On the well-posedness for the nonlinear radial Schrödinger equation with spatial variable coefficients

Nonlinear Analysis 182 (2019) 1–19

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On the well-posedness for the nonlinear radial Schrödinger equation with spatial variable coefficients Bo-wen Zheng College of Sciences, China Jiliang University, Hangzhou 310018, China

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abstract In this paper, we study the Cauchy problem for a generalized nonlinear Schrödinger equation with spatial variable coefficients. A new weighted Sobolev space W 1,q (R+ ) is introduced and the range of admissible pairs for the Strichartz estimates is improved. As an application, we show the local well-posedness result for the initial data in W 1,q (R+ ). © 2018 Elsevier Ltd. All rights reserved.

Article history: Received 17 August 2018 Accepted 28 November 2018 Communicated by Enzo Mitidieri MSC: 35B65 35Q40 35Q55 Keywords: Schrödinger equation with spatial variable coefficients Strichartz estimate Weighted Sobolev space Well-posedness

1. Introduction This paper is devoted to the Cauchy problem for the generalized nonlinear Schr¨odinger equation with spatial variable coefficients: ∫ r b d c −bκ i∂t v + Aµ v = λ1 r |v| v + λ2 |v| (r′ )−bκ−1 |v| dr′ , (1.1) 0 + v(r, 0) = v0 (r), (r, t) ∈ R × R, where v : R+ × R → C, r = |x|, (x ∈ Rn ) is the radius, κ := Aµ := −rp0 (∂rr +

2−p0 2 µ

p1 p2 ∂r − 2 ), r r

+

1−p1 2

and the operator

µ ≥ 0,

with p0 , p2 satisfy the assumptions p0 < 2, E-mail address: [email protected]. https://doi.org/10.1016/j.na.2018.11.016 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

p2 := (

2 − p0 2 p1 − 1 2 µ) − ( ) . 2 2

(1.2)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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The nonlinear term is denoted as g(r, v) := g˜1 (r, v) + g˜2 (r, v) for simplicity, and the indexes λ1 , λ2 ∈ R, b, c ≥ 1 satisfy b − d = c − 1. The elliptic operator Aµ = −rp0 (∂rr + pr1 ∂r − pr22 ) appearing in (1.1) plays a key role in many problems of physics and quantum mechanics. Schr¨ odinger type equations with variable coefficients have been of considerable interest among both mathematicians and physicists, and some remarkable progress on the well-posedness of the Cauchy problem has been made (see [9,11,14,15,23,26,28,31] and references therein). The mathematical interest in (1.1) comes mainly from the spatial variable coefficient rp0 , which arises in a model for the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system (HFSS) ⃗t (r, t) = ρ(r)S ⃗ × [S ⃗rr + n − 1 S ⃗r ] + ρr (r)[S ⃗ ×S ⃗r ], S (1.3) r ⃗ take values in S2 ⊂ R3 and × denotes the cross product in R3 , ρ(r) is a scalar function where the spin S and r = |x|. By a known geometrical process [9,25], the spin evolution equation (1.3) is equivalent to the following generalized nonlinear Schr¨ odinger equation n−1 n−1 2 ivt + ρ(vrr + vr − v + 2|v| v) + 2ρr vr 2 r ∫ rr ∫ r (1.4) n−1 ρ 2 ′ 2 ′ + [ρrr + ρr + 2 |v| dr ]v = 0, ρr′ |v| dr + 4(n − 1) ′ r 0 0 r and the integrability of (1.3) holds for the conditions ρ(r) = ϵ1 r−2(n−1) + ϵ2 r−(n−2) ,

(1.5)

where ϵ1 , ϵ2 are arbitrary constants. Obviously, the factor rp0 in the elliptic operator Aµ corresponds to the term ρ(r) in (1.4). Noticing that Eq. (1.1) includes (1.3) in the integrable case of (1.5) with the inhomogeneity ρ(r) = rp0 : b = c = 2. An essential prerequisite for the modern approach to the well-posedness of nonlinear Schr¨odinger equation is the development of Strichartz estimates. For the case p0 = 0, Eq. (1.1) reduces to the classical Schr¨odinger equation with the inverse-square potential under the assumption of the spherical symmetry: a i∂t u(x, t) − ∆u(x, t) + 2 u(x, t) = f (u), |x| (1.6) u(x, 0) = u0 (x), (x, t) ∈ Rn × R, of which the full range of Strichartz estimates were achieved by Burq, Planchon, Stalker and Tahvildar-Zadeh 2 in their papers [2,3] when a > − (n−2) . While the method of proof above is not applicable to the critical 4 2

case a = − (n−2) , Suzuki [27] and Mizutani [20] recently provided a complete answer to the Strichartz 4 estimates for all admissible pairs by splitting the solution into radial and non-radial parts in the framework of Lorentz spaces. Using these Strichartz inequalities, the local and global well-posedness theory for (1.6) has been extensively studied by many authors in recent years, see e.g. [8,10,16–18,21,27,33]. For the case p0 ̸= 0, as far as we know, there is few literature on the well-posedness of such type of Schr¨ odinger equation (1.1), which is a generalized version of (1.4), including (1.3) in the integrable case (1.5) with ρ(r) = rp0 : b = c = 2. Naturally, like (1.6), the broad goal of demonstrating well-posedness of (1.1) rests on a thorough understanding of such Strichartz estimates problems. However, in contrast to (1.6), the presence of the factor rp0 in Aµ prevents us from using the Fourier transform method to get explicit solution formula, which brings some difficulties to study the well-posedness of (1.1). The main purpose of this paper is to establish the well-posedness of the Cauchy problem (1.1) for initial data v0 ∈ W 1,q (R+ ). Indeed, we try to construct a closed subspace of L∞ ([0, T ]; W 1,q (R+ )) such that the operator defined by ∫ t

T(v)(t) = U(t)v0 (r) − i

U(t − τ )g(r, v(τ ))dτ, 0

(1.7)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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where U(t) denotes the solution to the linear problem i∂t v + Aµ v = 0, with initial data v0 , is stable and contractive in this space. Inspired by the results in [24,32] and [31], who recently obtained the Strichartz type estimates of (1.1) in some weighted Lebesgue spaces, we adapt a suitable Hankel transform (see Section 2) and employ some fine properties of the Hankel transform to improve the range of the admissible pairs in [31]. Based on it, Strichartz type estimates associated to the linear problem of (1.1) can be established, which are indispensable in the treatment of well-posedness in W 1,q (R+ ) intersected with an appropriate Strichartz space. To state the main results precisely, we need some notations. We say that the triplet (m, l, q) is an Lqκ,σ γq admissible if 1 < q ≤ l < γ−1 and satisfy 1 1 1 = γ( − ), (1.8) m q l where γ :=

2κ+p1 −p0 +1 . 2−p0

Given any admissible triplets (m, l, q), we define the following Strichartz space X0 (I) = L∞ (I; Lqκ,σ (R+ )) ∩ Lm (I; Llκ,σ (R+ )), equipped with the norm ∥v∥X0 (I) = sup ∥v∥Lh (I;Lpκ,σ (R+ )) , Λ = {(∞, q), (m, l)},

(1.9)

(h,p)∈Λ

and the dual Strichartz norm ∥v∥X ′ (I) = 0

sup (h,p)∈Λ0

∥v∥Lh (I;Lpκ,σ (R+ )) , Λ0 = (

m ql , ). b bq + l

Note that, for 1 ≤ p, h ≤ ∞, the norm of the weighted Lebesgue space Lpκ,σ (R+ ) and space–time space Lh (I; Lpκ,σ ) of function v are defined as ∫ 1 p p ∥f ∥Lκ,σ (R+ ) = ( |f (r)| r−κp dσr ) p < ∞, + ∫R 1 ∥v∥Lh (I;Lpκ,σ ) = ( ∥v∥hLp dt) h , I

κ,σ

with a usual modification when p or h is infinity, where dσr = r2κ+p1 −p0 dr is the Lebesgue measure. For ∫ ∫ 1 1 p p simplicity, ∥f ∥Lpσ (R+ ) = ( R+ |f (r)| dσr ) p and ∥f ∥Lp (R+ ) = ( R+ |f (r)| r−κp dr) p . κ,1 Our first result is mainly devoted to the Strichartz type estimates for Eq. (1.1). bq Theorem 1.1. Let (m, l, q) be any admissible triplets satisfying q ≥ bγ and l ≥ max{ q−1 , q}, the following statements hold.

