On the generalized porous medium equation in Fourier-Besov spaces
aa r X i v : . [ m a t h . A P ] D ec On the generalized porous medium equation in Fourier-Besovspaces
Weiliang Xiao Xuhuan Zhou ∗ School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, China;E-mail: [email protected] of Information Technology, Nanjing Forest Police College, Nanjing, 210023, China;E-mail: [email protected]
Abstract
We study a kind of generalized porous medium equation with fractional Laplacian andabstract pressure term. For a large class of equations corresponding to the form: u t + ν Λ β u = ∇ · ( u ∇ P u ), we get their local well-posedness in Fourier-Besov spaces for large initial data. If theinitial data is small, then the solution becomes global. Furthermore, we prove a blowup criterionfor the solutions.
Keywords porous medium equation, well-posedness, blowup criterion, Fourier-Besov spaces
MR (2010)
In this paper, we study the nonlinear nonlocal equation in R n of the form ( u t + ν Λ α u = ∇ · ( u ∇ P u ); u (0 , x ) = u . (1.1)Usually, u = u ( t, x ) ia a real-valued function, represents a density or concentration. Thedissipative coefficient ν > ν = 0 corresponds to theinviscid case. In this paper we study the viscid case and take ν = 1 for simplicity. Thefractional operator Λ α is defined by Fourier transform as (Λ α u ) ∧ = | ξ | α ˆ u . P is an abstactoperator.Equation (1.1) here comes from the same proceeding with that of the fractional porousmedium equation (FPME) introduced by Caffarelli and V´azquez [5]. In fact, equation (1.1)comes into being by adding the fractional dissipative term ν Λ α u to the continuity equation u t + ∇ · ( uV ) = 0, where the velocity V = −∇ p and the velocity potential or pressure p isrelated to u by an abstract operator p = P u .The absrtact form pressure term
P u gives a good suitability in many cases. The simplestcase comes from a model in groundwater in filtration [1, 19]: u t = △ u , that is: ν = 0 , P u = The research was supported by the NNSF of China under grant Xiao et al. u . A more general case appears in the fractional porous medium equation [5] when ν = 0and P u = Λ − s u, < s <
1. In the critical case when s = 1, it is the mean field equationfirst studied by Lin and Zhang [15]. Some studies on the well-posedness and regularity onthose equations we refer to [4, 6, 7, 17, 18, 20, 23] and the references therein.In the FPME, the pressure can also be represented by Riesz potential as P u = Λ − s u = K ∗ u, with kernel K = c n,s | y | s − n . Replacing the kernel K by other functions in this form: P u = K ∗ u , equation (1.1 ) also appears in granular flow and biological swarming, namedaggregation equation. The typical kernels are the Newton potential | x | γ and the exponentpotential − e −| x | .One of concerned problems on this equation is the singularity of the potential P u whichholds the well-posedness or leads to the blowup solution. Generally, smooth kernels at origin x = 0 lead to the global in time solution [3], meanwhile nonsmoooth kernels may lead toblowup phenomenon [14]. Li and Rodrigo [13, 14] studied the well-posedness and blowupcriterion of equation (1.1) with the pressure P u = K ∗ u , where K ( x ) = e −| x | in Sobolevspaces. Wu and Zhang [21] generalize their work to require ∇K ∈ W , which includesthe case K ( x ) = e −| x | . They take advantage of the controllability in Besov spaces of theconvolution K ∗ u under this condition, as