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arXiv:1211.0334v1 [math.AP] 2 Nov 2012

ON THE EXISTENCE AND CUSP SINGULARITY OF SOLUTIONS TO SEMILINEAR

GENERALIZED TRICOMI EQUATIONS WITH DISCONTINUOUS INITIAL DATA

Ruan, Zhuoping1,∗; Witt, Ingo2,∗∗; Yin, Huicheng1,∗

1. Department of Mathematics and IMS, Nanjing University, Nanjing 210093, P.R. China.

2. Mathematical Institute, University of G¨ottingen, Bunsenstr. 3-5, D-37073 G¨ottingen, Germany.

Abstract

In this paper, we are concerned with the local existence and singularity structure of low regularity solutions

to the semilinear generalized Tricomi equation ∂2

tu−tm∆u=f(t, x, u) with typical discontinuous initial

data (u(0, x), ∂tu(0, x)) = (0, ϕ(x)); here m∈N,x= (x1, ..., xn), n≥2, and f(t, x, u) is C∞smooth in its

arguments. When the initial data ϕ(x) is a homogeneous function of degree zero or a piecewise smooth function

singular along the hyperplane {t=x1= 0}, it is shown that the local solution u(t, x)∈L∞([0, T ]×Rn) exists

and is C∞away from the forward cuspidal cone Γ0=(t, x): t > 0,|x|2=4tm+2

(m+ 2)2and the characteristic

cuspidal wedge Γ±

1=(t, x): t > 0, x1=±2tm

2+1

m+ 2 , respectively. On the other hand, for n= 2 and piecewise

smooth initial data ϕ(x) singular along the two straight lines {t=x1= 0}and {t=x2= 0}, we establish

the local existence of a solution u(t, x)∈L∞([0, T ]×R2)∩C([0, T ], H m+6

2(m+2) −(R2)) and show further that

u(t, x)6∈ C2((0, T ]×R2\(Γ0∪Γ±

1∪Γ±

2)) in general due to the degenerate character of the equation under

study; here Γ±

2=(t, x): t > 0, x2=±2tm

2+1

m+ 2 . This is an essential diﬀerence to the well-known result for

solutions v(t, x)∈C∞(R+×R2\(Σ0∪Σ±

1∪Σ±

2)) to the 2-D semilinear wave equation ∂2

tv−∆v=f(t, x, v)

with (v(0, x), ∂tv(0, x)) = (0, ϕ(x)), where Σ0={t=|x|}, Σ±

1={t=±x1}, and Σ±

2={t=±x2}.

Keywords: Generalized Tricomi equation, conﬂuent hypergeometric function, hypergeometric function,

cusp singularity, tangent vector ﬁelds, conormal space

Mathematical Subject Classiﬁcation 2000: 35L70, 35L65, 35L67, 76N15

§1.Introduction

In this paper, we will study the local existence and the singularity structure of low regularity solution to

the following n-dimensional semilinear generalized Tricomi equation

∂2

tu−tm∆u=f(t, x, u),(t, x)∈[0,+∞)×Rn,

u(0, x) = 0, ∂tu(0, x) = ϕ(x),(1.1)

* Ruan Zhuoping and Yin Huicheng were supported by the NSFC (No. 10931007, No. 11025105), by the Priority Academic

Program Development of Jiangsu Higher Education Institutions, and by the DFG via the Sino-German pro ject “Analysis of PDEs

and application.” This research was carried out when Witt Ingo visited Nanjing University in March of 2012, and Yin Huicheng

was visiting the Mathematical Institute of the University of G¨ottingen in July-August of 2012.

** Ingo Witt was partly supported by the DFG via the Sino-German project “Analysis of PDEs and application.”

Typeset by A

M

S-T

EX

1

where m∈N,x= (x1, ..., xn), n≥2, ∆ =

n

X

i=1

∂2

i,f(t, x, u) is C∞smooth on its arguments and has a compact

support on the variable x, and the typical discontinuous initial data ϕ(x) satisﬁes one of the assumptions:

(A1)ϕ(x) = g(x, x

|x|), here g(x, y)∈C∞(Rn×Rn) and has a compact support in B(0,1) ×B(0,2);

(A2)ϕ(x) = ϕ1(x) for x1<0,

ϕ2(x) for x1>0,with ϕ1(x), ϕ2(x)∈C∞

0(Rn) and ϕ1(0) 6=ϕ2(0);

(A3) For n= 2, ϕ(x) =

ψ1(x) for x1>0, x2>0,

ψ2(x) for x1<0, x2>0,

ψ3(x) for x1<0, x2<0,

ψ4(x) for x1>0, x2<0,

with ψi(x)∈C∞

0(Rn)(1 ≤i≤4) and ψi(0) 6=ψj(0)

for some i6=j(1 ≤i < j ≤4).

It is noted that ϕ(x) = ψ(x)x1

|x|with ψ(x)∈C∞

0(B(0,1)) is a special function satisfying (A1), which has a

singularity at the origin.