(i) (Linear estimate) ∥U(t)φ∥X0 (I) ≤ C(µ, l, q)∥φ∥Lqκ,σ ,

(1.10)

(ii) (Inhomogeneous estimate) ∫ ∥ 0

t

U(t − τ )g(r, v(τ ))dτ ∥X0 (I) ≤ CT 1−

bγ q

∥g∥X ′ (I) , 0

where I = [0, T ], 0 < T ≤ ∞ and C := C(µ, l, q, m, b) is constant depending on µ, l, q, m, b. As an application, we obtain the local well-posedness of (1.1) in Lqκ,σ (R+ ) for d ≥ 1.

(1.11)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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(d−1)ql κ + 2κ + p1 − p0 + 1. Assume (m, l, q) is an arbitrary admissible triplet Theorem 1.2. Let n = (b−d)q+l bq satisfying q ≥ bγ, l ≥ max{ q−1 , q} and

(

(d − 1)ql + 2)κ ≥ p0 − p1 , (b − d)q + l

which leads to n ≥ 1. For v0 ∈ Lqκ,σ (R+ ), (i) there exist T > 0 and a unique solution v ∈ X0 ([0, T ]) to Eq. (1.1), where T = T (∥v0 ∥Lqκ,σ ) depends on ∥v0 ∥Lqκ,σ . (ii) Let I = [0, T ∗ ) be the maximal existence interval of the solution v to Eq. (1.1) such that v ∈ L∞ ([0, T ∗ ); Lqκ,σ (R+ )) ∩ Lm ([0, T ∗ ); Llκ,σ (R+ )) for q > bγ. Then ∥v(t)∥Lqκ,σ ≥

C (T ∗

1

γ

− t) b − q

,

(1.12)

where C depends on µ, q, l, m, b, n, λ1 , λ2 and ε > 0. It is worth mentioning that the theorem is an extension of a result by the authors [31], which asserts local well-posedness for Eq. (1.1) with λ2 = 0. Next, we treat the local well-posedness in a new weighted Sobolev space W 1,p (R+ ), which is defined as follows: Definition 1.3. For 1 ≤ p ≤ ∞, we define the weighted Sobolev space W 1,p (R+ ) by W 1,p (R+ ) = {u ∈ Lpκ,σ (R+ ) : Dr u ∈ Lp (R+ )}, κ,˜ σ endowed with the norm ∥u∥W 1,p (R+ ) = ∥u∥Lpκ,σ + ∥Dr u∥Lp , κ,˜ σ σr = r2κ+3p1 −2p0 dr is the Lebesgue measure. where the differential operator Dr := rp0 −p1 ∂r and d˜ The norm for the dual space (W 1,p )∗ of the weighted Sobolev space W 1,p (R+ ) is also defined by ∥f ∥(W 1,p )∗ :=

sup u∈W 1,p \θ

|⟨f, u⟩(W 1,p )∗ ,W 1,p | , ∥u∥W 1,p

f ∈ (W 1,p )∗ .

The main result in this paper is the following local well-posedness of (1.1) in W 1,q (R+ ) for d = 1. Theorem 1.4. Let p2 = 0, p0 = 2p1 ≤ 43 and n = 2κ + 1 − p20 . Assume (m, l, q) is an arbitrary admissible bq triplet satisfying q ≥ bγ and l ≥ max{ q−1 , q}. Then for v0 ∈ W 1,q (R+ ), there exist T = T (∥v0 ∥W 1,q ) > 0 and a unique solution v of Eq. (1.1) with v ∈ X(I) := L∞ (I; W 1,q (R+ )) ∩ Lm (I; W 1,l (R+ )), for the case d = 1, where X(I) is equipped with the norm ∥v∥X(I) = ∥v∥X0 (I) + ∥Dr v∥X0 (I) .

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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We remark that, to study the behavior of solution to (1.1) when v0 lies in W 1,q (R+ ), we need an assumption on the structure of nonlinearity with d = 1, which is weaker than Theorem 1.2. In the forthcoming paper [30], the authors proved the global existence for the integral equation (1.4) (i.e. ρ(r) = r2−n ) in one dimension with small initial data in weighted Sobolev space. Here we extend their result to any ρ(r) = rp0 , p0 ≤ 43 . Our proof of Theorem 1.4 mainly depends on the Hankel transform and the Strichartz estimate. Firstly, we employ Lqκ,σ (R+ )-solutions constructed in Theorem 1.2 as part of the weighted W 1,q (R+ )-solutions. Next, by using the first order partial differential operator Dr , which has special commutation relations with i∂t + Aµ , we derive a similar type of equation as (1.1) restricted to p2 = 0. This method was first applied by Klainerman [19] to prove the global existence theorem to the nonlinear wave equations with quadratic nonlinearities, and has been exploited in [6,12,22]. Finally, we obtain the uniform estimates that the spatial derivative of solution should satisfy on a basis of Hankel transform, which enables us to establish the well-posedness of Cauchy problem (1.1). The paper is organized as follows. Section 2 is devoted to some preliminaries, including the theory of Hankel transform associated with Aµ and Hankel convolution operator estimate. In Section 3, a detailed analysis of the kernel function implies the Strichartz estimates and consequently leads to Theorem 1.2. We show Theorem 1.4 in Section 4. If not specified, throughout this paper, we denote ≤α as ≤ C(α), where C(α) denotes various constants that only depend on α. 2. Preliminaries In this section, we present preliminary lemmas which will be used in the later sections. For the operator Aµ = −rp0 (∂rr +

p2 p1 ∂r − 2 ), r r

we introduce the following Hankel transform [32]: Definition 2.1. Suppose f (r) is an integrable function in R+ , we define the Hankel transform Hµ of f (r) as follows: ∫ 1−p1 2−p0 2 (2.1) Hµ [f (r)](λ) := B(λr)f (r)r−2κ dσr , B(z) := z 2 Jµ ( z 2 ), 2 − p0 R+ √ (p1 −1)2 +4p2 where Jµ (z) is the first Bessel function of order µ = defined as 2−p0 Jµ (r) =

(r/2)µ Γ (µ + 12 )π 1/2

∫

1

1

eiry (1 − y 2 )µ− 2 dy.

