SELF-SIMILAR SOLUTIONS, UNIQUENESS AND ... - Project Euclid

Differential and Integral Equations

Volume 19, Number 12 (2006), 1349–1370

SELF-SIMILAR SOLUTIONS, UNIQUENESS AND LONG-TIME ASYMPTOTIC BEHAVIOR FOR SEMILINEAR HEAT EQUATIONS ˜o de Freitas Ferreira1 Lucas Cata Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil Departamento de Matem´atica, CEP 50740-540 ´s Villamizar-Roa2 Elder Jesu Universidad Industrial de Santander-UIS, Escuela de Matem´ aticas A.A. 678, Bucaramanga-Santander, Colombia (Submitted by: Yoshikazu Giga) Abstract. We analyze the well-posedness of the initial-value problem for the semilinear equation in Marcinkiewicz spaces L(p,∞) . Mild solutions are obtained in spaces with the right homogeneity to allow the existence of self-similar solutions. As a consequence of our results we , Ω = Rn , prove that the class C([0, T ); Lp (Ω)), 0 < T ≤ ∞, p = n(ρ−1) 2γ has uniqueness of solutions (including large solutions) obtained in [19], [20] and [8]. The asymptotic stability of solutions is obtained, and as a consequence, a criterion for self-similarity persistence at large times is obtained.

1. Introduction In this paper we study the Cauchy problem for generalized semilinear heat equations on Rn , ut + (−Δ)γ u + f (u) = 0, u(0, x) = u0 ,

x ∈ Rn ,

(1.1)

x∈R ,

(1.2)

n

where n ≥ 1, γ > 0 and f : R → R satisfies   |f (a2 ) − f (a1 )| ≤ η |a2 − a1 | |a2 |ρ−1 + |a1 |ρ−1 ,

(1.3)

Accepted for publication: September 2006. AMS Subject Classifications: 35A05, 35D05, 35K05. 1 e-mail: lcff@dmat.ufpe.br. Supported by CAPES, Brazil 2 e-mail: [email protected]. Supported by Colciencias, Colombia, Proyecto BID-III etapa and Universidad Industrial de Santander-UIS. 1349

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L.C.F. Ferreira and E.J. Villamizar-Roa

for all a1 and a2 ∈ R with some constants η > 0 and 1 < ρ < ∞. Note that f (u) = |u|(ρ−1) u satisfies the condition (1.3). When γ = 1 the equation (1.1) is the well-known semilinear heat equation. Several important papers are devoted to existence and uniqueness of solutions for the Cauchy problem (1.1)-(1.2) (in the case γ = 1); see, for instance, [19], [20], [8], [4]. In [19], Weissler proved time-local existence and uniqueness of solutions with u0 ∈ Lp and p ≥ n(ρ−1) > 1. In [20] , Weissler proved the 2 p existence of global mild solutions in L when p = n(ρ−1) , provided the initial 2 data is sufficiently small. Let us remark that in [19], [20], Weissler proved the 1 n uniqueness with the additional condition limt→0 tα/2 uq = 0, α = ρ−1 − 2q and n(ρ−1) < q < nρ(ρ−1) . In [8], Giga has constructed a unique local regu2 2 r lar solution in L ((0, T ); Lq ) with restrictions in r and q to the norm being scaling invariant. In the same paper [8], Giga alerts the reader to the possibility that the class BC([0, T ); Lp ) with p = n(ρ−1) , is not sufficiently 2 in order to guarantee the uniqueness of solutions. In [16] a counterexample of nonuniqueness for the initial-boundary-value problem (1.1)-(1.2), with n γ = 1, p = n−2 , n ≥ 3 and with restricted domain being a ball, is given. The purpose of this paper is to show the existence and the uniqueness of global mild solutions and asymptotic stability results for the Cauchy problem (1.1)-(1.2) in Marcinkewicz spaces L(p,∞) in the principal case p = n(ρ−1) 2γ (zero-dimension norm). We also will make remarks about the time-local theory, obtaining the time-local version of theorems and also about the case (

n(ρ−1)

,∞)

p > n(ρ−1) (see Remark 3.7). The space L 2γ is invariant under scal2γ ing of the semilinear equation (1.1) and contains homogeneous functions 2γ of degree − ρ−1 . As a consequence of this fact, we obtain the existence of self-similar solutions for the initial-value problem (1.1)-(1.2). Let us remark that such a method to find self-similar solutions was initially analyzed by Giga and Miyakawa [10] in the study of the Navier-Stokes equations in vorticity-velocity formulation in Morrey spaces and, later, by other authors in several spaces (e.g see [1] and [22] in the framework of Lorentz spaces L(n,∞) ). In our estimates of nonlinear terms f (u) of semilinear equations, we do not use Kato-Fujita’s approach which uses two norms to prove the continuity of the nonlinear term in (1.1), to know the natural norm in (

E = BC((0, T ); L

n(ρ−1) ,∞) 2γ n(ρ−1) 2γ

(Rn )) and a time-dependent “regularizing norm”. (

n(ρ−1)

,∞)

Therefore, since L ⊂ L 2γ , we obtain the uniqueness of solutions (without a smallness assumption) in the strong version of the natural

Self-similar solutions

space E, i.e., C([0, T ); L L

n(ρ−1) 2γ

1351

(Rn )), with initial data in the Lebesgue space

n(ρ−1) 2γ

. We also remark that in the case 0 < γ ≤ 1 all theorems hold when we n , a bounded domain or an exterior domain in Rn , instead take Ω to be R+ n of only Ω = R . In the same case, we show that if f (u) = −uρ and u0 ≥ 0, then the solution obtained is positive. Let us stress that the approach with two norms (Kato-Fujita’s approach) just gets solutions that are instantaneously smoothed out and then it is indeed very restricted from the point of view of singular solutions and blowup. From another point of view, we show that L(p,∞) -spaces allow the existence of initially-singular solutions. These solutions are instantaneously smoothed out if we take the initial data a little smaller. However, we show the existence of global solutions in L(p,∞) -spaces with p = n(ρ−1) for which 2γ we do not know whether or not the singularity persists (see Remark 3.8). We also analyze the asymptotic stability of solutions and, as a consequence, a criteria for vanishing small perturbations of initial data at large time and the self-similarity persistence results for the solutions is obtained. n(ρ−1)

