted polytropic filtration operator and generalized mean curvature operator. In particular, the aforementioned generalized mean curvature operator satisfies (1.5) for all < ≤ p 1 2. We point out that the presence of various components such as the singular coefficient in the operator on the left-hand side of (1.1) and singular coefficient, weighted norm type nonlocal term, power like source, etc. on the right-hand side makes the study of Fujita-type nonexistence results very delicate. In particular, the derivation of a priori estimates is much more complicated because of the fact that several factors appear.
The main results are stated as follows.
then (1.1) does not admit nontrivial positive weak solutions in As a consequence of our main Theorems 1.1 and 1.2, by using the polytropic filtration type operator in problem (1.1) and rewriting it as , 1, we obtain the following corollary. then (1.10) has no nontrivial positive (or nonnegative) weak solution in If both q and r are not equal to 1, and then (1.10) has no nontrivial positive (or nonnegative) weak solution in   [6,10,14,25] and inequalities [11,12,17] with multipower-like local source terms. When ≠ s 0, taking = α 0 and considering the quasilinear parabolic equations with nonlocal source terms result in Corollary 1.1 covers the critical Fujita exponents classified by the parameter s given in the recent literature [8,[28][29][30], and is in full consistency with that given in [1] classified by q.
The analogous of Corollary 1.1 holds for the mean curvature case, which can be considered morally for the case = p 2 of the aforementioned corollary. Let . Then we obtain the following new corollary for the mean curvature problem (1.11). then (1.11) has no nontrivial positive (or nonnegative) weak solution in   [14] and the corresponding local inequality [17] when > N 2. In conclusion, comparison of the main results on the first critical Fujita exponents obtained earlier with the existing results is presented in Table 1.
On the other hand, it is obvious from [10,25] that the classical critical Fujita exponent q F is not optimal for the Cauchy problem of parabolic equation − △ = u u u t q . Thus, to identify the global and nonglobal solutions in the coexistence region, > q q F becomes really interesting and challenging. We refer the reader to [13] for the pioneering work in this subject. Here, for the nonlocal inequality (1.1), we derive the new second critical exponents corresponding to all the aforementioned Fujita-type results by virtue of the slow decay behavior of the initial data at spatial infinity. then (1.10) has no nontrivial positive (or nonnegative) weak solution belonging to the class We obtain a new second critical exponent  [9,13,[19][20][21]27]. If ≠ s 0 and considering nonlocal problems, we can deduce the second critical exponents for quasilinear parabolic equations with nonlocal source given in [15,16,[28][29][30]. then (1.11) has no nontrivial positive (or nonnegative) weak solution belonging to the class Indeed, the second critical Fujita exponent To sum up, we compare the main results on the second critical Fujita exponents with the existing results and obtain Table 2.
This article is organized as follows. In Section 2, we introduce the preparatory knowledge. In Section 3, we present several technical lemmas and obtain fine a priori estimates. Finally, we present the detailed proofs of main results in Section 4.

Preliminaries
In this section, we introduce some notations, definitions, and the careful selections of test functions.
Throughout this article, we denote C various constants independent of u, which may be different from line to line.
First, we define the weak solution of (1.1).  New Fujita-type theorems for a quasilinear parabolic differential inequality  7 Definition 2.1. For a weak solution of (1.1), we mean a nonnegative function ( ) u x t , defined on S, given by those functions such that for any nonnegative test function Obviously, when the equal sign holds in (1.1), we can consider test functions ∈ φ C 0 1 not necessarily nonnegative.
When necessary, we make use of the following weak formulation of (1.1) for any nonnegative test function Next, we construct the test function in the form of separated variables with parameters carefully. Let ( ) B 0 R be the ball of N , centered at the origin and with radius , a nonnegative cutoff function in S, given by .
. Meanwhile, we give the notations for the supports by Then, we give the test functions in two cases.
is a nonnegative weak solution of the differential inequality (1.1) on S, ( ) > ξ ε ε 0 is a standard family of mollifiers, and there is no need to introduce the parameter τ given in case 1, since we deal with positive solutions, namely,

A priori estimates
In this section, we present a priori estimates of the solution needed in the proofs of the main results. To this end, we will give several technical preliminaries in Subsection 3.1 and derive the corresponding a priori estimates in two cases in Subsection 3.2.

Technical lemmas
In this subsection, we prepare some technical preparatory lemmas in order to obtain a priori estimates.
1 and > k 0, the following inequality holds: Proof. Rewrite the integrand in the integral on the left-hand side of (3.1) as such that the exponent of u in the third term on the right-hand side is 0, and σ 2 satisfying It follows that We claim σ , rσ s 1 1 and > σ 1 2 . In fact, by calculating directly, we obtain  By substituting it into the left-hand side of (3.1), we have Then, by Young's inequality with exponents σ 1 , ′ σ 1 , and parameter ε 1 , we derive which completes the proof of Lemma 3.1. □ , then for all < < + − d p m 0 1 and > k 0, the following inequality holds: where Proof. The integrand in the integral on the left-hand side of (3.2) can be rewritten as follows: such that the exponent of u in the third term on the right-hand side is 0, and take σ 4 satisfying It implies that   By inserting the aforementioned formula into the left-hand side of (3.2), we have Then, by applying Young's inequality with exponents σ 3 , ′ σ 3 , and parameter ε 2 to the right-hand side of the last inequality, we obtain  Lemma 3.2 is proved. □ and > k 0, the following inequality holds: Proof. We rewrite the integrand in the integral on the left-hand side of (3.3) as follows: such that the exponent of u in the third term on the right-hand side of the aforementioned formula is 0, and an appropriate σ 6 satisfying + + = σ s rσ σ 1 1 1.