−1

For the Hankel transform (2.1), one can see that both the inversion theorem and Plancherel’s formula hold, which are stated as follows: Lemma 2.2 ([32]). The Hankel transform Hµ satisfies: (i) Hµ [Aµ ϕ](λ) = λ2−p0 Hµ [ϕ](λ), (ii) Hµ is an L2 isometry, i.e. ∥Hµ ϕ∥L2κ,σ (R+ ) = ∥ϕ∥L2κ,σ (R+ ) , (iii) Hµ = Hµ−1 , where the operator Hµ−1 is the inverse operator of Hµ . Based on the Hankel transform (2.1), we will establish an analogous theory associated with certain convolutions defined for functions on R+ .

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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If f (r) and g(r) are measurable functions on R+ , the generalized Hankel convolution f ♯g of f and g is defined by ∫ (f ♯g)(r) = f (s)(τ s g)(r)s−2κ dσs , (2.2) R+ s

where the Hankel translation τ f of f is (τ s f )(r) =

∫

s, r ∈ R+ ,

f (z)dWr,s (z), R+

and the measure dWr,s (z) supported on [|r dWr,s (z) = (

2−p0 2

−s

2−p0 2

|, r

2−p0 2

+s

2−p0 2

(2.3)

] is given by

2 − p0 1−µ △2µ+1 (rsz)1−p1 −κ z −2κ dσz , ) 4 Γ (µ + 21 )π 1/2 2−p0

where △ denotes the area of a triangle with sides r 2 , s is zero. It is easy to check that τ s satisfies ∫ ∫ (τ s f )(r)g(r)r−2κ dσr = R+

2−p0 2

,z

2−p0 2

if there is such a triangle and otherwise

(τ s g)(r)f (r)r−2κ dσr ,

(2.4)

R+

which is quite straightforward for the measure Wr,s (z)dσr is symmetric with respect to r and z, i.e. dWr,s (z)dσr = dWz,s (r)dσz . The ♯-convolution for Hankel transform was first investigated by Cholewinski [7] and Hirschman [13]. The Hankel transform (2.1) behaves in regard to the convolution “♯” exactly as does the usual Fourier transform in regard to ordinary convolution “ ∗ ” on R+ . More precisely, we have the following basic formula. Lemma 2.3.

Assume f (r), g(r) are integrable functions on R+ , then Hµ [(f ♯g)](ρ) = ρ−κ Hµ [f ](ρ)Hµ [g](ρ).

(2.5)

Proof . Before proceeding to the proof of (2.5), we claim that τ s [B(ρz)](r) = ρ−κ B(ρr)B(ρs). In fact, by hypothesis κ =

2−p0 2 µ

1−p1 2 ,

one can see that ∫ Hµ [ρ−κ B(ρr)B(ρs)](z) = B(ρz)B(ρr)B(ρs)ρ−3κ dσρ R+ ∫ 1−p1 2 − p0 1−µ = (¯ rs¯z¯) 2−p0 Jµ (¯ ρz¯)Jµ (¯ ρr¯)Jµ (¯ ρs¯)( ρ¯) d¯ ρ 2 R+ 2 − p0 1−µ △2µ+1 =( ) (rsz)1−p1 −κ χ[|¯r−¯s|, r¯+¯s] , 4 Γ (µ + 12 )π 1/2 2−p0

+

(2.6)

2−p0

(2.7)

2 where ρ¯ = 2−p ρ 2 , y¯ = y 2 , y = r, s, z, and the last equality holds by using the integral formula for 0 Bessel function in [29]: ∫ 2µ−1 △(a, b, c)2µ−1 Jµ (at)Jµ (bt)Jµ (ct)t1−µ dt = χ[|a−b|, a+b] . Γ (µ + 12 )π 1/2 (abc)µ R+

Applying Lemma 2.2 (iii) to (2.7), we obtain ∫ ρ−κ B(ρr)B(ρs) = B(ρz)dWr,s (z) = τ s [B(ρz)](r), R+

which yields the claim (2.6).

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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Next, from (2.2)–(2.3), by changing the order of integration, we have ∫ ∫ −2κ Hµ [(f ♯g)](ρ) = f (s)s ( (τ s g)(r)B(ρr)r−2κ dσr )dσs + + R R ∫ ∫ = f (s)s−2κ ( τ s [B(ρ·)](r)g(r)r−2κ dσr )dσs , R+

R+

where the last equality is a direct consequence of (2.4). Using the result (2.6), we can easily verify that ∫ ∫ Hµ [(f ♯g)](ρ) = ρ−κ ( f (s)B(ρs)s−2κ dσs )( g(r)B(ρr)r−2κ dσr ) R+

= ρ−κ Hµ [f ](ρ)Hµ [g](ρ),

R+

which is the desired result. □ By taking into account the behavior of the Hankel transform on weighted Ll -space, it is easy to see that the ♯-convolution defines a bilinear bounded mapping from Lhκ,σ (R+ ) × Lqκ,σ (R+ ) into Llκ,σ (R+ ), which is the following result: Lemma 2.4.

Let 1 ≤ l, h, q < ∞ and 1 +

1 l

= ∫∫

1 h

(f ♯g)(r) =

+ 1q . If f ∈ Llκ,σ (R+ ), g ∈ Lqκ,σ (R+ ), then the integral

f (z)g(s)s−2κ dσs dWr,s (z)

R+

converges for almost all r and satisfies ∥(f ♯g)(r)∥Llκ,σ ≤ Γ (µ + 1)−1 (2 − p0 )−µ ∥f (z)∥Lhκ,σ ∥g(s)∥Lqκ,σ .

(2.8)

Proof . Suppose q ′ , h′ satisfy the relation 1 1 1 + + ′ = 1, q′ l h

h h + ′ = 1, l q

q q + ′ = 1, l h

by using the H¨ older inequality, we have ∫ g(s) q g(s) q (τ s f )(r) h (τ s f )(r) qh′ |(f ♯g)(r)| = | | κ | h′ {| κ | l | | l }| | dσs | s s sκ sκ R+ ∫ h q 1 g(s) (τ s f )(r) h ′ ′ ≤ ∥g(s)∥Lhq ∥(τ s f )(r)∥Lq h ( | κ |q | | dσs ) l κ κ,σ κ,σ s s R+ := I1 · I2 · I3 .

(2.9)

˜r,s (z)z −2κ dσz in this argument for convenience, then the Jensen’s inequality We rewrite dWr,s (z) = W leads to ∫ ∫ ˜r,s (z) f (z) W h q′ (I2 ) = | (rs)κ z κ+p1 −p0 dz| s−hκ dσs κ (rs)κ R+ R+ z ∫ ∫ ˜ f (z) h Wr,s (z) −κ ≤ C1h−1 ( z dσz )s−hκ dσs | κ (rs)κ | z (rs)κ R+ R+ ∫ ∫ f (z) h κ(h−1) h−1 ˜r,s (z)s−κ dσs )z −κ dσz = C1 | κ | r ( W z R+ R+ ∫ f (z) h = C1h rκh | κ | dσz , (2.10) z + R

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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where C1 := Γ (µ + 1)−1 (2 − p0 )−µ and the last equality is obtained by the following assertion: ∫ (rz)κ ˜r,s (z)dσs = . s−κ W Γ (µ + 1)(2 − p0 )µ R+ Indeed, by multiplying ρ−κ to (2.6) in Lemma 2.3, we obtain ∫ ˜r,s (z)z −2κ dσz = ρ−2κ B(ρr)B(ρs). ρ−κ B(ρz)W

(2.11)

(2.12)

