Local and Global Well-Posedness for Fractional Porous Medium Equation in Critical Fourier-Besov Spaces

: In this paper, we study the Cauchy problem for the fractional porous medium equation in R n for n ≥ 2 . By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 < β ≤ 2 , the local solution v ∈ X T := L ∞ T


Introduction
In this article, we investigate the existence of mild solutions for the initial value problem of the following fractional porous medium equation (FPME): where n ≥ 2, v = v(x, t) denotes the density or concentration, v 0 is the initial data, µ > 0 is the dissipative coefficient, κ = ±1, and here for simplify the notation, we take µ = κ = 1.The operator Λ β is the Fourier multiplier with symbol |ξ| β , and p represents the gas pressure which releted to v by an abstract operator; p = P v.When κ = −1 and 0 < m < 1, the system (1.1) was first formulated by Caffarelli and Vázquez [5].Indeed, the system (1.1) is created by adding to the continuity equation where V = ∇p is the velocity, the fractional dissipative term µΛ β v.In that work, they demonstrated that a weak solution exists when v 0 is a bounded function with exponential decay at infinity.Please see literature [20] for more information on the solution of Equation (1.2).When κ = 1, β = 2 and m = 1, Equation (1.1) corresponds to the following classical Keller-Segel equation: which describes a model of chemotaxis.The system (1.3) was introduced by Keller and Segel [11].The well-posedness of the system (1.3) has been studied by several researchers in various spaces, such as Corrias et al. in the Lebesgue space L 1 (R n ) ∪ L n 2 (R n ) [7], Kozono and Sugiyama in the Sobolev space L 1 (R n ) ∪ W 2,2 (R n ) [13], Ogawa and Shimizu in the Hardy space H 1 R 2 [17] and in the Besov space Ḃ0 1,2 R 2 [18], Iwabuchi in the Fourier-Herz Ḃ−2 2 (R n ) [10], for more results, please refer to Lemarié-Rieusset [15] and the references therein.
For the case κ = 1, 1 < β < 2 and m = 1, Equation (1.1) was initially analyzed by Escudero [9].It was utilized to characterize the spatiotemporal patterns exhibited by a population density consisting of individuals that perform Lévy flights.Furthermore, in that paper, it has been established that Equation (1.1) in this case, has global in time solutions.Biler and Karch [2] have established, in the critical Lebesgue space L n β (R n ) , the existence of both local and global solutions of Equation (1.1) with small initial data.Additionally, they have demonstrated the finite-time blowup of non-negative solutions with specific initial data that satisfy high-concentration or large-mass conditions.In the critical Besov spaces Ḃ1−β 2,q R 2 , it has been proved global well-posedness with small initial data of Equation (1.1) by Biler and Wu [3].Zhai [24] has demonstrated the global existence, uniqueness, and stability of solutions with a general potential type nonlinear term in the critical Besov spaces, given that the initial data is sufficiently small.Certain aspects of these results were also extended to the fractional power bipolar type drift-diffusion system.Further information on this topic can be found in [3,19] and the relevant references cited therein.
Inspired by some results presented in [22,25], this article aims to prove the well-posedness results of In addition, we prove them in the limit cases β = 1 and p = ∞.To address the system (1.1), we think about the following integral equations: where e −tΛ β := F −1 e −t|ξ| β F , F and F −1 are the Fourier transform and the inverse Fourier transform, respectively.We can solve (1.4) by applying the contraction mapping argument to the following mapping: Throughout this paper, we use F Ḃs p,r to denote the homogeneous Fourier-Besov spaces, C will represent constants that may differ at different places, A B denotes the existence of a constant C > 0 such that A ≤ B, and p 0 is the conjugate of p ∈ [1, ∞] (i.e., 1  p + 1 p ′ = 1).Our first theorem is as follows: Then we have the following results: Local and Global Well-Posedness for Fractional Porous Medium Equation 31.(For Then there is a T = T (v 0 ) > 0 such that the system (1.1) admits a unique solution v ∈ X T , where Moreover, if T * denotes the maximal existence time of v, is small enough.
Then the system (1.1) admits a unique solution v satisfying

Remark 1.2. The results of this work remain valid if we take the Fourier-Herz space Ḃs
It is further worth noting that, in the special case m = 1, Equation (1.1) becomes the generalized Keller-Segel system.
Corresponding to Theorem 1.1, in the case β = 1, We get the following theorem: Then we have the following results:

Preliminaries
This section introduces some basic knowledge of Littlewood-Paley theory and Fourier-Besov spaces and reviews some lemmas that are pertinent to our purposes.
We start by recall the Littlewood-Paley decomposition (see [1] for more details).Let ϕ ∈ S(R n ) be a smooth radial function such that and we denote ϕ j (ξ) = ϕ(2 −j ξ).Then for every u ∈ S ′ (R n ), we define the frequency localization operators for all j ∈ Z, as follows with F −1 the inverse Fourier transform.Here, we observe that the almost orthogonality property of the Littlewood-Paley decomposition is satisfied, i.e. for any where P is the set of all polynomials on R n .Throughout the paper, the following Bony paraproduct decomposition will be used: With the decomposition stated above, the homogeneous Fourier-Besov space can be defined as follows: Then the homogeneous Fourier-Besov space F Ḃs p,r (R n ) is defined by Definition 2.2.[21] For s ∈ R, 0 < T ≤ ∞ and 1 ≤ p, r, ρ ≤ ∞.We define the mixed time-space We notice that we have F Ḃs 1,r = Ḃs r and F Ḃs 1,1 = χ s , where Ḃs r and χ s are the Fourier-Herz space [6] and the Lei-Lin space [16], respectively.
Due to Minkowski's inequality, we have 1. (Bernstein's inequality) For any multiindex β and 1 ≤ r ≤ p ≤ ∞ the following inequality is valid: (2.4) Lemma 2.4.[8] Let g be a homogeneous smooth function on R n \ {0} of degree m.Then for every s ∈ R and 1 ≤ p, q ≤ ∞, the operator g(D) is continuous from 3. Well-posedness for 1 < β ≤ 2: Proof of Theorem 1.1 In this section, we establish well-posedness of the system (1.1) in critical Fourier-Besov spaces , and 1 ≤ p, r ≤ ∞.

The case p < ∞
We first consider the fractional power dissipative equation, for which we give the following result: Next, we get the following key bilinear estimate.
. There holds Proof.Using the following paraproduct decomposition due to J. M. Bony [4], for fixed j, where So we can write, . We estimate the above three terms one by one.First, applying Young's inequality (2.4), Holder's inequality, Bernstein's inequality (2.3) and Lemma 2.4, when ε > 0, one has Multiplying by 2 sj , and taking l r −norm of both sides in the above estimate, we get Similarly, when 2m + ε − 1 > 0, we prove that Now for the third term, using Young's inequality (2.4), Holder's inequality, Bernstein's inequality (2.3) and Lemma 2.4, one has Multiplying by 2 sj , and taking l r −norm of both sides in the above estimate, when s > 0, we get We can now start to prove the first assertion of Theorem 1.1.For t ∈ [0, T ], we define the following map: in the metric space: Using Proposition 3.1 and Lemma 3.2 by choosing γ = ρ = β 2β−2 , for any v, w ∈ E T , we obtain and Using the standard contraction mapping argument ( [14]) with these two estimates (3.10) and (3.11), we can demonstrate that if T is appropriately small, then Ψ is a contraction mapping from (E T , d) into itself, and here, we omit the details.Hence there is v ∈ E T such that Ψ(v) = v, which is a unique solution of Equation (1.1).Furthermore, Proposition 3.1 has given us, (3.12) ) is small enough, we can directly choose T = ∞ in (3.10) and (3.12), which gives that T * = ∞.i.e., the solution v is global.
Let T 0 ∈ (0, T * ), and for t ∈ [T 0 , T ), we consider the following integer equation As before, we can show that Using again the contraction mapping argument as in (3.10), which yields that the solution exists on [T, T * ].Choosing T sufficiently close to T * , then the solution existing on a time larger than T * , which is a contradiction.This completes the proof of the first assertion of Theorem 1.1.

The case p = ∞
In this subsection, we study the limit case p = ∞.The following is the essential bilinear estimate. .There holds ) .
We consider the resolution space , in order to demonstrate the second assertion of Theorem 1.1.Returning to the mapping (3.9), and according to Proposition 3.1 with γ = 1, Lemma 3.3, we have ) .(3.18) Due to the standard contraction mapping argument as in Subsection is small enough, we can prove that Equation (1.1) has a unique solution v satisfying The proof of Theorem 1.1 is complete.

Well-posedness for β = 1: Proof of Theorem 1.3
In this section, we will establish the global well-posedness for the system (1.1) in the limit cases β = 1, with initial data in critical Fourier-Besov paces

The case p < ∞
In this case, the crucial estimate is the following: . (4.1) Proof.We get the estimates of the terms J j i (i = 1, 2, 3) by making a slight modification to the proof of Lemmas 3.2 and 3.3, as follows: . Thus, we have Similarly, when m > 1 2 , there holds and for the the last term, one has This completes the proof of Lemma 4.1.
We are now in a position to demonstrate the first assertion of Theorem 1.3.We consider the resolution space L ∞ t (F Ḃ1−2m+ n p ′ p,1 (R n )) and returning to the mapping (3.9).Proposition 3.1 (with β = 1, γ = ∞)