Moreover, we will show that if the initial data has small norm in L 2γ and lies on the closure of the intersection of two L(p,∞) -spaces, which we will n(ρ−1)

specify later ( in particular in Lebesgue space L 2γ ), the solution vanishes when time is larger in the norm of space. We will focus on the case 0 < γ < n2 . The basic properties of L(p,∞) spaces will be reviewed in the next section. Section 3 will be devoted to show well-posedness theorems. Self-similarity, positive solutions and decay rates when we take more regular initial data will be discussed in Section 4. Finally, in Section 5 we will analyze the asymptotic stability of solutions and we will give a criterion of vanishing small perturbation of initial data at large time. 2. Function spaces and definitions In this section, we introduce the functional spaces relevant to our study of solutions of the Cauchy problem for problem (1.1)-(1.2), we list some facts about convolution and we discuss the notion of solution in these spaces. 2.1. Lorentz Spaces: L(p,q) . We introduce some preliminaries about the Lorentz spaces; for details, see for instance, [11], [18]. For each Lebesgue measurable function f on Rn , the rearrangement f ∗ and the average function

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L.C.F. Ferreira and E.J. Villamizar-Roa

f ∗∗ are, respectively, defined by f ∗ (t) = inf{s > 0 : m({x ∈ Rn : |f (x)| > s}) ≤ t}, t > 0,  1 t ∗ f ∗∗ (t) = f (s)ds, t > 0, t 0 where m is the Rn -Lebesgue measure. The Lorentz space L(p,q) (Rn ) ≡ L(p,q) consists of all measurable functions f such that f ∗(p,q) < ∞, with ⎧  1 ⎨ q  ∞ p1 ∗ q q [t f (t)] dt/t , 0 < p < ∞,0 < q < ∞, p 0 f ∗(p,q) = 1 ⎩ 0 < p ≤ ∞ , q = ∞. supt>0 t p f ∗ (t), The quantity f ∗(p,q) does not satisfy the triangle inequality. A natural way of metrizing the space L(p,q) is to define the norm f (p,q) as ⎧  1 ⎨ q  ∞ p1 ∗∗ q q [t f (t)] dt/t , if 1 < p < ∞, 1 ≤ q < ∞, p 0 f (p,q) = 1 ⎩ if 1 < p ≤ ∞, q = ∞. supt>0 t p f ∗∗ (t), The spaces L(p,q) endowed with the norm f (p,q) are Banach spaces and p f ∗(p,q) ≤ f (p,q) ≤ f ∗(p,q) p−1 holds. Lorentz spaces have the same scaling relation as Lp -spaces; that is, −n for all λ > 0 we have f (λx)(p,q) = λ p f (p,q) , where 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. We observe that Lp (Rn ) = L(p,p) (Rn ) and L(p,q1 ) (Rn ) ⊂ L(p,q2 ) (Rn ), q1 ≤ q2 , with continuous injection. Furthermore, the Lorentz spaces can be constructed via the real interpolation [2]; that is, L(p,q) (Rn ) = (L1 (Rn ), L∞ (Rn ))1− 1 ,q , 1 < p < ∞. p

Lorentz spaces have the following interpolation property [3]: (L(p0 ,q0 ) (Rn ), L(p1 ,q1 ) (Rn ))θ,q = L(p,q) (Rn ), θ provided 0 < p0 < p1 < ∞, 0 < θ < 1, p1 = 1−θ p0 + p1 , 1 ≤ q0 , q1 , q ≤ ∞. If q = ∞, the space L(p,∞) is called the Marcinkiewics space or weakp L . Duals of Lorentz spaces are natural extensions of the Lp -spaces duality property (see [11]). For 1 < p < ∞, 1 < q < ∞, the dual space of L(p,q) is   L(p ,q ) , 1 < p < ∞, 1 < q < ∞, and hence, these spaces are reflexive. L(p,∞)

Self-similar solutions

1353



is the dual space of L(p ,1) provided 1/p + 1/p = 1 and L(p,1) is not the dual  space of L(p ,∞) . The next proposition shows that the L(p,∞) -spaces have a different behavior than Lp -spaces regarding density. For a proof see [7]. Proposition 2.1. Let 1 < p, q < ∞ and p = q. Then L(p,∞) ∩ L(q,∞) is not dense in L(p,∞) . Let us finally recall a property of product operators in Lorentz spaces. Proposition 2.2. [17] (generalized H¨ older’s inequality) Let 1 < p1 , p2 < ∞. Let f ∈ L(p1 ,q1 ) and g ∈ L(p2 ,q2 ) where p11 + p12 < 1, then the product h = f g belongs to L(r,s) where 1r = p11 + p12 , and s ≥ 1 is any number such that 1 1 1 q1 + q2 ≥ s . Moreover, h(r,s) ≤ r f (p1 ,q1 ) g(p2 ,q2 ) ,

(2.1)

r being the conjugate index of r. 3. Results of well posedness The aim of this section is to describe the results regarding the well posedness of problem (1.1)-(1.2) in the space L(p,∞) . We also consider well posedness in the Lebesgue space Lp . Definition 3.1. Let 1 < q ≤ ∞ and α = Banach spaces (

E ≡ BC((0, ∞), L

n(ρ−1) ,∞) 2γ

2 ρ−1

−

n γq .