Under the assumptions (A1)−(A3), we now state the main results in this paper.

Theorem 1.1. There exists a constant T > 0such that

(i) Under the condition (A1), (1.1) has a unique solution u(t, x)∈C([0, T ], H n

2+2

m+2 −(Rn)) ∩C((0, T ],

Hn

2+m+4

2(m+2) −(Rn)) ∩C1([0, T ], H n

2−m

2(m+2) −(Rn)) and u(t, x)∈C∞((0, T ]×Rn\Γ0), here Γ0={(t, x) : t >

0,|x|2=4tm+2

(m+ 2)2}.

(ii) Under the condition (A2), (1.1) has a unique solution u(t, x)∈L∞([0, T ]×Rn)∩C([0, T ], H m+6

2(m+2) −(Rn))

∩C((0, T ], H m+3

m+2 −(Rn)) ∩C1([0, T ], H 1

m+2 −(Rn)) and u(t, x)∈C∞((0, T ]×Rn\Γ+

1∪Γ−

1), here Γ±

1={(t, x) :

t > 0, x1=±2tm

2+1

m+ 2 }.

(iii) For n= 2, under the condition (A3), if m≤9, then (1.1) has a unique solution u(t, x)∈L∞([0, T ]×

R2)∩C([0, T ], H m+6

2(m+2) −(R2)) ∩C((0, T ], H m+3

m+2 −(R2))∩C1([0, T ], H 1

m+2 −(R2)). Moreover, in the general case,

u(t, x)6∈ C2((0, T ]×R2\Γ0∪Γ±

1∪Γ±

2), here Γ0and Γ±

1have been deﬁned in (i) and (ii) respectively, and

Γ±

2={(t, x) : t > 0, x2=±2tm

2+1

m+ 2 }.

Remark 1.1. In order to prove the C∞property of solution in Theorem 1.1.(i) and (ii), we wil l show

that the solution of (1.1) is conormal with respect to the cusp characteristic conic surface Γ0or the cusp

characteristic surfaces Γ±

1respectively in §6below. And the deﬁnitions of conormal spaces will be given in §4.

Remark 1.2. Since we only focus on the local existence of solution in Theorem 1.1, it does not lose the

generality that the initial data ϕ(x)in (A1)−(A3)are assumed to be compactly supported. In addition, the

initial data (u(0, x), ∂tu(0, x)) = (0, ϕ(x)) in (1.1) can be replaced by the general forms (u(0, x), ∂tu(0, x)) =

(φ(x), ϕ(x)), where Dxφ(x)satisﬁes (A1)when ϕ(x)satisﬁes (A1),φ(x)is C1piecewise smooth along {t=x1=

0}when ϕ(x)satisﬁes (A2), and φ(x)is C1piecewise smooth along the lines {t=x1= 0}and {t=x2= 0}

when ϕ(x)satisﬁes (A3), respectively.

Remark 1.3. The initial data problem (1.1) under the assumptions (A2)and (A3)is actually a special case

of the multidimensional generalized Riemann problem for the second order semilinear degenerate hyperbolic

equations. For the semilinear N×Nstrictly hyperbolic systems of the form ∂tU+

n

X

j=1

Aj(t, x)∂jU=

F(t, x, U )with the piecewise smooth or conormal initial data along some hypersurface ∆0⊂ {(t, x) : t=

0, x ∈Rn}(including the Riemann discontinuous initial data), the authors in [19-20] have established the local

well-posedness of piecewise smooth or bounded conormal solution with respect to the Npairwise transverse

characteristic surfaces Σjpassing through ∆0. With respect to the Riemann problem of higher order semilinear

2

degenerate hyperbolic equations, we will establish the related results in our forthcoming paper.

Remark 1.4. The reason that we pose the restriction on m≤9in (iii) of Theorem 1.1 is due to the require-

ment for utilizing the Sobolev’s imbedding theorem to derive the boundedness of solution (one can see details

in (5.7) of §5below), otherwise, it seems that we have to add some other conditions on the nonlinear function

f(t, x, u)since the solution w(t, x)∈L∞([0, T ]×R2)does not hold even if w(t, x)satisﬁes a linear equation

∂2

tw−tm∆w=g(t, x)with (w(0, x), ∂tw(0, x)) = (0,0) and g(t, x)∈L∞([0, T ]×R2)∩L∞([0, T ], H s(R2))

with 0≤s < 1for large m. Firstly, this can be roughly seen from the following explicit formula of w(t, x)in

Theorem 3.4 of [24]:

w(t, x) = 1

π(4

m+ 2)m

m+2 Zt

0

dτ Zφ(t)−φ(τ)

0

dr1∂rZB(x,r)

g(τ, y )

pr2− |x−y|2dyr=r1

(r1+φ(t) + φ(τ))−γ

×(φ(τ)−r1+φ(t))−γFγ, γ ; 1; (−r1+φ(t)−φ(τ))(−r1−φ(t) + φ(τ))