well as the controllability of its gradient ∇K ∗ u .In this article we study the well-posedness and blowup criterion of equation (1.1) inFourier-Besov spaces under an abstract pressure condition k \ △ k ( ∇ P u ) k L p ≤ C kσ k d △ k u k L p . (1.2)In Fourier-Besov spaces, it is the localization express of the norm estimate k∇ P u k F ˙ B sp,q ≤ C k u k F ˙ B s + σp,q . (1.3)Corresponding to the FPME, i.e. P u = Λ − s , we get σ = 1 − s obviously. And if P u = K∗ u , K ∈ W , in the aggregation equation, we get σ = 1 when K ∈ L and σ = 0 when ∇K ∈ L .The Fourier-Besov spaces we use here come from Konieczny and Yoneda [11] when dealwith the Navier-Stokes equation (NSE) with Coriolis force. Besides, Fourier-Besov spaceshave been widely used to study the well-posedness, singularity, self-similar solution, etc. ofFluid Dynamics in various of forms. For instance, the early pseudomeasure spaces P M α in which Cannone and Karch studied the smooth and singular properties of Navier-Stokesequations [8]. The Lei-Lin spaces X σ deal with global solutions to the NSE [12] and tothe quasi-geostrophic equations (QGE) [2]. The Fourier-Herz spaces B σq in the Keller-Segelsystem [9], in the NSE with Coriolis force [10] and in the magneto-hydrodynamic equations(MHD) [16].In the case of Besov spaces, we gain some well-posedness and blow-up results of equation(1.1) under an corresponding condition to (1.3) in [24]. Due to the difficulty in deal withthe nonlinear term ∇ · ( u ∇ P u ), in that case we need a little strict initial condition: u ∈ ˙ B n/p + σ − βp, ∩ ˙ B n/p + σ − β +1 p, . However, in this paper, we find that Fourier-Besov spaces are eneralized PME in Fourier-Besov spaces Theorem 1.1.
Let p, q ∈ [1 , ∞ ] , max { , σ + 1 } < α < n (1 − p ) + σ + 2 . Then for any u ∈ ˙ F B n (1 − p ) − α + σ +1 p,q , equation (1.1) admits a unique mild solution u and u ∈ C (cid:0) [0 , T ); ˙ F B n (1 − p ) − α + σ +1 p,q (cid:1) ∩ ˜ L ([0 , T ); ˙ F B n (1 − p )+ σ +1 p,q ) . Moreover, there exists a constant C = C ( α, p, q ) such that for u satisfying k u k ˙ F B n (1 − p ) − α + σ +1 p,q ≤ C , the solution u is global, and k u k L ∞ T ( ˙ F B n (1 − p ) − α + σ +1 p,q ) + k u k ˜ L T ( ˙ F B n (1 − p )+ σ +1 p,q ) ≤ C k u k ˙ F B n (1 − p ) − α + σ +1 p,q . Theorem 1.2.
Let T ∗ denote the maximal time of existence of u in L ∞ T ( ˙ F B βp,q ) ∩ ˜ L T ( ˙ F B β + αp,q ) .Here β = n (1 − p ) − α + σ + 1 . If T ∗ < ∞ , then k u k ˜ L ([0 ,T ∗ ); ˙ F B β + αp,q ) = ∞ . Let us introduce some basic knowledge on Littlewood-Paley theory and Fourier-Besov spaces.Let ϕ ∈ C ∞ c ( R n ) be a radial positive function such that supp ϕ ⊂ { ξ ∈ R n : 34 ≤ | ξ | ≤ } , X j ∈ Z ϕ (2 − j ξ ) = 1 for any ξ = 0 . Define the frequency localization operators as follows: △ j u = ϕ j ( D ) u ; S j u = ψ j ( D ) u, here ϕ j ( ξ ) = ϕ (2 − j ξ ) and ψ j = P k ≤ j − ϕ j .By Bony’s decomposition we can split the product uv into three parts: uv = T u v + T v u + R ( u, v ) , with T u v = X j S j − u ∆ j v, R ( u, v ) = X j ∆ j u ˜∆ j v, ˜∆ j v = ∆ j − v + ∆ j v + ∆ j +1 v. Let us now define the Fourier-Besov space as follows.
Xiao et al.
Definition 2.1.
For β ∈ R , p, q ∈ [1 , ∞ ] , we define the Fourier-Besov space ˙ F B βp,q as ˙ F B βp,q = { f ∈ S ′ / P : k f k ˙ F B βp,q = (cid:16) X j ∈ Z jβq k ϕ j ˆ f k qL p (cid:17) /q < ∞} . Here the norm changes normally when p = ∞ or q = ∞ , and P is the set of all polynomials. Definition 2.2.