R+

Since limρ→0 ρ−κ B(ρz) =

zκ Γ (µ+1)(2−p0 )µ ,

let ρ → 0 on both sides of (2.12), then the relation (2.11) follows

easily. It is obvious to check that the assertion (2.11) also holds for replacing the variable s with r or z. Hence, combining (2.9) with (2.10), we have ∫ ∫ q g(s) q (τ s f )(r) h ′ l ∥(f ♯g)∥Llκ,σ ≤ ∥g(s)∥Lhq |I2 | r−κl ( | κ | | | dσs )dσr κ,σ s sκ R+ R+ lh ′

h ′

q ′

≤ C1q ∥g(s)∥Lhq ∥f (r)∥Lq h κ,σ ∫ κ,σ ∫ h g(s) q −κh h κl( −1) ( |(τ s f )(r)| r q′ dσr )dσs , × | κ | s s R+ R+ 1 h where (2.13) can be bounded by ( Γ (µ+1)(2−p µ) 0)

∥(f ♯g)∥lLl

κ,σ

≤ C1l (

∫ | R+

∫

q

R+

| g(s) sκ | dσs

∫

R+

(2.13)

h

| fz(z) κ | dσz , then

∫ l l f (z) h g(s) q q( | dσ ) | κ | dσz ) h , s sκ z + R

which is the desired result. □ We next recall the Marcinkiewicz interpolation theorem due to [1], which will be useful in our work. Lemma 2.5. Let (U, ν) and (V, ω) be measure spaces and let q0 ̸= q1 . Assume that there exist two positive constants M0 and M1 such that ∥T (f )∥Ll0 ,∞ (V ) ≤ M0 ∥f ∥Lq0 (U )

for all f ∈ Lq0 (U ),

∥T (f )∥Ll1 ,∞ (V ) ≤ M1 ∥f ∥Lq1 (U )

for all f ∈ Lq1 (U ),

Put

1 1−θ θ = + , q q0 q1

1 1−θ θ = + , l l0 l1

and assume q ≤ l. Then ∥T (f )∥Ll (V ) ≤ M ∥f ∥Lq (U ) , with M satisfying M ≤ Cθ M01−θ M1θ . 3. Local well-posedness in Lqκ,σ Applying the Hankel transform (2.1) to Eq. (1.1), we have { i∂t (Hµ v)(λ, t) + λ2−p0 (Hµ v)(λ, t) = (Hµ g)(λ, t) (Hµ v)(λ, 0) = (Hµ v0 )(λ), where g(r, v) := g˜1 (r, v) + g˜2 (r, v).

(2.14)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

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Solving the ODE above and inverting the Hankel transform with Lemma 2.2, the solution of (1.1) can be given by ∫ t 2−p0 2−p0 v(r, t) = Hµ [eitλ (Hµ v0 )(λ)](r) − Hµ [i ei(t−τ )λ (Hµ g)(λ, τ )dτ ](r). 0

Using Lemma 2.3, the solution can be expressed in terms of ♯-convolution ∫ t v(r, t) = Kµ (r, t)♯v0 (r) − i Kµ (r, t − τ )♯g(r, v(τ ))dτ, 0 κ itλ2−p0

and the kernel Kµ (r, t) := Hµ [λ e

](r).

We define the solution semigroup U(t) := eitAµ by U(t)φ = Kµ (r, t)♯φ, then the solution of (1.1) can be written in the following integral form ∫ t v(r, t) = U(t)v0 (r) − i U(t − τ )g(r, v(τ ))dτ.

(3.1)

0

3.1. Strichartz estimate In this section, we shall prove Theorem 1.1. The key ingredients are to exploit weighted Ll − Lq time decay estimates of the fundamental solution to (1.1), which is of the following form: Lemma 3.1.

Assume 1 ≤ q ≤ l ≤ ∞, then U(t)φ satisfies 1

1

∥U(t)φ∥Llκ,σ ≤ C(µ, l, q)tγ( l − q ) ∥φ∥Lqκ,σ ,

(3.2)

where C(µ, l, q) is a constant only depending on µ, l, q. Proof . Since U(t)φ = Kµ (r, t)♯φ, using Lemma 2.4 with 1 +

1 l

=

1 h

+ 1q , we have

∥U(t)φ∥Llκ,σ ≤ Γ (µ + 1)−1 (2 − p0 )−µ ∥Kµ (r, t)∥Lhκ,σ ∥φ∥Lqκ,σ . 2−p0 2

˜=λ By changing the variable λ ∫ Kµ (r, t) =

(3.3)

, we get 2−p0

B(λr)λκ eitλ dωλ ∫ 1−p1 2−p0 2 2 ˜ λ ˜ µ+1 eitλ˜ 2 dλ, ˜ = r 2 Jµ ( r 2 λ) 2 − p0 2 − p0 R+ −1 r2−p0 =( )µ+1 rκ exp( ), (2 − p0 )it (2 − p0 )2 it R+

where the last equality is obtained by means of the identity in ([29], chapter 13): ∫ 2 2 aµ −a 4p . Jµ (at)e−pt tµ+1 dt = e (2p)µ+1 R+ It follows that ∥Kµ (r, t)∥Lhκ,σ = (2 − p0 )

4κ+2p1 −p0 −µ−1 (2−p0 )h

Therefore, ∥U(t)φ∥Llκ,σ ≤ C(µ, l, q)t 1

1

γ −h

where C(µ, l, q) = [(2 − p0 )2µ+1 Γ (µ + 1)] l − q h

γ

2κ+p1 −p0 +1 1 1 ( l −q) 2−p0

. □

1

1

h− h Γ (µ + 1) h tγ( h −1) . ∥φ∥Lqκ,σ ,

(3.4)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

10

As a consequence of Lemma 3.1, we give the proof of Theorem 1.1. Proof of Theorem 1.1. (i) From the definition (1.9) ∥v∥X0 (I) = sup ∥v∥Lh (I;Lpκ,σ (R+ )) , Λ = {(∞, q), (m, l)}, (h,p)∈Λ

it suffices to prove the bound of ∥v∥Lh (I;Lpκ,σ (R+ )) with Λ = {(∞, q), (m, l)}. When Λ = (∞, q), it is a direct consequence of Lemma 3.1. When Λ = (m, l), since l ≥ q from the definition of admissible triplet, together with Lemma 3.1, it suffices to consider the case l > q. We first define the operator T : Lqκ,σ (R+ ) → Lm (I) with (T φ)(t) = ∥U(t)φ∥ ˜l . Lκ,σ Assume (m, ˜ ˜ l, q˜) be an admissible triplet, from Lemma 3.1, it is obvious that (T φ)(t) = ∥U(t)φ∥ ˜l ≤ C(µ, ˜ l, q)∥φ∥ ˜l , Lκ,σ Lκ,σ

(3.5)

for q ≤ ˜ l ≤ ∞, which means that T is a (˜ l, ∞) type operator. Since q˜ ≤ ˜ l ≤ ∞, then γ( 1 − 1 )

1

− q ∥φ∥ ˜ ∥φ∥ ˜ , = C(µ, ˜ l, q˜)t m (T φ)(t) ≤ C(µ, ˜ l, q˜)t ˜l ˜ q ˜ q L L κ,σ

κ,σ

which yields that

|{t : |(T φ)| > τ }| ≤ |{t : t < ( ≤(

C(µ, ˜ l, q˜)∥φ∥ ˜ q L

κ,σ

τ

C(µ, ˜ l, q˜)∥φ∥ ˜ q L

κ,σ

τ

˜ }| )m

˜, )m

that is, ∥(T φ)(t)∥ m ≤ C(µ, ˜ l, q˜)∥φ∥ ˜ , q L ˜ ,∞ (I) L

(3.6)