We define the following

)}, Eq ≡ {u ∈ E : tα/2 u ∈ BC((0, ∞); L(q,∞) )},

with the norms in E and Eq ,defined, respectively, as hE = sup h( n(ρ−1) ,∞) , t>0

2γ

hEq = hE + sup tα/2 h(t)(q,∞) . t>0

Here, we have denoted by BC the class of bounded and continuous functions from the corresponding interval onto a Banach space. 3.1. The linearized equation and mild solutions. Let us define Gγ (t) as the convolution operator with kernel gγ given by means of its Fourier 2γ transform g γ (ξ, t) = e−|ξ| t . Equivalently, Gγ (t) is the fundamental solution of the linear equation ∂t φ + (−Δ)γ φ = 0, (3.1) r where the operator (−Δ) is defined as usual through the Fourier transform as ((−Δ)r φ)ˆ (ξ) = |ξ|2r φ(ξ),

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L.C.F. Ferreira and E.J. Villamizar-Roa



with the convention ˆ φ(ξ) =

e−ix·ξ φ(x)dx.

Rn

Note the following homogeneity properties of gγ (x, t): (∇jx gγ )(x, t) = t

− n+j 2γ

1 − 2γ

(∇jx gγ )(xt

, 1) for all j ∈ {0} ∪ N.

(3.2)

Now, we state some estimates about the semi-group Gγ (t) in Lorentz spaces, which generalizes the well-known estimate of the heat kernel (γ = 1). Lemma 3.2. Let 1 ≤ d ≤ ∞ and 1 < p ≤ r < ∞. There exists a constant C = C(r, n, γ, p) > 0 such that, for all ϕ ∈ L(p,d) (Rn ) and for all t > 0,

j j n 1 1

∇x Gγ (t)ϕ ≤ Ct− 2γ ( p − r )− 2γ ϕ , (3.3) p,d r,d where j ∈ {0} ∪ N. Proof. Let 1 < p ≤ r < ∞ and l ≥ 1 such that 1r = 1l + p1 − 1. By Young’s inequality in Lp -spaces and inequalities (3.2)-(3.3), we estimate



j



n+j 1

− 2γ j

∇ Gγ (t)ϕ ≤ ∇jx gy ϕ p = t− 2γ

(∇ g )(xt , 1)

ϕLp x γ L l Lr l

n

n 1 − n+j + − ( − 1 )− j j ≤ t 2γ 2γl (∇x gγ )(x, 1) l ϕLp ≤ Ct 2γ p r 2γ ϕLp , which proves the estimate (3.3) in the case of Lp = L(p,p) . The general case follows using interpolation arguments.  Let us now make precise the notion of mild solution for problem (1.1)(1.2). Definition 3.3. A global mild solution of the initial-value problem (1.1)(1.2) in E and Eq is a function u(t) in the corresponding space satisfying  t Gγ (t − s)f (u) (s) ds, (3.4) u(t) = Gγ (t)u0 + B(u)(t) ≡ Gγ (t)u0 − 0

and u(t) u0 when t → (

of L

n(ρ−1) ,∞) 2γ

0+ ,

where the limit is taken in the weak-∗ topology

.

Let us remark that an analogous definition of functional spaces E and Eq , and of solution, can be done in Lebesgue spaces Lp with the weak-∗ n(ρ−1)

convergence replaced by strong convergence in L 2γ . Time-local versions of definitions of spaces E, Eq , and of mild solutions also, can be considered in the same way. We prove the following well-posedness results for mild solutions in the sense of Definition 3.3 .

Self-similar solutions

Theorem 3.4. (Well posedness) Let 0 < γ
0 and ε = ε(δ) > 0 (ε = Cδ, C > 0 is the constant obtained in the Lemma 3.2 when j = 0) such that if u0 ( n(ρ−1) ,∞) < δ, then the initial-value problem (1.1)-(1.2) has a global 2γ

mild solution u(t, x) ∈ E in the sense of Definition 3.3. n(ρ−1)

n ∩ L(p,∞) with 1 < p < 2γ . There Furthermore, assume u0 ∈ L 2γ exists δp > 0, 0 < δp ≤ δ, such that if u0 ( n(ρ−1) ,∞) < δp , then the previous (

,∞)

2γ

solution u(t, x) ∈ BC((0, ∞), L(p,∞) ). The previous theorem provides a well-posedness theory for the semi(

n(ρ−1)

,∞)

linear equation in a one-space norm in L 2γ . Due to Lemma 3.9, the uniqueness of solutions in the previous theorem is obtained only in B2 := B(u0 , 2) ⊂ E. Theorem 3.5. (Regularization) Under the assumptions of Theorem 3.4, for nρ(ρ−1) any n(ρ−1) 2γ < q < 2γ , there exists 0 < δq < δ such that if u0 ( n(ρ−1) ,∞) < 2γ

δq , then the solution u(t, x), of Theorem 3.4, belongs to Eq -space. The next theorem improves the uniqueness part of Theorem 3.4. n Theorem 3.6. (Uniqueness) Assume 0 < γ < n/2, n−2γ < ρ < ∞. Let u and v be two mild solutions of problem (1.1)-(1.2) in the class C([0, ∞);

L

n(ρ−1) 2γ

) with initial data u0 ∈ L

n(ρ−1) 2γ

. Then u = v.

Remark 3.7. (Generalizations) The restrictions 0 < γ < n2 and ∞ comes from Lemma 3.11 in which n+2γ n

n(ρ−1) 2γ

n n−2γ

0 t 2  · q , where  · q denotes the Lq − norm. In this way, we do not obtain the continuity of nonlinear terms in n(ρ−1)

the strong version of the natural space BC([0, ∞); L 2γ ). Therefore, when γ = 1, we obtain the previous results of [20]. Note that, in this case, the n(ρ−1)

initial data is taken in the strong sense, that is, in the L 2γ -norm. α Assuming that lim supt→0 t 2 Gγ (t)u0 (q,∞) is sufficiently small, the theorems above and their strong versions have a corresponding version for local