(−r1+φ(t) + φ(τ))(−r1−φ(t)−φ(τ)),

where φ(t) = 2tm

2+1

m+ 2 ,γ=m

2(m+ 2) , and F(a, b;c;z)is the hypergeometric function. It is noted that

∂rRB(x,r)

g(τ, y )

pr2− |x−y|2dy=∂rZr

0Z2π

0

g(τ, x1+pr2−q2cosθ, x2+pr2−q2sinθ)dqdθholds and thus

the L∞property of w(t, x)is closely related to the integrability of the ﬁrst order derivatives of g(t, x), which

is diﬀerent from the case in 2-D linear wave equation. On the other hand, the regularity of w(t, x)is in

C([0, T ], H s+2

m+2 (R2)) 6⊂ L∞([0, T ]×R2)for large mby Proposition 3.3 below and Sobolev’s imbedding theo-

rem.

Remark 1.5. By u(t, x)6∈ C2((0, T ]×R2\Γ0∪Γ±

1∪Γ±

2)in Theorem 1.1.(iii), we know that there exists

an essential diﬀerence on the regularity of solutions between the degenerate hyperbolic equation and strictly

hyperbolic equation with the same initial data in (A3)since v(t, x)∈C∞(R+×R2\Σ0∪Σ±

1∪Σ±

2)will hold

true if v(t, x)is a solution to the 2-D semilinear wave equation ∂2

tv−∆v=f(t, x, v)with (v(0, x), ∂tv(0, x)) =

(0, ϕ(x)), where Σ0={t=|x|},Σ±

1={t=±x1}, and Σ±

2={t=±x2}. The latter well-known result was

established in the references [1-2], [7-9] and [18] respectively under some various assumptions.

For m= 1, n = 1 and f(t, x, u)≡0, the equation in (1.1) becomes the classical Tricomi equation which

arises in transonic gas dynamics and has been extensively investigated in bounded domain with suitable

boundary conditions from various viewpoints (one can see [4], [17], [21-22] and the references therein). For

m= 1 and n= 2, with respect to the equation ∂2

tu−t△u=f(t, x, u) together with the initial data of

higher Hs(Rn)−regularity (s > n

2), M.Beals in [3] show the local existence of solution u∈C([0, T ], H s(Rn)) ∩

C1([0, T ], H s−5

6(Rn)) ∩C2([0, T ], H s−11

6(Rn)) for some T > 0 under the crucial assumption that the support

of f(t, x, u) on the variable tlies in {t≥0}. Meanwhile, the conormal regularity of Hs(Rn) solution u(t, x)

with respect to the characteristic surfaces x1=±2

3t3

2is also established in [3]. With respect to more general

nonlinear degenerate hyperbolic equations with higher order regularities, the authors in [10-11] studied the local

existence and the propagation of weak singularity of classical solution. For the linear degenerate hyperbolic

equations with suitable initial data, so far there have existed some interesting results on the regularities of

solution when Levi’s conditions are posed (one can see [12], [14-15] and the references therein). In the present

paper, we focus on the low regularity solution problem for the second order semilinear degenerate equation

with no much restrictions on the nonlinear function f(t, x, u) in (1.1) and typical discontinuous initial data.

We now comment on the proof of Theorem 1.1. In order to prove the local existence of solution to (1.1)

with the low regularity, at ﬁrst we should establish the local L∞property of solution v(t, x) to the linear

problem ∂2

tv−tm∆v=F(t, x) with (v(0, x), ∂tv(0, x)) = (ϕ0(x), ϕ1(x)) so that the composition function

f(t, x, v) makes sense. In this process, we have to make full use of the special structure of the piecewise

smooth initial data and the explicit expression of solution v(t, x) established in [23-24] since we can not apply

for the Sobolev imbedding theorem directly to obtain v(t, x)∈L∞

loc due to its low regularity (for examples,

3

in the cases of (A2)−(A3), the initial data are only in H1

2−(Rn)). Based on such L∞−estimates, together

with the Fourier analysis method and the theory of conﬂuent hypergeometric functions, we can construct

a suitable nonlinear mapping related to the problem (1.1) and further show that such a mapping admits a

ﬁxed point in the space L∞([0, T ]×Rn)∩C([0, T ], H s0(Rn) for suitable T > 0 and some number s0>0,

and then the local solvability of (1.1) can be shown. Next, we are concerned with the singularity structures

of solution u(t, x) of (1.1). It is noted that the initial data are suitably conormal under the assumptions