κ,σ

Given any admissible triplet (m, l, q), by invoking Marcinkiewicz interpolation theorem (2.14) between (3.5) and (3.6), we obtain ∥(T φ)(t)∥Lm (I) ≤ C(µ, l, q)∥φ∥Lqκ,σ , (3.7) where θ ∈ [0, 1] and 1 θ 1−θ 1 θ 1−θ = + , = + , ˜ l = l. ˜ q q˜ m m ˜ ∞ l Therefore, the linear estimate (i) is proved. (ii) To prove the inhomogeneous estimate (1.11), we denote ∫ t Θ(g)(r, t) := −i U(t − τ )g(r, v(τ ))dτ. 0

Similar to the argument of (i), we first consider the case Λ = (∞, q):

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

11

bq For l ≥ max{ q−1 , q}, we apply Lemma 3.1 to Θ(g) and use the H¨older inequality with respect to t, then t

∫ ∥Θ(g)∥

≤µ,l,q,b

q L∞ (I;Lκ,σ )

t∈I

≤µ,l,q,b

bq+l

(t − τ )

sup

γ( 1 q − ql ) ∥g∥

0

ql bq+l

dτ

∫ t bq+l 1 )s γ( 1 − sup( (t − τ ) q ql dτ ) s ∥g∥ t∈I

bγ q

m

ql

bq+l L b (I;Lκ,σ )

0

≤µ,l,q,b,m T 1−

(3.8)

Lκ,σ

∥g∥

,

ql

m

bq+l L b (I;Lκ,σ )

b . which holds for the hypothesis q ≥ bγ and 1s = 1 − m Next, we consider the case Λ = (∞, q): We invoke Lemma 3.1 again to get ∫ t bq+l γ( 1 − ) ∥Θ(g)∥Lm (I;Llκ,σ ) ≤µ,l,b ∥ (t − τ ) l ql ∥g∥ ql dτ ∥Lm (I) bq+l

0

≤µ,l,b

∫ (

Lκ,σ

T

τ

γ

(1−b)q−l s ql

1

dτ ) s ∥g∥

≤µ,l,b,m T 1−

m

ql

bq+l L b (I;Lκ,σ )

0 bγ q

∥g∥

,

ql

m

bq+l L b (I;Lκ,σ )

where the last but one inequality is bounded by using usual Young’s inequality with 1 = the proof of Theorem 1.1 is completed. □ b

1 s

+

b−1 m .

Hence,

b

Remark. Let g(r, u) = r−bκ |u| u, where f (u) = |u| u. The complex derivative of f is fu (u) =

b+2 b |u| , 2

fu¯ (u) =

b b−2 2 |u| u . 2

For u, v ∈ C, we have ∫ f (u) − f (v) =

1

[fu (v + t(u − v))(u − v) + fu¯ (v + t(u − v))(u − v)]dt. 0

Thus, b

b

|g(r, u) − g(r, v)| ≤ Cr−bκ (|u| + |v| )|u − v|.

(3.9)

3.2. Well-posedness In this section, we prove the local well-posedness of the solution to (1.1) via the usual Strichartz methodology. We first show the well-posedness in Lqκ,σ (R+ ) (Theorem 1.2) and then take into W 1,q (R+ ) (Theorem 1.4) account in Section 4. We begin with the following proposition. It provides an estimate for the model nonlinearity in the Strichartz space. Proposition 3.2.

Let n =

(d−1)ql (b−d)q+l κ + 2κ + p1

− p0 + 1 ≥ 1. Assume v ∈ X0 (I), then the nonlinear terms

g(r, v) in (1.1) satisfies the following estimate ∥g∥X ′ (I) ≤ C(n, λ1 , λ2 )(∥v∥bLm (I;Ll

κ,σ )

0

+∥v∥L∞ (I;Lqκ,σ ) ∥v∥bLm (I;Ll

∥v∥L∞ (I;Lqκ,σ )

κ,σ )

where the constant C(n, λ1 , λ2 ) only depends on n, λ1 , λ2 .

),

(3.10)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

12

Proof . Note that g(r, v) = g˜1 (r, v) + g˜2 (r, v), we have ∥g∥X ′ (I) ≤ ∥˜ g1 ∥X ′ (I) + ∥˜ g2 ∥X ′ (I) := B1 + B2 , 0

b

d

0

0

∫r

c

where g˜1 (r, v) = λ1 r−bκ |v| v, g˜2 (r, v) = λ2 |v| 0 (r′ )−bκ−1 |v| dr′ . For v ∈ X0 (I), using the H¨ older inequality with respect to r and t, we obtain ∫ ql bq+l ql −(b+1) bq+l κ+2κ+p1 −p0 B1 = |λ1 |∥( r | |v|b v| bq+l dr) ql ∥

m

L b (I)

R+

≤λ1 ∥ ∥v∥bLl ∥v∥Lqκ,σ ∥

m

L b (I)

κ,σ

≤λ1 ∥v∥bLm (I;Ll

κ,σ )

∥v∥L∞ (I;Lqκ,σ ) .

(3.11)

On the other hand, we invoke the H¨ older inequality twice to get ∫ r ∫ bq+l ql ql +2κ+p1 −p0 −κ | |v|d (r′ )−bκ−1 |v|c dr′ | bq+l dr) ql ∥ mb B2 ≤λ2 ∥( r bq+l L (I) 0 R+ ∫ ∫ r ql (b−d)q+l ≤λ2 ∥ ∥v∥dLl ( rn−1 | (r′ )−bκ−1 |v|c dr′ | (b−d)q+l dr) ql ∥ mb L (I) κ,σ + 0 ∫R ql cql (b−d)q+l n−1−bκ (b−d)q+l |v| (b−d)q+l dr) ql ≤λ2 ,n ∥∥v∥dLl ( r ∥ mb , κ,σ

L

R+

where the inequality (3.12) under the assumption n = following Hardy-type inequality [5]:

(d−1)ql (b−d)q+l κ

∥f ∥Lp (Rn ) ≤ C∥r∂r f ∥Lp (Rn ) ,

(I)

+ 2κ + p1 − p0 + 1 is a consequence of the

1 ≤ p < ∞,

(3.13)

∫ p for radial function f (|x|), x ∈ Rn , where ∥f ∥Lp (Rn ) := ( Rn |f (x)| dx)1/p . By hypothesis b − d = c − 1 and H¨ older inequality, we further have ∫ (b−d)q+l (b−d)q 1 q − (2κ+p1 −p0 ) l l B2 ≤λ2 ,n ∥∥v∥bLl ( r−dκq+(n−1) |v| dr) q ∥ κ,σ

(3.12)

m

L b (I)

R+

≤λ2 ,n ∥∥v∥bLl ∥v∥Lqκ,σ ∥

m

L b (I)

κ,σ

≤λ2 ,n ∥v∥bLm (I;Ll

κ,σ )

∥v∥L∞ (I;Lqκ,σ ) ,

which together with (3.11) imply the desired result.