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L.C.F. Ferreira and E.J. Villamizar-Roa

in time solutions, being the trade-off of the smallness of the existence time instead of the smallness assumption of the initial data. In the case of Lp α spaces, this condition is not necessary since lim supt→0 t 2 Gγ (t)u0 q = 0. Assuming that u0 ∈ L(r,∞) with r > n(ρ−1) 2γ , we can solve the initial-value (r,∞) problem in BC([0, T ); L ) and we also can take T = ∞, if we assume smallness of the initial data. The proof in this case is more simple. We can estimate directly the norm of the nonlinear term, without using duality and interpolation arguments or the approach with two norms. It is only necessary to use the semigroup estimate (3.3), with j = 0, and Holder’s inequality in Marcinkiewicz spaces L(r,∞) . In the case γ = 1, we can consider the initial-value problem (1.1)-(1.2) with n , a bounded domain, or an exterior domain in Rn , with smooth Ω being R+ boundary ∂Ω, where we assume that u0 = 0 on ∂Ω. In these cases, we obtain analogous results in the case Ω = Rn . For this, we must use the well-known estimates of the heat kernel in Ω (see for instance, [22] ). In particular, we re-obtain the result of Brezis and Cazenave [4], in the case when Ω is a bounded domain. Remark 3.8. [Smoothness] Adapting the arguments of T. Kato in [13], we can prove that the solutions of Theorem 3.5 are C ∞ -smooth instantly and are solutions of the initial-value problem (1.1)-(1.2) for t > 0, in the classical sense. However, we do not know if this property is true for solutions of Theorem 3.4. (It is possible that they are singular.) The essential point is that, for t > 0, the solutions given by Theorem 3.5 and their derivatives (

n(ρ−1)

,∞)

lie in L 2γ ∩ L(q,∞) ⊂ Lr ( by interpolation with n(ρ−1) < r < q) and 2γ this fact does not hold for the solutions of Theorem 3.4. 3.2. Proofs of the Well-posedness Theorems. We start with the following lemma in a generic Banach space which generalizes Theorem 13.2 of [14]. The proof is also based on the standard Picard iteration technique completed by the Banach fixed-point theorem. Lemma 3.9. Let 1 < ρ < ∞ and X be a Banach space with norm  · , and B : X → X be a map witch satisfies B(x) ≤ Kxρ and

(3.5)

 B(x) − B(z) ≤ Kx − z xρ−1 + zρ−1 . 2ρ Kaρ−1 −1

(3.6)

Let R > 0 be the unique positive root of the equation = 0. Given 0 < ε < R and y ∈ X, y = 0, such that y < ε, there exists a solution x ∈ X

Self-similar solutions

1357

for the equation x = y + B(x) such that x ≤ 2ε. The solution x is unique in the ball B2ε := B(0, 2ε). Moreover, the solution depends continuously on y in the following sense: If ˜ y  ≤ ε, x ˜ = y˜ + B(˜ x), and ˜ x ≤ 2ε, then 1 x − x ˜ ≤ y − y˜. (3.7) ρ 1 − 2 Kερ−1 Proof. Define F : X → X as F (x) = y + B(x). Note that, since 0 < ε < R, it follows that 2ρ Kερ−1 − 1 < 0. Firstly, we will prove that F (B2ε ) ⊂ B2ε . For this, if x ∈ B2ε , then we have that F (x) ≤ y + B(x) ≤ y + Kxρ ≤ ε + 2ρ Kερ < 2ε. Now, if x, z ∈ B2ε , we have

 F (x) − F (z) ≤ B(x) − B(z) ≤ Kx − z xρ−1 + zρ−1 ≤ K(2ρ ερ−1 )x − z < x − z,

where we have used that 2ρ Kερ−1 − 1 < 0. Therefore the application F is a contraction in B2ε . For the Banach’s fixed-point theorem, we conclude the proof of the existence and uniqueness. It is useful to remember that the solution is obtained by a successive approximation method. In fact, defining the sequence xn+1 = F (xn ), n = 1, 2, 3..., x1 = y, the solution x is obtained as the limit of the sequence {xn }. To prove the continuity of the solution operator on the parameter y, let x ˜ be x as in the statement of the theorem. Then, we have x − x ˜ ≤ y − y˜ + B(x) − B(˜ x) ≤ y − y˜ + 2ρ Kερ−1 x − x ˜, 

which is equivalent to (3.7).

3.2.1. Proof of Theorem 3.4. Since in Lemma 3.2 we already have proved the necessary estimates for the linear part of the integral form of problem (1.1)-(1.2), as a consequence of Lemma 3.9, we just need to verify the continuity of the nonlinear term in the integral form to obtain the well-posedness theorems. The weak- continuity at 0+ finishes the proof of well posedness of mild solutions. We first show the continuity of the nonlinear term B(u) in the mild formulation (3.4) defined as:  t B(u)(t) ≡ − Gγ (t − s)f (u) (s) ds. (3.8) 0

The next lemma is a generalization to 0 < γ < ∞ of the technical lemma found in [22] to the heat kernel (γ = 1).

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L.C.F. Ferreira and E.J. Villamizar-Roa

Lemma 3.10. Let 0 < γ < ∞ and 1 < p < q < ∞, then  ∞ 1 n ( − n )−1 t 2γ p q Gγ (t)φ(q,1) ds ≤ Cφ(p,1) .

(3.9)

0

n

( 1 − 1 )−1

Proof. Let us define ξ(t) = t 2γ p q Gγ (t)φ(q,1) and take p1 < p < p2 < q such that ( np − pn2 ) < 2γ. Using (3.3) for 1 < pk < q < ∞, k = 1, 2 and j = 0, we obtain 1

ξ(t) ≤ C t 2γ

(n − pn −2γ) p k

φ(pk ,1) .