(A1) and (A2), namely, Π1≤i,j≤n(xi∂j)kij ϕ(x)∈Hn

2−(Rn) for any kij ∈N∪ {0}in the case of (A1), and

(x1∂1)k1Π2≤i≤n∂ki

iϕ(x)∈H1

2−(Rn) for any ki∈N∪ {0}(i= 1, ..., n) in the case of (A2), then we intend to

use the commutator arguments as in [5-6] to prove the conormality of solution u(t, x) to (1.1). However, due

to the cusp singularities of surfaces Γ0,Γ±

1together with the degeneracy of equation, it seems that it is diﬃcult

to choose the smooth vector ﬁelds {Z1, ..., Zk}tangent to Γ0or Γ±

1as in [5-6] to deﬁne the conormal space

and take the related analysis on the commutators [∂2

t−tm∆, Zl1

1···Zlk

k] since this will lead to the violation

of Levi’s condition and bring the loss of regularity of Zl1

1···Zlk

ku(more detailed explanations can be found in

§4 below). To overcome this diﬃculty, motivated by [2-3] and [18], we will choose the nonsmooth vector ﬁelds

and try to ﬁnd the extra regularity relations provided by the operator itself and some parts of vector ﬁelds

to yield full conormal regularity of u(t, x) together with the regularity theory of second order elliptic equation

and further complete the proof of Theorem 1.1.(i) and (ii), here we point out that it is nontrivial to ﬁnd such

crucial regularity relations. On the other hand, in the case of n= 2 and assumption (A3), due to the lack of

the strong Huyghen’s principle, we can derive that the solution u(t, x)6∈ C2((0, T ]×R2\Γ0∪Γ±

1∪Γ±

2) of

(1.1), which yields a diﬀerent phenomenon from that in the case of second order strict hyperbolic equation as

pointed out in Remark 1.5.

This paper is organized as follows. In §2, for later uses, we will give some preliminary results on the regu-

larities of initial data ϕ(x) in various assumptions (A1)−(A3) and establish the L∞property of solution to

the related linear problem. In §3, by the partial Fourier-transformation, we can change the linear generalized

Tricomi equation into a conﬂuent hypergeometric equation, and then some weighted Sobolev regularity esti-

mates near {t= 0}are derived. In §4, the required conormal spaces are deﬁned and some crucial commutator

relations are given. In §5, based on the results in §2-§3, the local solvability of (1.1) is established. In §6, we

complete the proof on Theorem 1.1 by utilizing the concepts of conormal spaces and commutator relations in

§4 and taking some analogous analysis in Lemma 2.4 of §2 respectively.

In this paper, we will use the following notation:

Hs−(Rn) = {w(x) : w(x)∈Hs−δ(Rn) for any ﬁxed constant δ > 0.}

§2. Some preliminaries

In this section, we will give some basic lemmas on the regularities of initial data ϕ(x) in the assumptions

(A1)−(A3) and establish some L∞property of solution to the linear problem ∂2

tu−tm∆u=f(t, x) with

suitably piecewise smooth initial data.

With respect to the functions ϕ(x) given in (A1)−(A3) of §1, we have the following regularities in Sobolev

space.

Lemma 2.1. (i) If ϕ(x) = g(x, x

|x|), here g(x, y)∈C∞(Rn×Rn)and has a compact support in B(0,1) ×

B(0,2), then ϕ(x)∈Hn

2−(Rn).

(ii) If n= 2 and ϕ(x) =

ψ1(x)for x1>0,x2>0,

ψ2(x)for x1<0,x2>0,

ψ3(x)for x1<0,x2<0,

ψ4(x)for x1>0,x2<0,

where ψi(x)∈C∞

0(R2)(1 ≤i≤4), then

ϕ(x)∈H1

2−(R2).

(iii) If ϕ(x) = ϕ1(x)for x1<0,

ϕ2(x)for x1>0,, where ϕ1(x), ϕ2(x)∈C∞

0(Rn), then ϕ(x)∈H1

2−(Rn)and x1ϕ(x)∈

4

H3

2−(Rn).

Proof. (i) It follows from a direct computation that

|∂α

xϕ(x)| ≤ Cα|x|−|α|.(2.1)

Since ϕis integrable on Rn, we have that ˆϕ, the Fourier transform of ϕ, is continuous on Rn, which implies

that (1 + |ξ|)n

2−δˆϕ(ξ)∈L2({|ξ| ≤ 1}).

For |ξ|>1, we decompose ˆϕinto two parts

ˆϕ(ξ) = Z|x|<1

|ξ|

e−ix·ξϕ(x)dx +Z1

|ξ|≤|x|≤1

e−ix·ξϕ(x)dx

=I+II =I+

n

X

ℓ=1

χℓ(ξ)II, (2.2)

where {χℓ}n

ℓ=1 is a C∞conic decomposition of unity corresponding to the domain {ξ∈Rn:|ξ| ≥ 1}, moreover

ξℓ6= 0 in suppχℓ.

Obviously, the term Ican be dominated by the multiplier of |ξ|−n.