6
It follows that We claim σ , rσ s 5 5 and > σ 1 6 . Indeed, with ≥ r 1, > s 0 and > − p m 1 , simple calculations show that   By substituting it into the left-hand side of (3.3), we have Then, by applying Hölder inequality again with exponents σ 5 and ′ σ 5 to the right-hand side of the aforementioned formula, we derive 1 and > k 0, the following inequality holds: Proof. Rewrite the integrand in the integral on the left-hand side of (3.4) as follows: We take = + σ q s 7 such that the exponent of u in the third term on the right-hand side of the aforementioned formula is 0, and σ 8 satisfying We claim σ , rσ s 7 7 , and > σ 1 8 . In fact, > σ 1 7 holds clearly due to ≥ q 1, ≥ r 1, and > s 0. Meanwhile, we obtain by a simple calculation that Since ≥ q 1, ≥ r 1, > s 0, and both q and r do not equal to 1, we obtain > 1 rσ s 7 and > σ 1 8 obviously. Thus, by Hölder inequality with exponents σ , rσ s 7 7 and σ 8 , we obtain By substituting it into the left-hand side of (3.4), we have Then, by applying again Hölder inequality with exponents σ 7 and ′ σ 7 to the right-hand side of the aforementioned formula, we derive Lemma 3.4 is proved. □

A priori estimates
In this subsection, by using the preparatory lemmas obtained in Subsection 3.1, we give fine a priori estimates for the two cases = m 0 and ≠ m 0, namely, Propositions 3.1 and 3.3. Also, we obtain the additional a priori estimates, Propositions 3.2 and 3.4, to study the nonexistence of the solutions to problem (1.1) in the critical case. They play a crucial role in the proofs of our main results.
clearly and can be used as a test function in the weak formulation of (1.1), given by (2.1), so that   Since when > R R 0 , by combining with (3.9), we obtain Then, we estimate the last two terms by Lemmas 3.1 and 3.2 with = m 0 and derive  which gives, since

(3.11)
We next estimate the last two integrals in (3.11). The exponents of ζ in the two integrals are both Under some extra assumptions on the parameters, we obtain a refined a priori estimate for the critical case when = m 0.
Proof. By choosing ( ) We take > > R R 0 0 and use (1.5) to obtain to the last term in the aforementioned formula, we obtain   Since p q s q s p d q s q s 1 1 1 , we can estimate the second and fourth terms on the right-hand side of the aforementioned inequality in a similar approach as in Proposition 3.1 for (3.11) and obtain ( ) and inequality (3.12) is proved. □ Similar to Proposition 3.1, we derive a priori estimate of the positive solution to the differential inequality (1.1).
where δ 1 is the same as in Proposition 3.1 and Proof. The proof is analogous to that of Proposition 3.1 where we deal with positive solutions and take the test function as ( ) = − φ x t u ξ , ε d k since ≠ m 0. Repeat the proof of Proposition 3.1, we obtain, in place of (3.10)    and Proof. The proof here is similar to that of Proposition 3.2, but now we deal with the positive solution since ≠ m 0 so that (3.13) is replaced by By substituting (3.15) into the second term on the right-hand side and applying Lemmas 3.4 and 3.3 with ≠ m 0 for the first and third terms, we derive

Proofs of main theorems
In this section, we show the process of the proofs for the main Theorems 1.1-1.4 in detail.

First critical Fujita exponent
This subsection is devoted to prove Theorems 1.1 and 1.2 on the first critical Fujita exponents.
To make it easier to discuss the exponents of R in (4.1), we rewrite δ 1 and δ 2 as follows: holds.
In the following, we discuss the exponents of R on the right-hand side of (4.1) in two cases. By using Proposition 3.2, we obtain another a priori estimate, namely, (3.12) Taking , a simple calculation yields 1 0 a n d 0 . . Next, we discuss the exponents of R on the right-hand side of (4.3) in two cases.

(4.6)
One can see that the expression on the right-hand side in (4.6) arrives its minimum at

(4.8)
The expression on the right-hand side in (4.8) arrives its minimum at By (1.14), let → +∞ R , the contradiction can be derived. So (1.1) has no nontrivial positive weak solution belonging to S d . Theorem 1.4 is proved. □ Acknowledgment: The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
Funding information: The work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2019MA072).
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.

Conflict of interest:
The authors declare that there is no conflict of interests regarding the publication of this article.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.