Let us define = − + 2γ) satisfying 0 < l1 < 1 < l2 . Therefore, the previous property on ξ(t) can be restated as ξ(t) ∈ Llk ,∞ (0, ∞) and ξ(t)(lk ,∞) ≤ Cφ(pk ,1) , for k = 1, 2. Now take λ ∈ (0, 1) such that 1 λ 1−λ λ 1−λ p = p1 + p2 and thus, 1 = l1 + l2 . Therefore, by interpolation theorems in Lorentz spaces (see [3]), we have ξ(t)L1 (0,∞) ≤ Cφ(p,1) , because  p ,1 n  L 1 (R ), Lp2 ,1 (Rn ) λ,1 = Lp,1 (Rn ) and Ll1 ,∞ (0, ∞), Ll2 ,∞ (0, ∞) λ,1 = L1 (0, ∞).  1 lk

1 n 2γ ( pk

n p

Now, let us consider the linear operator  ∞ Gγ (s)(f (h))(s) ds. C(h)(x) = 0

The next lemma deals with the continuity of the operator C(h), which we will use to prove the continuity of the nonlinear term B(u) of the mild formulation (3.4). Lemma 3.11. Let

n n−2γ

< ρ < ∞ and h ∈ L∞ ((0, ∞); L

(

C(h)( n(ρ−1) ,∞) ≤ K sup h(t)ρn(ρ−1) 2γ

(

t>0

Moreover, if h ∈ L∞ ((0, ∞); L

(

n(ρ−1) ,∞) 2γ

Proof. For simplicity we denote l =

(

,∞)

2γ

,∞)

n(ρ−1) 2γ .

).Then

.

∩ L(p,∞) ), with 1 < p
0

2γ

n(ρ−1) ,∞) 2γ

n 2γ ,

then

sup h(t)(p,∞) . t>0

Since

n n−2γ

< ρ < ∞, then ( l ,∞)

1 < ρ < l. Now, note that if h ∈ L(l,∞) , then |f (h)| ≤ η |h|ρ ∈ L ρ  l > l = f (h)( l ,∞) ≤ Chρ(l,∞) . Let φ ∈ L(l ,1) and note that l−ρ ρ

n 1 2γ ( l

−

l−ρ l )

, with and

l l−1

− 1 = 0. We use duality, H¨ older’s inequality and (3.9) in order

Self-similar solutions

to obtain C(h)(l,∞) ≤ = = ≤

sup



∞

sup



∞

sup



φ(l ,1) =1

φ(l ,1) =1

φ(l ,1) =1

Rn

0



C(h)φ(x)dx

Rn

Rn

0

1359

(Gγ (s)f (h))(x)φ(x)dxds

f (h)(Gγ (s)φ)dxds

∞

f (h)( ρl ,∞) Gγ (s)φ(

sup

φ(l ,1) =1 0

≤ C sup h(t)ρ(l,∞) t>0

sup

φ(l ,1) =1

l ,1) l−ρ

ds

φ(l ,1) = C sup h(t)ρn(ρ−1) t>0

(

2γ

,∞)

.

1 r = (r,∞) L ,

Next, we deal with the last assertion of the lemma. Let r > 1 such that 1 p

+

2γ n.

n(ρ−1) ( 2γ ,∞)

∩ L(p,∞) , then |f (h)| ≤ η |h|ρ ∈

Note that if h ∈ L

with f (h)(r,∞) ≤ Ch(p,∞) hρ−1 n(ρ−1) (

2γ

,∞)

. Now, we proceed in the same

way as the proof of the first part and conclude the proof.



First, we remember that f (0) = 0. We are now in position to prove the continuity of the nonlinear term B(u) defined by B(u) = C(h) with h(s, ·) = u(t − s, ·), if 0 ≤ s ≤ t and h(s, .) = 0, otherwise. In the following K or Kp will denote generic constants that may change from line to line. Using Lemma 3.11, we estimate B(u)(t)( n(ρ−1) ,∞) ≤ K sup h(t)ρn(ρ−1) 2γ

(

t>0

≤ K sup u(t) t>0

2γ

,∞)

ρ (

n(ρ−1) ,∞) 2γ

= KuρE

(3.10)

and B(u)(p,∞) ≤ Kp sup h(t)ρ−1 n(ρ−1) (

t>0

2γ

,∞)

ρ−1

≤ Kp sup u(t) t>0

(

n(ρ−1) ,∞) 2γ

sup h(t)(p,∞) t>0

sup u(t)(p,∞) .

(3.11)

t>0

The proof of Theorem 3.4, which gives the well posedness of the integral equation (3.4) in the L(p,∞) -spaces, is an application of the above lemmas. We start by applying Lemma 3.9 to the integral equation (3.4) with X = E and y = Gγ (t)u0 . In fact, let 0 < δ < ε/C < R/C, where C and K are the constants of Lemma 3.2 and Lemma 3.11 respectively, and R is as in

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L.C.F. Ferreira and E.J. Villamizar-Roa

Lemma 3.9. Hence, making use of Lemma 3.2, we have Gγ (t)u0 E < ε, provided u0 ( n(ρ−1) ,∞) < δ. 2γ

Using Lemma 3.11, we have proved the estimate (3.10) which corresponds with the property (3.5) in the space E. Since f satisfies the inequality (1.3), proceeding analogously as in the proof of Lemma 3.11, it is not difficult to prove that the nonlinear term  t B(u) = − Gγ (t − s)f (u) (s) ds 0

satisfies assumption (3.6) of the Lemma 3.9; i.e,   ρ−1 B(u) − B(v)E ≤ K u − vE uρ−1 + v . E E

(3.12)

Moreover, in an analogous way to the proof of Lemma 3.11 (second part), we can obtain   ρ−1 sup B(u) − B(v)(p,∞) ≤ Kp sup u − v(p,∞) uρ−1 + v . (3.13) E E t>0

t>0

Now, Lemma 3.9 implies the existence of a global mild solution u ∈ E. This solution is unique in the ball B2ε := B(0, 2ε) ⊂ E. Let us remark that Lemma 3.9 guarantees the uniqueness of solutions only in B2 := B(u0 , 2) ⊂ E. In order to check that solutions of the integral equation are indeed mild solutions in the sense of Definition 3.3, it remains to show that the solution (

u 0 when t → 0+ in weak-star topology of L  l = n(ρ−1) and note that, for all ϕ ∈ L(l ,1) , we have 2γ

n(ρ−1) ,∞) 2γ

. For this, let

|Gγ (t)u0 − u0 , ϕ| = |u0 , Gγ (t)ϕ − ϕ| ≤ u0 (l,∞) Gγ (t)ϕ − ϕ(l ,1) → 0, since Gγ (t)ϕ − ϕ(l ,1) → 0 when t → 0+ . We omit the proof of B(u) 0, because it follows in a similar way to the proof of Gγ (t)u0 0. The final part of the theorem, i.e., u ∈ BC((0, ∞); L(p,∞) ) for initial data n(ρ−1)

n u0 ∈ L 2γ ∩ L(p,∞) , with 1 < p < 2γ , can be proved as follows. Since the solution obtained by Lemma 3.9 is constructed by recursion: (