On the other hand, for any 1 ≤ℓ≤n,

χℓ(ξ)II =χℓ(ξ)

|ξ|nZ1≤|x|≤|ξ|

e−ix·ξ

|ξ|g(x

|ξ|,x

|x|)dx

=χℓ(ξ)

|ξ|n

1

iξℓ

|ξ|Z1≤|x|≤|ξ|

e−ix·ξ

|ξ|∂ℓg(x

|ξ|,x

|x|)dx

+χℓ(ξ)

|ξ|n

1

−iξℓ

|ξ|Z|x|=1

e−ix·ξ

|ξ|g(x

|ξ|,x

|x|)cos(~n, xℓ)dS

=χℓ(ξ)

|ξ|n

1

(iξℓ

|ξ|)mZ1≤|x|≤|ξ|

e−ix·ξ

|ξ|∂m

lg(x

|ξ|,x

|x|)dx

+

m−1

X

k=0

χℓ(ξ)

|ξ|n

1

(iξℓ

|ξ|)k

1

(−iξℓ

|ξ|)Z|x|=1

e−ix·ξ

|ξ|∂k

ℓg(x

|ξ|,x

|x|)cos(~n, xℓ)dS

=III +IV. (2.3)

Due to

|∂k

ℓg(x

|ξ|,x

|x|)| ≤

k

X

j=0 X

|α|≤k−j

Cαj |(∂j

ℓ∂α

yg)( x

|ξ|,x

|x|)| |x|−(k−j)|ξ|−j,

then from (2.1), I V is dominated by the multiplier of |ξ|−n, and moreover,

|III| ≤ C

|ξ|nX

α+β=m

Cαβ

1

|ξ|αZ1≤|x|≤|ξ||x|−βdx

≤

CX

α+β=m

Cαβ 1

|ξ|m+1

|ξ|n+αif β6=n;

C

|ξ|nln|ξ|if β=n.

(2.4)

Therefore, for m≥nand |ξ| ≥ 1, we have |ˆϕ(ξ)| ≤ C(1 + ln|ξ|)

|ξ|nby (2.2)-(2.4), which derives (1 +

|ξ|)n

2−δˆϕ(ξ)∈L2({|ξ| ≥ 1}) for any δ > 0, and further completes the proof of (i).

5

(ii) Without loss of generality, we assume supp ψi(x)⊂[−1,1; −1,1] (1 ≤i≤4).

Since

|ˆϕ(ξ)| ≡Z1

0Z1

0

ψ1(x)e−ix·ξdx +Z0

−1Z1

0

ψ2(x)e−ix·ξdx

+Z0

−1Z0

−1

ψ3(x)e−ix·ξdx +Z1

0Z0

−1

ψ4(x)e−ix·ξdx

≤

C

|ξ1| |ξ2|for |ξ1| ≥ 1,|ξ2| ≥ 1;

C

|ξ1|for |ξ1| ≥ 1,|ξ2|<1;

C

|ξ2|for |ξ1|<1,|ξ2| ≥ 1;

Cfor |ξ1|<1,|ξ2|<1,

then from the fact 1 + |ξ| ≤ (1 + |ξ1|)(1 + |ξ2|) one has for any 0 < δ < 1

2

ZR2

(1 + |ξ|)1−δ|ˆϕ(ξ)|2dξ

≤C

2

Y

i=1 Z∞

1

(1 + |ξi|)1−δ

|ξi|2dξi+C

2

X

i=1 Z∞

1

(1 + |ξi|)1−δ

|ξi|2dξi+C

≤C.

Thus, the proof of (ii) is completed.

(iii) The proof procedure is similar to that in (ii), we omit it here.

Remark 2.1. By the similar proof procedure as in Lemma 2.1.(i), we can also prove: If f(x)∈C∞(Rn\{0})

and has compact support, moreover, |∂αf(x)| ≤ Cα|x|r−|α|for x6= 0 and r > −n

2, then f(x)∈Hn

2+r−(Rn).

Remark 2.2. Under the assumption (A2), for any α∈(N∪ {0})n−1, we can also have that ∂α

x′ϕ(x)∈

H1

2−δ(Rn)for any δ > 0small, here x′= (x2, ..., xn). Thus, (1 + |ξ|)1

2−δ(1 + |ξ′|)|α|ˆϕ(ξ)∈L2(Rn), where

ξ′= (ξ2,···, ξn).

Lemma 2.2. If u(t, x)∈C([0, T ], H 1

2−(Rn)) is a solution of the following linear equation

∂2

tu−tm∆u= 0,(t, x)∈[0,+∞)×Rn,

u(0, x) = ψ(x), ∂tu(0, x) = ϕ(x),(2.5)

where ϕ(x)satisﬁes the assumption (A2),∂α

x′ψ(x)∈H3

2−(Rn)for all 0≤ |α| ≤ [n

2] + 1, then u(t, x)∈

L∞([0, T ]×Rn).