In this paper, we need two kinds of mixed time-space norm defined asfollows: For s ∈ R , ≤ p, q ≤ ∞ , I = [0 , T ) , T ∈ (0 , ∞ ] , and let X be a Banach space withnorm k · k X , k f ( t, x ) k L r ( I ; X ) := (cid:0) Z I k f ( τ, · ) k rX dτ (cid:1) r , k f ( t, x ) k ˜ L r ( I ; ˙ F B βp,q ) := (cid:0) X j ∈ Z jβq k ϕ j ˆ f k qL r ( I ; L p ) (cid:1) q . By Minkowski’ inequality, there holds L r ( I ; F ˙ B sp,q ) ֒ → ˜ L ( I ; F ˙ B sp,q ) , if r ≤ q and ˜ L r ( I ; F ˙ B sp,q ) ֒ → L r ( I ; F ˙ B sp,q ) , if r ≥ q. (2.1) Lemma 2.1.
Let X be a Banach space with norm k · k X and B : X × X X be a boundedbilinear operator satisfying k B ( u, v ) k X ≤ η k u k X k v k X , for all u, v ∈ X and a constant η > . Then for any fixed y ∈ X satisfying k y k X < ǫ < η ,the equation x := y + B ( x, x ) has a solution ¯ x in X such that k ¯ x k X ≤ k y k X . Also, thesolution is unique in ¯ B (0 , ǫ ) . Moreover, the solution depends continuously on y in thesense: if k y ′ k X < ǫ, x ′ = y ′ + B ( x ′ , x ′ ) , k x ′ k X < ǫ , then k ¯ x − x ′ k X ≤ − ǫη k y − y ′ k X . Lemma 2.2. [22] If r = − θr + − θr , then k u k ˜ L rT ( ˙ F B β + θαp,q ) ≤ k u k − θ ˜ L r T ( ˙ F B βp,q ) k u k θ ˜ L r T ( ˙ F B β + αp,q ) . Now we prove a priori estimate which will be used in our proof. Consider the followingdissipative equation: ∂ t u + Λ α u = f ( t, x ) , u (0 , x ) = u ( x ) , t > , x ∈ R n . (2.2) Lemma 2.3.
Let < T ≤ ∞ , β ∈ R and ≤ r ≤ ∞ . Assume u ∈ ˙ F B βp,q , f ∈ ˜ L ( I ; ˙ F B βp,q ) .Then the solution u ( t, x ) to (2.2) satisfies k u k ˜ L r ( I ; ˙ F B β + αrp,q ) ≤ C ( k u k ˙ F B βp,q + k f k ˜ L ( I ; ˙ F B βp,q ) ) (2.3) eneralized PME in Fourier-Besov spaces Proof.
Consider the integral form of (2.2) u ( t, x ) = e − t Λ α u + Z t e − ( t − τ )Λ α f ( τ, x )d τ := Lu + Gf.
For the linear part, k ϕ j F ( Lu ) k L p ≤ e − t jα (3 / α k ϕ j c u k L p . Hence k Lu k ˜ L rT ( ˙ F B β + αrp,q ) ≤ (cid:13)(cid:13) j ( β + αr ) k e − t jα (3 / α k L rT k ϕ j c u k L p (cid:13)(cid:13) l q ≤ C k u k ˙ F B βp,q . On the other hand, for the integral part, k ϕ j F ( Gf ) k L p ≤ Z t e − ( t − τ )( j ) α k ϕ j ˆ f k L p d τ Taking L r -norm with respect to time, by Minkowskii’s inequality k ϕ j F ( Gf ) k L rT ( L p ) ≤ C − jαr k ϕ j ˆ f k L T ( L p ) . Hence k Gf k ˜ L rT ( ˙ F B β + αrp,q ) = (cid:13)(cid:13) j ( β + αr ) k ϕ j F ( Gf ) k L rT ( L p ) (cid:13)(cid:13) l q ≤ C k f k ˜ L T ( ˙ F B βp,q ) . Combine the above estimates, we obtain our desire inequality.