,∞)

u1 (t, x) = Gγ (t)u0 (x), uk+1 (t, x) = u1 (t, x) + B(uk ), where k ∈ N , we can use Lemma 3.2 and the estimate (3.11) to get, respectively,  0 (p,∞) , sup u1 (t)(p,∞) ≤ Cu t>0

Self-similar solutions

1361

and  0 (p,∞) + Kp sup uk (t)(p,∞) sup uk (t)ρ−1 sup uk+1 (t)(p,∞) ≤ Cu n(ρ−1) t>0

t>0

(

t>0

2γ

,∞)

.

(3.14) Now, let us choose 0 < δp ≤ δ such that 0 < εp ≤ ε and 2ρ (εp )ρ−1 Kp < 1 (εp = Cδp and ε = Cδ). We assume that u0 ( n(ρ−1) ,∞) < εp . The proof of 2γ

Lemma 3.9 and the proof of the first part of Theorem 3.4 show that sup uk (t)( n(ρ−1) ,∞) ≤ 2εp . t>0

2γ

Now, we consider the sequence {wk }k≥2 defined as wk+1 = uk+1 − uk = B(uk ) − B(uk−1 ). Using the estimate (3.13) we have

  ρ−1 + u (t) sup wk+1 (p,∞) ≤ Kp sup wk (t)(p,∞) uk (t)ρ−1 k−1 E E t>0

≤2

ρ

t>0 (εp )ρ−1 Kp sup wk (t)(p,∞) . t>0

(3.15)

Let us denote Mk = supt>0 wk (t)(p,∞) . By inequality (3.14), we know that 0 ≤ M2 < ∞. The sequence {Mk }k≥2 satisfies Mk+1 ≤ 2ρ (εp )ρ−1 Kp Mk . Taking A = 2ρ (εp )ρ−1 Kp < 1, we can write Mk ≤ Ak−2 M2 → 0 when k → ∞. Therefore the sequence {uk } is a Cauchy sequence in BC((0, ∞); L(p,∞) ) and thus, it converges to u (t, x) ∈ BC((0, ∞); L(p,∞) ). Now, the uniqueness of limits in the distribution sense permits us to finish the proof of the theorem.  3.2.2. Proof of Theorem 3.5. Let α =

2 ρ−1

−

n γq

with

n(ρ−1) 2γ

0 t 2 ·(q,∞) of linear part of integral equation as Gγ (t)u0 (q,∞) ≤ t

2γ n − 2γ ( n(ρ−1) − 1q )

α

u0 ( n(ρ−1) ,∞) ≤ Ct− 2 u0 ( n(ρ−1) ,∞) . 2γ

2γ

Now, in order to prove Theorem 3.5 by applying Lemma 3.9, we just need to show properties (3.5)-(3.6) for the nonlinear term B(u) in the space Eq . Indeed is sufficient to prove the property (3.5) since (3.6) follows in a similar way. Since we already have proved (3.5) in the norm of the space E, it α remains to prove that property in the norm supt>0 t 2 ·(q,∞) .

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L.C.F. Ferreira and E.J. Villamizar-Roa

For this, first note that f (u)( q ,∞) ≤ C uρ(q,∞) . Since 1 < follows that 1
0 0  ρ  ρ n ρ α α α − 2γ ( q − 1q )−ρ α +1 2 sup t 2 u(q,∞) =t = t− 2 sup t 2 u(q,∞) , t>0

t>0



and this finishes the proof of the theorem.

3.2.3. Proof of Theorem 3.6.. Let u and v be two mild solutions of (1.1)n(ρ−1)

n(ρ−1)

(1.3) in C([0, ∞); L 2γ ), with initial data u0 ∈ L 2γ . In order to prove Theorem 3.6, it is sufficient to prove that u = v in [0, T ], 0 < T < ∞, T small enough. We denote by w = u − v, w1 = Gγ (t)u0 − u, w2 = Gγ (t)u0 − v. Then we can obtain the following estimate

 t



w(t)( n(ρ−1) ,∞) =

Gγ (t − s)(f (u)(s) − f (v)(s))ds n(ρ−1) 2γ

(

0

 t



≤

Gγ (t − s)|w(s)|(|u(s)|ρ−1 + |v(s)|ρ−1 )ds

(

0

(

0

(

0

,∞)

n(ρ−1) ,∞) 2γ

 t



≤ C

|Gγ (t − s)||w(s)|(|w1 (s)|ρ−1 + |w2 (s)|ρ−1 )ds

 t



ρ−1

+ 2C

|Gγ (t − s)||w(s)|Gγ (s)u0 | ds

2γ

n(ρ−1) ,∞) 2γ

n(ρ−1) ,∞) 2γ

= I1 + I2 .

Working in an analogous way to estimate (3.13) and using the fact that Lr ⊂ Lr,∞) , 1 < r ≤ ∞, we obtain ρ−1 I1 ≤ CK sup w( n(ρ−1) ,∞) ( sup w1 ρ−1 n(ρ−1) + sup w2  n(ρ−1) ). 2γ

0 0 and x ∈ Rn . Proof. Looking at the proof of Lemma 3.9, the solution is obtained by a successive approximation method. In fact, we have the sequence: u1 (t, x) = Gγ (t)u0 (x),

uk+1 (t, x) = u1 (t, x) + B(uk ),

where k = 1, 2, . . . It is easy to verify that u1 (t, x) satisfies 2γ

u1 (t, x) = λ ρ−1 u1 (λx, λ2γ t). Using an induction argument we prove that un has the property un (t, x) = 2γ

λ ρ−1 un (λx, λ2γ t), for all n. Therefore, as the mild solution u(t, x) is obtained as the limit of the sequence {un } in the space E, we have that u(t, x) must satisfy 2γ

u(t, x) = λ ρ−1 u(λx, λ2γ t), for all λ > 0, all t > 0 and almost every x ∈ Rn . Now, let f (u) = −uρ , 0 < γ ≤ 1 and u0 ≥ 0. We can prove the last affirmation in the theorem, noting that the elements of the previous sequence {uk }k∈N are all positives and that uk → u in E. Indeed, since the semigroup Gy (t) is a convolution operator with positive kernel and u0 ≥ 0 then u1 ≥ 0. Analogously, B(u1 ) ≥ 0 and therefore u2 ≥ 0 as the sum of two non-negative terms; and so on. 