Proof. Set y(t, ξ ) = RRnu(t, x)e−ix·ξdx with ξ∈Rnand y′′(t, ξ)≡∂2

ty(t, ξ), then it follows from the

equation of (2.5) that

y′′(t, ξ ) + tm|ξ|2y(t, ξ) = 0.(2.6)

Let τ=2tm

2+1|ξ|

m+ 2 and v(τ)≡y(t, |ξ|), then

d2v

dτ2+m

(m+ 2)τ

dv

dτ +v= 0.(2.7)

6

As in [25], taking z≡2iτ =4i

m+ 2 tm+2

2|ξ|and w(z) = v(z

2i)ez

2yields for t > 0 and |ξ| 6= 0

zw′′ (z) + ( m

m+ 2 −z)w′(z)−m

2(m+ 2) w(z) = 0.(2.8)

(2.8) has two linearly independent solutions w1(z) = Φ( m

2(m+2) ,m

m+2 ;z) and w2(z) = z2

m+2 Φ( m+4

2(m+2) ,m+4

m+2 ;z)

by [13], which are called the conﬂuent hypergeometric functions.

By (2.6)-(2.8) and [23], we have for t≥0 and ξ∈Rn

y(t, ξ) = V1(t, |ξ|)ψ∧(ξ) + V2(t, |ξ|)ϕ∧(ξ)

≡y1(t, ξ) + y2(t, ξ ) (2.9)

with (V1(t, |ξ|) = e−z

2Φ( m

2(m+2) ,m

m+2 ;z),

V2(t, |ξ|) = te−z

2Φ( m+4

2(m+2) ,m+4

m+2 ;z).(2.10)

Since Φ( m

2(m+2) ,m

m+2 ;z) and Φ( m+4

2(m+2) ,m+4

m+2 ;z) are analytic functions of z, then |Φ( m

2(m+2) ,m

m+2 ;z)|and

|Φ( m+4

2(m+2) ,m+4

m+2 ;z)| ≤ CMfor |z| ≤ M. For suﬃciently large |z|, we have from formula (9) in pages 253 of [13]

that

|Φ( m

2(m+ 2) ,m

m+ 2;z)| ≤ C|z|−m

2(m+2) 1+O|z|−1,|Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2 ;z)| ≤ C|z|−m+4

2(m+2) 1+O|z|−1.

(2.11)

From Remark 2.2, we have that for 0 ≤ |α| ≤ [n

2] + 1 and 0 < δ < 1

2

ˆϕ(ξ) = gα(ξ)

(1 + |ξ1|)1

2−δ(1 + |ξ′|)|α|,(2.12)

where gα(ξ)∈L2(Rn), ξ′= (ξ2, ...., ξn).

Therefore, for any t∈(0, T ], we have

ZRn|y2(t, ξ)|dξ ≤C t ZRn

e−2i

m+2 tm+2

2|ξ|Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2;4i

m+ 2 tm+2

2|ξ|)ϕ∧(ξ)

dξ

≤Ctm+ 2

2tm+2

2nZRn

Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2 ; 2i|η|)ϕ∧((m+ 2)η

2tm+2

2

)

dη

≤Ctm+ 2

2tm+2

2nZRn

1

(1 + |η|2)m+4

4(m+2) |ϕ∧((m+ 2)η

2tm+2

2

)|dη (by (2.11))

≤Cαtm+ 2

2tm+2

2n

2ZRn

1

(1 + |η|2)m+4

2(m+2)

1

(1 + |η1|

tm+2

2

)1−2δ

1

(1 + |η′|

tm+2

2

)2|α|dη1

2

(by (2.12))

≤CαtZRn

1

(1 + tm+2|η|2)m+4

2(m+2)

1

(1 + |η1|)1−2δ

1

(1 + |η′|)2|α|dη1

2

(choosing |α|= [ n

2] + 1 >n

2)

≤CTt1−3(m+2)δ

4ZR

1

(1 + |η1|)1+δdη11

2

(choosing δ < m+ 4

3(m+ 2) )

≤CT(choosing δ < 4

3(m+ 2) ).

7

Similarly,

ZRn|y1(t, ξ)|dξ ≤Cm+ 2

2tm+2

2nZRn

Φ( m

2(m+ 2) ,m

m+ 2 ; 2i|η|)ψ∧((m+ 2)η

2tm+2

2

)

dη

≤Cαm+ 2

2tm+2

2n

2ZRn

1

(1 + |η|2)m

2(m+2)

1

(1 + |η1|

tm+2

2

)3−2δ

1

(1 + |η′|

tm+2

2

)2|α|dη1

2

≤CαZRn

1

(1 + |η1|)3−2δ

1

(1 + |η′|)2|α|dη1

2

≤CT(choosing |α|= [ n

2] + 1 and 0 < δ < 1).

Thus, |u(t, x)| ≤ RRn|y(t, ξ)|dξ ≤RRn|y1(t, ξ)|dξ +RRn|y2(t, ξ)|dξ ≤CTfor (t, x)∈(0, T ]×Rn, and then

Lemma 2.2 is proved.