In this section we prove our main Theorem. First we know that the integral form of u is asfollows u = e − t Λ α u + Z t e − ( t − τ )Λ α ∇ · ( u ( τ ) ∇ P u ( τ ))d τ := S ( t ) u + H ( u, u ) . (3.1)Here S ( t ) u = F − ( e − t | ξ | α ˆ u ), and H ( u.v ) = R t e − ( t − τ )Λ α ∇ · ( u ( τ ) ∇ P v ( τ ))d τ . Now we getthe following estimate Proposition 3.1.
Let γ, p, q ≥ , ǫ > max { , − σ } , β > , γ = γ + γ , there holds k u∂ i P v k ˜ L γt ( ˙ F B βp,q ) ≤ C k u k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k v k ˜ L γ t ( ˙ F B β + σ + ǫp,q ) + C k v k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k u k ˜ L γ t ( ˙ F B β + σ + ǫp,q ) . Xiao et al.
Proof.
By the Bony’s decomposition, it is easy to get that∆ j ( u∂ i P v ) = X | k − j |≤ ∆ j ( S k − u ∆ k ( ∂ i P v )) (3.2)+ X | k − j |≤ ∆ j ( S k − ( ∂ i P v )∆ k u ) + X k ≥ j − ∆ j (∆ k u ˜∆ k ( ∂ i P v )) (3.3)We can get the following estimates:2 jβ k X | k − j |≤ ϕ j F ( S k − u ∆ k ( ∂ i P v )) k L γt ( L p ) ≤ jβ X | k − j |≤ kF ( S k − u ) ∗ ϕ k F ( ∂ i P v ) k L γt ( L p ) ≤ C jβ X | k − j |≤ (cid:13)(cid:13) X l ≤ k − ln (1 − p ) k ϕ l ˆ u k L p kσ k ϕ k ˆ v k L p (cid:13)(cid:13) L γt ≤ C jβ X | k − j |≤ (cid:13)(cid:13) X l ≤ k − ( l − k ) ǫ nl (1 − p ) − ǫl k ϕ l ˆ u k L p k ( σ + ǫ ) k ϕ k ˆ v k L p (cid:13)(cid:13) L γt ≤ C k u k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) j ( β + σ + ǫ ) k ϕ j ˆ v k L γ t ( L p ) . Similarly, we can estimate2 jβ k X | k − j |≤ ϕ j F ( S k − ( ∂ i P v )∆ k u ) k L γt ( L p ) ≤ C k v k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) j ( β + σ + ǫ ) k ϕ k ˆ u k L γ t ( L p ) . And when β >
0, we can also get that for some k a j k l q = 1,2 jβ kF ( X k ≥ j − ∆ j (∆ k ( ∂ i P v ) ˜∆ k u ) k L γt ( L p ) ≤ jβ k X k ≥ j − ϕ j F (∆ k ( ∂ i P v )) ∗ ( ˜ ϕ k ˆ u ) k L γt ( L p ) ≤ C jβ (cid:13)(cid:13) X k ≥ j − kn (1 − p )+ kσ k ϕ k ˆ v k L p k ˜ ϕ k ˆ u k L p (cid:13)(cid:13) L γt ≤ C X k ≥ j − ( j − k ) β (cid:13)(cid:13) kn (1 − p ) − ǫk k ϕ k ˆ v k L p k ( β + σ + ǫ ) k ˜ ϕ k ˆ u k L p (cid:13)(cid:13) L γt ≤ C X k ≥ j − ( j − k ) β k v k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k ( β + σ + ǫ ) k ˜ ϕ k u k ˜ L γ t ( L p ) ≤ Ca j k v k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k u k ˜ L γ t ( ˙ F B β + σ + ǫp,q ) . eneralized PME in Fourier-Besov spaces k u∂ i P v k ˜ L γt ( ˙ F B βp,q ) ≤ C k u k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k v k ˜ L γ t ( ˙ F B β + σ + ǫp,q ) + C k v k ˜ L γ t ( ˙ F B n (1 − p ) − ǫp,q ) k u k ˜ L γ t ( ˙ F B β + σ + ǫp,q ) . Proposition 3.2.