(3.14) □

Our task now is to show Theorem 1.2. Proof of Theorem 1.2. Let T(v)(t) = U(t)v0 + Θ(g)(t), where Θ(v) is the nonlinear term in (3.1). We define Y(I) = {v ∈ X0 (I) : ∥v∥X0 (I) ≤ M }, where I = [0, T ] and M, T are positive constants to be determined later. We follow the standard fixed point argument to prove this result. It means that for appropriate values of T we shall show that T defines a contraction map on Y(I). Since ∥T(v)∥X0 (I) ≤ ∥U(t)v0 ∥X0 (I) + ∥Θ(g)(t)∥X0 (I) , we apply Strichartz inequalities (1.10) and (1.11) to obtain ∥T(v)∥X0 (I) ≤ C(µ, l, q)∥v0 ∥Lqκ,σ + CT 1−

bγ q

∥g∥X ′ (I) , 0

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

13

where C = C(µ, l, q, m, b). Moreover, Proposition 3.2 yields that ∥T(v)∥X0 (I) ≤ C(µ, l, q)∥v0 ∥Lqκ,σ + CT 1−

bγ q

(∥v∥bLm (I;Ll

κ,σ )

+∥v∥L∞ (I;Lqκ,σ ) ∥v∥bLm (I;Ll

κ,σ )

≤ C(µ, l, q)∥v0 ∥Lqκ,σ + CT 1−

bγ q

∥v∥L∞ (I;Lqκ,σ )

)

∥v∥b+1 X0 (I) .

where C = C(µ, l, q, m, b, n, λ1 , λ2 ). Hence, for v ∈ Y(I), ∥T(v)∥X0 (I) ≤ C(µ, l, q)∥v0 ∥Lqκ,σ + C(µ, l, q, m, b, n, λ1 , λ2 )T 1−

bγ q

M b+1 .

Choosing M = 2C(µ, l, q)∥v0 ∥Lqκ,σ and T > 0 such that CT 1−

bγ q

Mb ≤

1 , 4

(3.15)

we conclude that T(v) ∈ Y(I). Now we prove that T is a contraction. Again using Strichartz inequalities (1.10) and (1.11) in Theorem 1.1, we deduce ∫ t ∥T(u) − T(v)∥X0 (I) ≤ ∥ U(t − τ )[g(r, u) − g(r, v)]dτ ∥X0 (I) 0

≤ C(µ, l, q)T 1−

bγ q

∥g(r, u) − g(r, v)∥X ′ (I) ,

(3.16)

0

then the rest of the argument are to deal with the bound of (3.16). Similarly for the estimates (3.11) and (3.14), we use (3.9) to obtain ∥˜ g1 (r, u) − g˜1 (r, v)∥X ′ (I) 0

b

≤λ1 ∥r−(b+1)κ |u| |u − v|∥

b

≤λ1 ∥∥u∥bLl ∥u − v∥Lqκ,σ ∥

m

L b (I)

κ,σ

≤λ1 (∥u∥bLm (I;Ll

κ,σ )

+ ∥r−(b+1)κ |v| |u − v|∥

ql m bq+l L b (I;Lσ )

+ ∥∥v∥bLl ∥u − v∥Lqκ,σ ∥ κ,σ

+ ∥v∥bLm (I;Ll

κ,σ )

m

ql

bq+l L b (I;Lσ ) m

L b (I)

)∥u − v∥L∞ (I;Lqκ,σ ) .

(3.17)

The second nonlinear term ∥˜ g2 (r, u) − g˜2 (r, v)∥X ′ (I) ∫ r0 d c c ≤λ2 ∥r−κ |u| (r′ )−bκ−1 (|u| − |v| )dr′ ∥ 0 d

d

+ ∥r−κ (|u| − |v| )

∫

r

c

(r′ )−bκ−1 |v| dr′ ∥

where the first term (3.18) can be controlled by ∫ r d c−1 ∥r−κ |u| (r′ )−bκ−1 |u| |u − v|dr′ ∥ + ∥r

|u|

∫

m

r

c−1

(r′ )−bκ−1 |v|

|u − v|dr′ ∥

≤n (∥u∥bLm (I;Ll

κ,σ )

m

ql

bq+l L b (I;Lσ )

b−d ≤n ∥∥u∥dLl (∥u∥b−d + ∥v∥L )∥u − v∥Lqκ,σ ∥ l Ll κ,σ

(3.19)

ql

0 κ,σ

,

bq+l L b (I;Lσ )

0 d

ql

m

bq+l ) L b (I;Lσ

0

−κ

(3.18)

ql

m

bq+l L b (I;Lσ )

+ ∥u∥dLm (I;Ll

κ,σ )

m

L b (I)

κ,σ

∥v∥b−d Lm (I;Ll

κ,σ )

)∥u − v∥L∞ (I;Lqκ,σ )

(3.20)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

14

d

d

and by replacing v with |u| − |v| in (3.14), the second term (3.19) becomes ∫ r d−1 c (r′ )−bκ−1 |v| dr′ ∥ m ∥r−κ |u| |u − v| ql bq+l ) L b (I;Lσ

0

d−1

+ ∥r−κ |v|

∫

r

|u − v|

c

(r′ )−bκ−1 |v| dr′ ∥

ql

m

bq+l ) L b (I;Lσ

0

b−d ≤n ∥(∥u∥d−1 + ∥v∥d−1 )∥u − v∥Llκ,σ ∥v∥L ∥v∥Lqκ,σ ∥ l Ll Ll κ,σ

κ,σ

≤n (∥u∥d−1 Lm (I;Ll

κ,σ )

×

+ ∥v∥d−1 Lm (I;Ll

κ,σ )

m

L b (I)

κ,σ

)∥u − v∥Lm (I;Llκ,σ )

∥v∥b−d ∥v∥L∞ (I;Lqκ,σ ) . Lm (I;Llκ,σ )

(3.21)

Now, by collecting (3.20) and (3.21), we obtain ∥˜ g2 (r, u) − g˜2 (r, v)∥X ′ (I) 0

≤λ2 ,n (∥u∥bLm (I;Ll

κ,σ )

+ ∥u∥dLm (I;Ll

κ,σ )

+ (∥u∥d−1 Lm (I;Ll

κ,σ )

× ∥v∥b−d Lm (I;Ll

κ,σ )

b−d ∥v∥L m (I;Ll

d−1 + ∥v∥L m (I;Ll

κ,σ )

κ,σ )

)∥u − v∥L∞ (I;Lqκ,σ )

)∥u − v∥Lm (I;Llκ,σ )

∥v∥L∞ (I;Lqκ,σ ) .

(3.22)

Therefore, from (3.17) and (3.22), we conclude that ∥g(r, u) − g(r, v)∥X ′ (I) ≤

2 ∑

0

∥˜ gj (r, u) − g˜j (r, v)∥X ′ (I) 0

j=1

≤λ1 ,λ2 ,n (∥u∥bLm (I;Ll

κ,σ )

+ ∥u∥dLm (I;Ll

κ,σ )

× ∥u − v∥L∞ (I;Lqκ,σ ) + × ∥v∥b−d Lm (I;Ll

κ,σ )

b−d ∥v∥L m (I;Ll

κ,σ )

d−1 (∥u∥L m (I;Ll κ,σ )

+

+ ∥v∥bLm (I;Ll

κ,σ )

d−1 )∥u ∥v∥L m (I;Ll κ,σ )

)

− v∥Lm (I;Llκ,σ )

∥v∥L∞ (I;Lqκ,σ )

b−d ≤λ1 ,λ2 ,n (∥u∥bX0 (I) + ∥u∥dX0 (I) ∥v∥X + 2∥v∥bX0 (I) 0 (I) b−d+1 + ∥u∥d−1 X0 (I) ∥v∥X0 (I) )∥u − v∥X0 (I)

(3.23)

Finally, combining (3.23) with (3.16), we get the estimate ∥T(u) − T(v)∥X0 (I) ≤ C(µ, l, q, λ1 , λ2 , n)T 1−

bγ q

5M b ∥u − v∥X0 (I)

provided u, v ∈ Y(I). Hence, the inequality (3.15) implies that ∥T(u) − T(v)∥X0 (I) ≤

1 ∥u − v∥X0 (I) , 2

that is, T is a contraction on Y(I). Theorem 1.2(i) is proved. (ii) By standard argument, one is able to show lim ∥v(t)∥Lqκ,σ = ∞.