Lemma 2.3. If f(t, x)∈C([0, T ], H s(Rn)) and ∂α

x′f(t, x)∈L∞([0, T ], H s(Rn)) with s > 1

2and |α| ≤ [n

2]+1,

v(t, x)is a solution to the following problem

∂2

tu−tm∆u=f(t, x),

u(0, x) = ∂tu(0, x) = 0,(2.13)

then u(t, x)∈L∞([0, T ]×Rn).

Proof. By the assumptions on f(t, x), we have

f∧(t, ξ) = gα(t, ξ)

(1 + |ξ1|)s(1 + |ξ′|)|α|,

where gα(t, ξ)∈L∞([0, T ], L2(Rn)) and |α|= [ n

2] + 1.

From (2.13), we have

u(t, x) = Zt

0

(V2(t, |ξ|)V1(τ, |ξ|)−V1(t, |ξ|)V2(τ , |ξ|))f∧(τ, ξ)dτ ∨

(t, x),

where the expressions of V1(t, |ξ|) and V2(t, |ξ|) are given in (2.10).

It is noted that

|u∧(t, ξ)| ≤ Zt

0|V2(t, |ξ|)V1(τ, |ξ|)f∧(τ , ξ)|dτ +Zt

0|V1(t, |ξ|)V2(τ, |ξ|)f∧(τ , ξ)|dτ

≡I+II. (2.14)

Set η=2

m+ 2 tm+2

2ξ, we have

|I| ≤ Ct Zt

0|Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2 ; 2i|η|)||Φ( m

2(m+ 2) ,m

m+ 2 ; 2i(τ

t)m+2

2|η|)||f∧(τ, (m+ 2)η

2tm+2

2

)|dτ

≤CαtZt

0

(1 + |η|)−m+4

2(m+2) (1 + ( τ

t)m+2

2|η|)−m

2(m+2) |gα(τ, (m+2)η

2tm+2

2

)|

(1 + |η1|

tm+2

2

)s(1 + |η′|

tm+2

2

)|α|dτ

8

and thus

ZRn|I|dξ ≤Cαt1−(m+2)n

4Zt

0

dτZRn

1

(1 + |η|)m+4

m+2 (1 + ( τ

t)m+2

2|η|)m

m+2 (1 + |η1|

tm+2

2

)2s(1 + |η′|

tm+2

2

)2|α|dη1

2

≤CαtZt

0

dτZRn

1

(1 + |η1|)2s(1 + |η′|)2|α|dη1

2

≤Cαt2.(2.15)

On the other hand, due to

|II| ≤ CαZt

0

(1 + |η|)−m

2(m+2) (1 + ( τ

t)m+2

2|η|)−m+4

2(m+2)

τ|gα(τ, (m+2)η

2tm+2

2

)|

(1 + |η1|

tm+2

2

)s(1 + |η′|

tm+2

2

)|α|dτ,

then we can obtain as in (2.15)

ZRn|II |dξ ≤C t2.(2.16)

Substituting (2.15) and (2.16) into (2.14) yields

ZRn|u∧(t, ξ)|dξ ≤Ct2.

Consequently, |u(t, x)| ≤ RRn|u∧(t, ξ)|dξ ≤C t2, and the proof on Lemma 2.3 is completed.

Finally, we study the L∞property of solution to the 2-D linear problem (1.1) under the assumption (A3).

Lemma 2.4. If u(t, x)∈C([0, T ], H 1

2−(R2)) is a solution of the following linear problem

∂2

tu−tm∆u= 0,(t, x)∈[0, T ]×R2,

u(0, x) = 0, ∂tu(0, x) = ϕ(x),(2.17)

where ϕ(x)satisﬁes the assumption (A3), then u∈L∞([0, T ]×R2).

Remark 2.3. Due to ϕ(x)∈H1

2−(R2)by Lemma 2.1.(ii), then the optimal regularity of the solution u(t, x)

to (2.17) is L∞([0, T ], H 1

2+2

m+2 −(R2)) (see Proposition 3.3 in §3below). Thus, for m≥2, we can not derive

u(t, x)∈L∞([0, T ]×R2)directly by the Sobolev imbedding theorem. On the other hand, the proof procedure on

Lemma 2.4 will be rather useful in analyzing the singularity structure of u(t, x)in §6below.

Proof. In terms of Corollary 3.5 in [24], we have the following expression for the solution of (2.17)

u(t, x) = 2tCm(φ(1))φ(1)F(γ, γ; 1; 1) Z1

0

(1 −s2)−γ(∂tv)(sφ(t), x)ds, (2.18)

where Cm= ( 2

m+ 2)m

m+2 2−2

m+2 ,γ=m

2(m+ 2) ,F(γ, γ; 1; 1) = F(γ, γ; 1; z)|z=1 with F(γ, γ ; 1; z) a hypergeo-

metric function, which satisﬁes z(1 −z)ω′′(z) + (1 −(2γ+ 1)z)ω′(z)−γ2ω(z) = 0, and v(t, x) is a solution to

the following linear wave equation

∂2

tv−∆v= 0, v(0, x) = 0, ∂tv(0, x) = ϕ(x).(2.19)