Let p, q ≥ , ǫ > max { , − σ } , β > − . If u, v ∈ ˜ L T ( ˙ F B n (1 − p ) − ǫp,q ) ∩ ˜ L T ( ˙ F B β + σ + ǫ +1 p,q ) , we have k H ( u, v ) k ˜ L ∞ T ( ˙ F B βp,q ) + k H ( u, v ) k ˜ L T ( ˙ F B β + αp,q ) ≤ C k u k ˜ L T ( ˙ F B n (1 − p ) − ǫp,q ) k v k ˜ L T ( ˙ F B β + σ + ǫ +1 p,q ) + C k v k ˜ L T ( ˙ F B n (1 − p ) − ǫp,q ) k u k ˜ L T ( ˙ F B β + σ + ǫ +1 p,q ) . Proof.
We note that H ( u, v ) is a solution to equation (2.2) with u = 0 , f = ∇ · ( u ∇ P v ).So by Lemma 2.3 there holds k H ( u, v ) k ˜ L ∞ T ( ˙ F B βp,q ) + k H ( u, v ) k ˜ L T ( ˙ F B β + αp,q ) ≤ C k∇ · ( u ∇ P v ) k ˜ L T ( ˙ F B βp,q ) . Then Proposition 3.1 gives the estimate.
Theorem 3.1.
Let p, q ∈ [1 , ∞ ] , { , σ + 1 } < α < n (1 − p ) + σ + 2 . Then for any u ∈ ˙ F B n (1 − p ) − α + σ +1 p,q , equation (1.1) admits a unique mild solution u and u ∈ C (cid:0) [0 , T ); ˙ F B n (1 − p ) − α + σ +1 p,q (cid:1) ∩ ˜ L ([0 , T ); ˙ F B n (1 − p )+ σ +1 p,q ) . Moreover, there exists a constant C = C ( α, p, q ) such that for u satisfying k u k ˙ F B n (1 − p ) − α + σ +1 p,q ≤ C , the solution u is global, and k u k L ∞ T ( ˙ F B n (1 − p ) − α + σ +1 p,q ) + k u k ˜ L T ( ˙ F B n (1 − p )+ σ +1 p,q ) ≤ C k u k ˙ F B n (1 − p ) − α + σ +1 p,q . Proof.
First suppose t ∈ [0 , T ], T fixed. Let ǫ = α − σ − β = n (1 − p ) − α + σ + 1, by theabove proposition k H ( u, v ) k L ∞ T ( ˙ F B n (1 − p ) − α + σ +1 p,q ) + k H ( u, v ) k ˜ L T ( ˙ F B n (1 − p )+ σ +1 p,q ) ≤ C k u k ˜ L T ( ˙ F B n (1 − p ) − α σ +1 p,q ) k v k ˜ L T ( ˙ F B n (1 − p ) − α σ +1 p,q ) . Define X = ˜ L T ( ˙ F B n (1 − p ) − α + σ +1 p,q ), by Lemma 2.2 k H ( u, v ) k X ≤ C k u k X k v k X . Xiao et al.