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L.C.F. Ferreira and E.J. Villamizar-Roa (

5. Asymptotic Stability in L

n(ρ−1) ,∞) 2γ

In this section we study the large-time behavior of solutions of Section 3. Our results are the following: Theorem 5.1. Assume that u and v are solutions of problem (1.1)-(1.2) (

given by Theorem 3.4 corresponding to initial conditions u0 , v0 ∈ L respectively, satisfying

n(ρ−1) ,∞) 2γ

lim Gγ (t)(u0 − v0 )( n(ρ−1) ,∞) = 0.

t→∞

2γ

Then lim u(t) − v(t)( n(ρ−1) ,∞) = 0.

t→∞

2γ

Furthermore, if we assume u and v are solutions of problem (1.1)-(1.2) given by Theorem 3.5 (i.e., in Eq ) corresponding to initial conditions u0 and v0 (

∈L

n(ρ−1) ,∞) 2γ

satisfying α

lim t 2 Gγ (t)(u0 − v0 )(q,∞) = 0,

t→∞

then

α

lim t 2 u(t) − v(t)(q,∞) = 0.

t→∞

The next corollary proves that mild solutions, with small initial data in n(ρ−1)

the Lebesgue space L 2γ , have a simple long-time diffusive behavior since all the solutions decay to 0 as t → ∞ ( see Corollary 5.2). n(ρ−1)

Corollary 5.2. Let u0 ∈ L 2γ (or more generally u0 ∈ L(p,∞) ∩ L(l,∞) with l = n(ρ−1) 2γ ) be as in the version in Lebesgue space of Theorem 3.5 (see Remark 3.7). Then the corresponding solution satisfies lim u(t)

t→∞

L

n(ρ−1) 2γ

= 0.

As a consequence, the unique self-similar solution in L is the null solution.

n(ρ−1) 2γ

with small norm

Remark 5.3. One of the key points in the proof of the previous corollary is that the intersection between any two strong Lp -spaces is dense in any of them. However, this property is not true for L(p,∞) (see Proposition 2.1). This fact is important because it does not allow limt→∞ u(t)( n(ρ−1) ,∞) = 2γ

0 (see the proof of Corollary 5.2 below). If the previous assertion were true, then no nontrivial self-similar solution of (1.1)-(1.2) would exist in

Self-similar solutions (

n(ρ−1)

1367

,∞)

L 2γ . This is another reason why L(p,∞) -spaces are more suitable to study self-similar solutions. Remark 5.4. [Self-similarity persistence] Let us remark that asymptotic stability applied to the particular case in which one solution is self-similar implies the existence of a bassin of attraction for each self-similar solution, which is a consequence of the fact that the linear part of the solutions has to match for large times. This complicated dynamics was already shown for the three-dimensional Navier-Stokes equations in [5]. Remark 5.5. [Generalizations] Under the hypotheses of Theorem 5.1 and assuming that lim Gγ (t)(u0 − v0 )(p,∞) = 0, t→∞

the asymptotic stability can be also obtained in the norm ·(p,∞) . (

n(ρ−1)

Taking v0 = 0, as a consequence of the last remark, if u0 ∈ Lp ∩ L 2γ then limt→∞ u(t)Lp = 0. It is a decay that does not follow from Theorem 4.1. 5.1. Proof of Theorem 5.1. Taking the norm .( n(ρ−1) ,∞) of the difference 2γ

between two mild solutions u and v, and splitting it in three parts, we obtain u(t) − v(t)( n(ρ−1) ,∞) ≤ Gγ (t)(u0 − v0 )( n(ρ−1) ,∞) 2γ 2γ

 δt



+

Gγ (t − s)(f (u) − f (v))ds ( n(ρ−1) ,∞) 2γ 0

 t



+

Gγ (t − s)(f (u) − f (v))ds ( n(ρ−1) ,∞) = I0 + I1 + I2 , 2γ

δt

where the small constant δ will be chosen later. Making the simple change of variables s = tz, we estimate I1 as  δt   ρ−1 I1 ≤ C (t − s)−1 u(s)ρ−1 + v(s) n(ρ−1) n(ρ−1) (

0

2γ

(

,∞)

2γ

,∞)

× u(s) − v(s)( n(ρ−1) ,∞) ds 2γ  δ ≤ 2ρ ερ−1 C (1 − z)−1 u(tz) − v(tz)( n(ρ−1) ,∞) dz. 2γ

0

Using Lemma 3.11 we bound I2 by  I2 ≤ K sup u(s)ρ−1 n(ρ−1) δt≤s≤t

(

2γ

,∞)

+ v(s)ρ−1 n(ρ−1) (

2γ

 ,∞)

(5.1)

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L.C.F. Ferreira and E.J. Villamizar-Roa

×



sup u(s) − v(s)( n(ρ−1) ,∞) 2γ

δt≤s≤t ρ ρ−1

=2 ε



K sup u(s) − v(s)( n(ρ−1) ,∞) . 2γ

δt≤s≤t

Therefore, we get u(t) − v(t)( n(ρ−1) ,∞) ≤ Gγ (t)(u0 − v0 )( n(ρ−1) ,∞) 2γ 2γ  δ + 2ρ ερ−1 C (1 − s)−1 u(ts) − v(ts)( n(ρ−1) ,∞) ds 2γ