(3.24)

t→T ∗

From the definition of Y(I) and the condition (3.15), we rewrite Y(I) = {v ∈ X0 (I) : ∥v∥X0 (I) ≤ 2C1 ∥v0 ∥Lqκ,σ , (2C1 )b C2 T 1− where the constants C1 = C(µ, q, l), C2 = C(µ, l, q, m, b, n, λ1 , λ2 ).

bγ q

∥v0 ∥bLq

κ,σ

≤

1 }, 4

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

15

Moreover, for all t < s < T ∗ with ∥v(t)∥Lqκ,σ < ∞, by following a similar procedure as (i), we can find the unique solution in Y([t, s]) = {v ∈ X0 ([t, s]) : ∥v∥X0 ([t,s]) ≤ 2C1 ∥v(t)∥Lqκ,σ , (2C1 )b C2 |s − t|

bγ

1− q

∥v(t)∥bLq

κ,σ

≤

1 }. 4

(3.25)

Hence, there exists ε > 0 such that ε ≤ (2C1 )b C2 |s − t|

bγ

1− q

∥v(t)∥bLq

κ,σ

≤

1 , 4

which yields that ∥v(t)∥Lqκ,σ ≥ (

ε (2C1 )b C2 |s −

bγ

1− t| q

˜ C

)1/b ≥

1−γ q

,

|T ∗ − t| b

˜ := C(µ, l, q, m, b, n, λ1 , λ2 , ε). The proof of (ii) is completed. where C

□

4. Local well-posedness in W 1,q The aim of this section is to prove the local well-posedness in W 1,q (R+ ) with q ≥ bγ (i.e. Theorem 1.4). As mentioned in Section 3, we construct a closed subspace of L∞ (I; Lqκ,σ (R+ )) such that the operator T defined by ∫ t

(Tv)(t) = U(t)v0 (r) − i

U(t − τ )g(r, v(τ ))dτ,

(4.1)

0

is contractive in this space. The fundamental tool is the Strichartz estimates (see Theorem 1.1). However, the contraction of the operator T in the derivative space X(I) cannot be achieved immediately due to the fact that Dr and U(t) are not commutative. To resolve it, we employ the operator Dr = rp0 −p1 ∂r to act on (1.1) restricted to p2 = 0, then we derive that i∂t (Dr v) + A˜ (Dr v) = Dr g(r, v), µ (4.2) Dr v(r, 0) = Dr v0 , (r, t) ∈ R+ × R, where A˜ := −rp0 (∂r2 + µ

3p1 −p0 ∂r r

−

(p0 −2p1 )(p1 −1) ) r2

and the nonlinearity

b b d−2 Dr g(r, v) = [ λ1 r−bκ |v| + dλ2 ℜ(|v| v 2

∫

r

c

(r′ )−bκ−1 |v| dr′ )]Dr v

0 b

b

−(bκλ1 − λ2 )r−bκ−1+p0 −p1 |v| v + λ1 r−bκ vDr (|v| ) 4 ∑ := gj (r, v),

(4.3)

j=1

here ℜ(u) denotes the real part of the function u. If for T > 0 sufficiently small, there exists the solution v of (1.1) such that v(t) = (Tv)(t), then by (4.2) and Duhamel formula, we have ∫ ˜ Dr [T(v)] = U(t)[D r v0 ] − i

˜ − τ )Dr g(r, v)(τ )dτ, U(t 0

˜ = e−itA˜ µ. with the semigroup operator U(t)

t

(4.4)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

16

4.1. Strichartz estimates Through the above consideration and using Theorem 1.1, we try to establish Strichartz type estimates for (4.2). Compared with (1.1), the elliptic operator in Eq. (4.2) becomes A˜ = −rp0 (∂r2 + µ

(p0 − 2p1 )(p1 − 1) 3p1 − p0 ) ∂r − r r2

with the similar notations √ µ ˜=

(3p1 − p0 − 1)2 + 4(p0 − 2p1 )(p1 − 1) |p0 − p1 − 1| = , 2 − p0 2 − p0

2 − p0 1 + p0 − 3p1 2˜ κ + 3p1 − 2p0 + 1 µ ˜+ , γ ˜= 2 2 2 − p0 We note that, the assumption p0 = 2p1 in Theorem 1.4 implies that κ ˜=

κ ˜=κ=

2 − p0 , 2

d˜ σr = dσr ,

γ ˜ = γ.

From the above observation, we obtain the following Strichartz type estimates for Eq. (4.2) analogous to Theorem 1.1. bq Lemma 4.1. Let p0 = 2p1 . Assume (m, l, q) is any admissible triplet satisfying q ≥ bγ and l ≥ max{ q−1 , q}, q + then for Dr v0 ∈ Lκ,σ (R ), the following statements hold.

(i) (Linear estimate) ˜ ∥U(t)D r v0 ∥X0 (I) ≤ C(l, q)∥Dr v0 ∥Lqκ,σ ,

(4.5)

(ii) (Inhomogeneous estimate) ∫ t bγ ˜ − τ )Dr g(r, v(τ ))dτ ∥X (I) ≤ CT 1− q ∥Dr g∥X ′ (I) , U(t ∥ 0 0

0

(4.6)

where I = [0, T ], 0 < T ≤ ∞ and C := C(l, q, m) is constant depending on l, q, m. In order to prove the well-posedness of (1.1) in W 1,q (R+ ), we need to establish some useful estimates for the nonlinearity g(r, v). We first consider the nonlinearity in the space X0′ (I) and in the sequel in the space Dr−1 X0′ (I), that is, we estimate the norm ∥g(r, v)∥X ′ (I) and ∥Dr g(r, v)∥X ′ (I) for any Lqκ,σ -admissible 0 0 triplets (m, l, q). Based on result of Proposition 3.2, we only need to estimate the nonlinear terms Dr g of (4.2), which is stated as follows: Proposition 4.2. Let p0 = 2p1 ≤ 34 and n = 2κ + 1 − Dr g of (4.2) with d = 1 satisfy the following estimates

p0 2 .

Assume v ∈ X(I), then the nonlinear terms

∥Dr g∥X ′ (I) ≤ C(2∥Dr v∥L∞ (I;Lqκ,σ ) ∥v∥bLm (I;Ll

κ,σ )

0

b−1 +∥v∥L∞ (I;Lqκ,σ ) ∥Dr v∥Lm (I;Llκ,σ ) ∥v∥L m (I;Ll

κ,σ )

)

(4.7)

where C = C(b, λ1 , λ2 , n, p0 ) are constants depending on b, λ1 , λ2 , n, p0 . Proof . For v ∈ X(I), since Dr g =

∑4

j=1 gj (r, v),

we divide the estimate ∥Dr g∥X ′ (I) into three cases, indeed

∥Dr g∥X ′ (I) ≤

0

4 ∑

0

∥gj ∥X ′ (I) . 0

j=1

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

17

Case 1: j = 1 or 4. Utilizing the H¨ older inequality, we obtain ∥gj ∥X ′ (I) ≤λ1 ,b ∥∥v∥bLl ∥Dr v∥Lqκ,σ ∥ 0

κ,σ

≤λ1 ,b ∥v∥bLm (I;Ll

κ,σ )

m

L b (I)

∥Dr v∥L∞ (I;Lqκ,σ ) .

(4.8)

Case 2: j = 2. Since d = 1, we have ∥g2 ∥X ′ (I) ≤λ2 ∥ ∥Dr v∥Llκ,σ 0 ∫ ∫ r ql (b−1)q+l ×( ∥ r2κ+3p1 −2p0 | (r′ )−bκ−1 |v|c dr′ | (b−1)q+l dr) ql R+

m

L b (I)

0

.

(4.9)

In view of the Hardy-type inequality (3.13), in order to show that the second norm in the right hand side of (4.9) is bounded we need n = 2κ + 3p1 − 2p0 + 1, which is implied by hypothesis p0 = 2p1 , so we deduce ∫ cql bκql (b−1)q+l n−1− (b−1)q+l |v| (b−1)q+l dr) ql ∥ ∥g2 ∥X ′ (I) ≤λ2 ,n ∥∥Dr v∥Llκ,σ ( r 0

R+ ≤λ2 ,n ∥∥Dr v∥Llκ,σ ∥v∥b−1 ∥v∥Lqκ,σ ∥ mb . Llκ,σ L (I)

m

L b (I)

Therefore, invoking the H¨ older inequality with respect to t, we obtain ∥g2 ∥X ′ (I) ≤λ2 ,n ∥Dr v∥Lm (I;Llκ,σ ) ∥v∥b−1 Lm (I;Ll

κ,σ )

0

∥v∥L∞ (I;Lqκ,σ ) .

(4.10)

Case 3: j = 3. For the norm ∥g3 ∥X ′ (I) , it follows from H¨older inequality that 0 ∫ 1 q |rp0 −p1 −κ−1 v| rn−1 dr) q , ∥g3 ∥ ql ≤λ1 ,λ2 ,p0 ,b ∥v∥bLl ( κ,σ

bq+l

Lκ,σ

R+

which together with the Hardy-type inequality (3.13) yield that ∥g3 ∥X ′ (I) ≤λ1 ,λ2 ,b,n,p0 ∥∥v∥bLl ∥Dr v∥Lqκ,σ ∥ 0

m

L b (I)

κ,σ

≤λ1 ,λ2 ,b,n,p0 ∥v∥bLm (I;Ll

κ,σ )

∥Dr v∥L∞ (I;Lqκ,σ ) .

Hence, the result (4.7) follows from the estimates (4.8), (4.10) and (4.11). □ 4.2. Well-posedness We now have all tools to prove the main result of this section, Theorem 1.4. Proof . The Strichartz space X(I) is defined by X(I) = L∞ (I; W 1,q (R+ )) ∩ Lm (I; W 1,l (R+ )), for any admissible triplets (m, l, q), and ∥v∥X(I) = ∥v∥X0 (I) + ∥Dr v∥X0 (I) . We shall show that T defined in (4.4) is a contraction on the complete metric space Y (I) = {v ∈ X(I) : ∥v∥X(I) ≤ M }

(4.11)

B.-w. Zheng / Nonlinear Analysis 182 (2019) 1–19

18

with the metric d(u, v) = ∥u − v∥X0 (I) . for suitable M > 0 and T > 0. First, we claim that Y (I) with the metric d(u, v) is a complete metric space. Indeed, the proof follows similar arguments as in [4]. Since Y (I) ⊂ X(I), and X(I) is a complete space, it suffices to show that Y (I), with the metric d(u, v) , is closed in X(I). Let vn ∈ Y (I) such that d(vn , v) → 0 as n → ∞, we need to show that v ∈ Y (I). If vn ∈ L∞ (I; W 1,q (R+ )), then for almost all t ∈ I, vn (t) bounded in W 1,q (R+ ) and so vn (t) ⇀ u(t) in W 1,q ,

∥u(t)∥W 1,q ≤ lim inf ∥vn ∥W 1,q ≤ M.

(4.12)

n→+∞

On the other hand, the hypothesis d(vn , v) → 0 implies that vn → v in Lm (I; Llκ,σ ) for all (m, l, q) admissible triplets. Then we get vn (t) → v(t) in Lqκ,σ for almost all t ∈ I. Therefore, by uniqueness of the limit we deduce that v(t) = u(t). From (4.12), we also obtain that ∥v(t)∥W 1,q ≤ M. That is, v ∈ L∞ (I; W 1,q (R+ )). From similar arguments, if vn ∈ Lm (I; W 1,l (R+ )), we obtain v ∈ Y (I). This completes the proof of the claim. Returning the proof of the theorem, let T(v) = U(t)v0 + Θ(g)(t) as (4.1), it follows from the Strichartz inequalities (1.10), (1.11) and Lemma 4.1 that ∥T(v)∥X0 (I) ≤ C(l, q, p0 )∥v0 ∥Lqκ,σ + C(l, q, m, b, p0 )T 1−

bγ q

and ∥Dr T(v)∥X0 (I) ≤ C(l, q, p0 )∥Dr v0 ∥Lqκ,σ + C(l, q, m, p0 )T 1−

∥g∥X ′ (I) ,

(4.13)

∥Dr g∥X ′ (I) ,

(4.14)

0

bγ q

0

where g = g˜1 (r, v) + g˜2 (r, v). Similarly as in the proof of Theorem 1.2, we deduce using Propositions 3.2 and 4.2 ∥g∥X ′ (I) ≤n,λ1 ,λ2 ∥v∥bLm (I;Ll

κ,σ )

0

∥v∥L∞ (I;Lqκ,σ ) + ∥v∥L∞ (I;Lqκ,σ ) ∥v∥bLm (I;Ll

κ,σ )

≤n,λ1 ,λ2 ∥v∥b+1 X(I) , and ∥Dr g∥X ′ (I) ≤ 3C(b, λ1 , λ2 , n, p0 )∥v∥b+1 X(I) , 0

Hence, if v ∈ Y (I), then ∥T(v)∥X(I) ≤ C(l, q, p0 )∥v0 ∥W 1,q + C(λ1 , λ2 , n, b, q, l, m)T 1−

bγ q

M b+1 .

(4.15)

Now, choosing M = 2C(l, q, p0 )∥v0 ∥W 1,q and T > 0 such that bγ 1 CT 1− q M b ≤ , (4.16) 4 we conclude that T(v) ∈ Y (I). Such calculations establish that T is well defined on Y (I). To prove T is a contraction mapping from Y (I) to Y (I), we use (3.9) and an analogous argument as before d(T(u), T(v)) ≤l,q,λ1 ,λ2 ,n T 1− ≤l,q,λ1 ,λ2 ,n

bγ q

∥g(r, u) − g(r, v)∥X ′ (I) 0

bγ

T 1− q

(∥u∥bX(I)

b + ∥u∥X(I) ∥v∥b−1 X(I) + 3∥v∥X(I) )d(u, v)

and so, taking u, v ∈ Y (I), we get d(T(u), T(v)) ≤ C(l, q, λ1 , λ2 , n)T 1−

bγ q

5M b d(u, v).

Therefore, from the inequality (4.16), T is a contraction on Y (I) and by the contraction mapping theorem we have a unique fixed point v ∈ Y (I) of T. The proof of Theorem 1.4 is completed. □

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19

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