By Lemma 2.1 we know that if k e − t Λ α u k X < C , then (3.1) has a unique solution in B (0 , C ). Here B (0 , C ) := { x ∈ X : k x k X ≤ C } .To conclude k e − t Λ α u k X < C , we first note that e − t Λ α u is the solution to (2.2) with f = 0 , u = u , by Lemma 2.3, we can obtain k e − t Λ α u k X ≤ C k u k ˙ F B n (1 − p ) − α + σ +1 p,q . (3.4)Hence if k u k ˙ F B n (1 − p ) − α + σ +1 p,q ≤ C , (3.1) has a unique global solution in X . Moreover, k u k X ≤ C k u k ˙ F B n (1 − p ) − α + σ +1 p,q .On the other hand, denote u = F − χ {| ξ |≤ λ } c u + F − χ {| ξ | >λ } c u := u + u , where λ = λ ( u ) > u converges to 0 in ˙ F B n (1 − p ) − α + σ +1 p,q as λ → + ∞ . By (3.4) there exists λ large enough such that k e − t Λ α u k X ≤ C .
For u , k e − t Λ α u k X = (cid:13)(cid:13) j ( n (1 − p ) − α + σ +1) k ϕ j e − t | ξ | α χ {| ξ |≤ λ } c u k L T ( L p ) (cid:13)(cid:13) l q ≤ (cid:13)(cid:13) j ( n (1 − p ) − α + σ +1) k sup | ξ |≤ λ e − t | ξ | α | ξ | α k L T k ϕ j | ξ | − α ˆ u k L p (cid:13)(cid:13) l q ≤ Cλ α T k u k ˙ F B n (1 − p ) − α + σ +1 p,q . Thus for arbitrary u in ˙ F B n (1 − p ) − α + σ +1 p,q , (3.1) has a unique local solution in X on [0 , T )where T ≤ (cid:16) C λ α k u k ˙ F B n (1 − p ) − α + σ +1 p,q (cid:17) . The continuity with respect to time is standard.Next we give a blowup criterion as following:
Theorem 3.2.
Let T ∗ denote the maximal time of existence of u in L ∞ T ( ˙ F B βp,q ) ∩ ˜ L T ( ˙ F B β + αp,q ) .Here β = n (1 − p ) − α + σ + 1 . If T ∗ < ∞ , then k u k ˜ L ([0 ,T ∗ ); ˙ F B β + αp,q ) = ∞ . (3.5) Proof.
Supposing k u k ˜ L ([0 ,T ∗ ); ˙ F B β + αp,q ) dt < ∞ , then we can find 0 < T < T ∗ satisfying k u k ˜ L ([ T ,T ∗ ); ˙ F B β + αp,q ) < . eneralized PME in Fourier-Besov spaces t ∈ [ T , T ∗ ], s ∈ [ T , t ], by Lemma 2.3 k u ( s ) k ˙ F B βp,q + k u k ˜ L ([ T ,s ); ˙ F B β + αp,q ) ≤ k u ( T ) k ˙ F B βp,q + k u k L ∞ ([ T ,s ); ˙ F B βp,q ) k u k ˜ L ([ T ,s ); ˙ F B β + αp,q ) ≤ k u ( T ) k ˙ F B βp,q + 12 k u k L ∞ ([ T ,s ); ˙ F B βp,q ) . So, sup T ≤ s ≤ t k u ( s ) k ˙ F B βp,q ≤ k u ( T ) k ˙ F B βp,q + 12 k u k L ∞ ([ T ,t ); ˙ F B βp,q ) . Put M = max(2 k u ( T ) k ˙ F B βp,q , max t ∈ [0 ,T ] k u k ˙ F B βp,q ) , we can get k u ( t ) k ˙ F B βp,q ≤ M, ∀ t ∈ [0 , T ∗ ] . On the other side, u ( t ) − u ( t ) = e − t | D | α u − e − t | D | α u + Z t e − ( t − τ ) | D | α ∇ · ( uP u )( τ )d τ − Z t e − ( t − τ ) | D | α ∇ · ( uP u )( τ )d τ = [ e − t | D | α u − e − t | D | α u ] + [ Z t e − ( t − τ ) | D | α ( e − ( t − t ) | D | α − ∇ · ( uP u )( τ )d τ ]+ [ Z t t e − ( t − τ ) | D | α ∇ · ( uP u )( τ )d τ ]:= L + L + L . k L k ˙ F B βp,q = (cid:13)(cid:13) jβ k ϕ j ( e − t | ξ | α − e − t | ξ | α )ˆ u k L p (cid:13)(cid:13) l q ≤ (cid:13)(cid:13) jβ k ϕ j ( e − ( t − t ) | ξ | α − u k L p (cid:13)(cid:13) l q ≤ k u k ˙ F B βp,q . k L k ˙ F B βp,q ≤ (cid:13)(cid:13) jβ Z t k ϕ j e − ( t − τ ) | ξ | α (1 − e − ( t − t ) | ξ | α ) F ( ∇ · ( uP u )( τ )) k L p d τ (cid:13)(cid:13) l q ≤ (cid:13)(cid:13) j ( β +1) Z t k ϕ j ( e − ( t − t ) | ξ | α − F ( uP u )( τ ) k L p d τ (cid:13)(cid:13) l q . k L k ˙ F B βp,q ≤ (cid:13)(cid:13) jβ Z t t k ϕ j e − ( t − τ ) | ξ | α F ( ∇ · ( uP u )( τ )) k L p d τ (cid:13)(cid:13) l q Xiao et al. ≤ (cid:13)(cid:13) j ( β +1) Z t t k ϕ j F ( uP u )( τ ) k L p d τ (cid:13)(cid:13) l q . By the dominated convergence theorem, we can getlim sup t ,t ր T ∗ ,t
Taking respectively r and r ′ in Lemma 2.3, and using the proof of Proposition3.2, we get for ǫ > max { , − σ } , β > − k H ( u, v ) k ˜ L rT ( ˙ F B β + αrp,q ) + k H ( u, v ) k ˜ L r ′ T ( ˙ F B β + αr ′ p,q ) ≤ C k u k ˜ L rT ( ˙ F B n (1 − p ) − ǫp,q ) k v k ˜ L r ′ T ( ˙ F B β + σ + ǫ +1 p,q ) + C k v k ˜ L rT ( ˙ F B n (1 − p ) − ǫp,q ) k u k ˜ L r ′ T ( ˙ F B β + σ + ǫ +1 p,q ) . Set ǫ = αr ′ − σ − β = n (1 − p ) − α + σ + 1, we then gain the important bilinear estimate H ( u, v ) ≤ C k u k X k v k X under the condition: r ′ max { , σ + 1 } < α < n (1 − p ) + σ + 2 . Thus by Lemma 2.1 we know that if k e − t Λ α u k X < C , then equation (3.1) admits a uniquesolution in X . Step 2:
Now we need to derive k e − t Λ α u k X < C . Since e − t Λ α u is the solution to (2.2)with f = 0 , u = u , by Lemma 2.3 we gain the global solution in X for small initial data.On the other hand, we can also obtain the local solution on [0 , T ) in X by the same methodin Theorem 3.1 for arbitrary initial data, where T ≤ min n(cid:16) C λ αr k u k ˙ F B βp,q (cid:17) r , (cid:0) C λ αr ′ k u k ˙ F B βp,q (cid:17) r ′ o . eneralized PME in Fourier-Besov spaces Step 3:
We have proved equation has an unique solution in X under the condition: r ′ max { , σ + 1 } < α < n (1 − p ) + σ + 2, 1 < r < ∞ . Using the integral form (3.1)and Lemma 2.3, we can deduce k u k ˜ L ∞ T ( ˙ F B βp,q ) + k u k ˜ L T ( ˙ F B β + αp,q ) ≤ C ( k u k ˙ F B βp,q + k u k X ) . Hence u is also belongs to C ([0 , T ); ˙ F B βp,q ) ∩ ˜ L T ( ˙ F B β + αp,q ). Since 1 < r < ∞ in X can bechose to be a sufficiently large number, we in fact improve the index in Theorem 3.1 tomax { , σ + 1 } < α < n (1 − p ) + σ + 2 . Besides, this improvement make no difference to the proof of Theorem 3.2.
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