0

ρ ρ−1

+2 ε

K sup u(s) − v(s)( n(ρ−1) ,∞) ,

(5.2)

2γ

δt≤s≤t

for all t > 0. Next, we define Γ = lim sup u(t) − v(t)( n(ρ−1) ,∞) = t→∞

2γ

sup u(t) − v(t)( n(ρ−1) ,∞) .

lim

k∈N ,k→∞ t≥k

2γ

We claim that Γ = 0. By the Lebesgue dominated convergence theorem, we obtain  δ  1  lim sup (1 − s)−1 u(ts) − v(ts)( n(ρ−1) ,∞) ds ≤ Γ log . 2γ 1−δ t→∞ 0 As sup sup u(s) − v(s)( n(ρ−1) ,∞) ≤ t≥k δt≤s≤t

2γ

sup u(s) − v(s)( n(ρ−1) ,∞) ,

δk≤s 0 small enough and conclude that Γ = 0. We omit the proof of the second part of Theorem 5.1 , because we can obtain its proof using similar arguments and the estimates in the regularizing α norm t 2  · (q,∞) , which have been proved in Section 3. 

Self-similar solutions

1369

5.1.1. Proof of Corollary 5.2. Remark 5.5 says that Theorem 5.1 holds when we consider the strong version of the functional space Eq (i.e the version in the Lebesgue space of Eq ). We will prove that lim Gγ (t)u0 

n(ρ−1) L 2γ

t→∞

= 0, when u0 ∈ L

n(ρ−1) 2γ

.

Thus, we will obtain the desired result after letting v0 = 0 in Theorem 5.1. In order to prove the claim, we take a sequence u0,k ∈ Lp ∩ L 1 < p < n(ρ−1) 2γ . By Lemma 3.2, we have Gγ (t)u0,k 

n(ρ−1) L 2γ

2γ n 1 − 2γ ( p − n(ρ−1) )

= Ct

n(ρ−1) 2γ

, with

u0 Lp → 0, n(ρ−1)

n(ρ−1)

when t → ∞. Now, since the intersection Lp ∩ L 2γ is dense in L 2γ , we conclude this part of the corollary. In the more general case u0 ∈ L(p,∞) ∩ L(l,∞) with l = n(ρ−1) 2γ , the proof is analogous. Assume now that u(t) ∈ BC((0, ∞); L u(t, x) = uλ (t, x) = λ −

n(ρ−1) 2γ 2γ ρ−1

) is a self-similar solution; i.e,

u(λx, λ2γ t).

(5.3)

1

By substituting λ = t 2γ into (5.3), we get u(t, x) = t Hence, taking the limit t → ∞ we obtain 0 = lim u(t) t→∞

n(ρ−1) L 2γ

= lim t t→∞

1 − ρ−1

u(t

1 − 2γ

x, 1) L

n(ρ−1) 2γ

1 − ρ−1

u(t

1 − 2γ

= u(x, 1)

L

−

1

−

x, 1).

n(ρ−1) 2γ

.

1

Therefore, u(x, 1) = 0 almost everywhere, and u(t, x) = t ρ−1 u(t 2γ x, 1) = 0.  Acknowledgements. The authors are grateful to Professor J. A. Carrillo for his comments. References [1] O.A. Barraza, Self-similar solutions in weak Lp −spaces of the Navier-Stokes equation, Rev. Mat. Iberoamericana, 12 (1996), 411–439. [2] C. Bennett and R. Sharpley, “Interpolation of Operators,” Academic Press, Pure and Applied Mathematics, 129 (1988). [3] J. Bergh and J. Lofstrom, “Interpolation Spaces,” Springer, Berlin-Heidelberg-New York, (1976). [4] H. Brezis and T. Cazenave, A nonlnear heat equation with sngular initial data, J. d’ Anal. Math, 68 (1996), 277–304. [5] M. Cannone and G. Karch, About the regularized Navier-Stokes equations, J. Math. Fluid Mech, 7 (2005), 1–28.

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[6] T. Cazenave and F.B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schr¨ odinger and heat equations, Math. Z., 228 (1998), 83–120. [7] L.C.F. Ferreira, Solu¸c˜ oes Auto-Similares para a Equa¸c˜ ao Quase-Geostr´ ofica e Comportamento Assint´ otico, Ph.D. Thesis Universidade Estadual de Campinas, Brazil, (2005). [8] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186–212. [9] Y. Giga and T. Kambe, Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Comm. Math. Phys., 117 (1988), 549– 568. [10] Y. Giga and T. Miyakawa, Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces, Comm. Partial Diff. Eq., (14) (1989), 577–618. [11] R. Hunt, On L(p, q) spaces, L’Enseignement Math´ematique, t., 12 (1966), 249–276. [12] T. Kato, Strong Lp solutions of the Navier-Stokes equations in the Rm with applications, Math. Z., 187 (1984), 471–480. [13] T. Kato, Strong solutions of the Navier-Stokes equations in Morrey Spaces, Bol. Soc. Bras. Mat., 22 (1992), 127–155. [14] P. Lemari´e-Rieusset, “Recent Developments in the Navier-Stokes Problem,” Chapman & Hall/ CRC Press, Boca Raton, (2002). [15] Y. Meyer, “Wavelets, Paraproducts and Navier-Stokes Equations,” Current developments in Mathematics (1996), International Press, 105–212 Cambridge, MA 022382872 (1999). [16] W. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc., 287 (1985), 657–671. [17] R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J., 30 (1963), 126– 142. [18] E.M. Stein and G. Weiss, “Introduction to Fourier Analysis on Euclidean Spaces,” Princeton University Press, Princeton, N.J. (1971). [19] F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in Lp , Indiana Univ. Math. J., 29 (1980), 79–102. [20] F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29–40. [21] F.B. Weissler, Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation, Arch. Rational Mech. Anal., 91 (1985), 231–245. [22] M. Yamazaki, The Navier-Stokes equations in the weak-Ln spaces with time-dependent external force, Math. Ann., 317 (2000), 635–675.