From (2.19), we have

v(t, x) = 1

2πZB(x,t)

ϕ(ξ)

pt2−(x1−ξ1)2−(x2−ξ2)2dξ. (2.20)

9

Let vi(t, x) be the smooth solution to the linear wave equation ∂2

tvi−∆vi= 0 with the initial data (vi(0, x),

∂tvi(0, x)) = (0, ψi(x)). Then it follows from (2.20) and a direct computation that for t > 0 and x1>0, x2>0

(in other domains, the expressions are completely analogous)

v(t, x) =

v1(t, x) for x1

t≥1, x2

t≥1;

v1(t, x) + I1(t, x) for x1

t≤1, x2

t≥1;

v1(t, x) + I2(t, x) for x2

t≤1, x1

t≥1;

v1(t, x) + I1(t, x) + I2(t, x) for 0 < x1< t, 0 < x2< t,|x|> t;

v1(t, x) + I1(t, x) + I2(t, x) + I3(t, x) for x1>0, x2>0, |x|< t

(2.21)

with

I1(t, x) = 1

2πZt

x1

rdr Zarccos(x1

r)

−arccos(x1

r)

(ψ2−ψ1)(x−rω)

√t2−r2dθ,

I2(t, x) = 1

2πZt

x2

rdr Zarccos(x2

r)

−arccos(x2

r)

(ψ4−ψ1)(x−rω)

√t2−r2dθ,

I3(t, x) = 1

2πZt

|x|

rdr Zarccos(x2

r)

arcsin(x1

r)

(ψ1+ψ3−ψ2−ψ4)(x−rω)

√t2−r2dθ,

where ω= (cosθ, sinθ), r=p|x1−ξ1|2+|x2−ξ2|2and (x1−ξ1, x2−ξ2) = (rcosθ, rsinθ).

Due to ϕ(x)∈H1

2−(R2) by Lemma 2.1.(ii), then it follows from the regularity theory of solution to linear

wave equation that

v(t, x)∈C([0, T ], H 3

2−(R2)) ∩C1([0, T ], H 1

2−(R2)) ⊂W1,1([0, T ]×R2).

Thus, we can take the ﬁrst order derivative ∂tvpiecewisely for t > 0 and x1>0, x2>0 as follows

∂tv(t, x) =

∂tv1(t, x) for x1

t≥1, x2

t≥1;

∂tv1(t, x) + ∂tI1(t, x) for x1

t≤1, x2

t≥1;

∂tv1(t, x) + ∂tI2(t, x) for x2

t≤1, x1

t≥1;

∂tv1(t, x) + ∂tI1(t, x) + ∂tI2(t, x) for 0 < x1< t, 0 < x2< t,|x|> t;

∂tv1(t, x) + ∂tI1(t, x) + ∂tI2(t, x) + ∂tI3(t, x) for x1>0, x2>0, |x|< t.

(2.22)

Here we only treat the term ∂tI3in (2.22) since the treatments on ∂tI1and ∂tI2are analogous or even

simpler in their corresponding domains.

If we set ψ=ψ1+ψ3−ψ2−ψ4, then it follows from a direct computation that for x1>0, x2>0 and

|x|< t

I3(t, x) = Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

ψ(ξ)

pt2− |x−ξ|2dξ2.(2.23)

Taking the transformations x=ty and ξ=tη in (2.23) yields

I3(t, ty) = tJ (t, x

t)

where

J(t, z) = Z0

z1−√1−z2

2

dη1Z0

z2−√1−(z1−η1)2

ψ(tη)

p1− |z−η|2dη2for 0 <|z|<1 and z1, z2>0,

10

which derives

∂1I3(t, x) = (∂1J)(t, x

t), ∂2I3(t, x) = (∂2J)(t, x

t)

and thus

∂tI3(t, x) = I3(t, x)

t+1

tZ0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

ξ· ∇ξψ(ξ)

pt2− |x−ξ|2dξ2−x· ∇xI(t, x)

t.(2.24)

It is noted that for x1>0, x2>0 and |x|< t,

|∂1I3(t, x)|=|lim

h→0

I3(t, x1+h, x2)−I3(t, x1, x2)

h|

=Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

∂1ψ(ξ)

pt2− |x−ξ|2dξ2−Z0

x2−√t2−x2

1

ψ(0, ξ2)

pt2−x2

1−(x2−ξ2)2dξ2

=Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

∂1ψ(ξ)

pt2− |x−ξ|2dξ2+Zx2

√t2−x2

1

1

ψ(0, x2−sqt2−x2

1)d(arcsins)

=Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

∂1ψ(ξ)

pt2− |x−ξ|2dξ2+ψ(0,0)arcsin(x2

pt2−x2

1

)

−π

2ψ(0, x2−qt2−x2

1) + qt2−x2

1Zx2

√t2−x2

1

1

∂2ψ(0, x2−sqt2−x2

1)arcsins ds

≤CT1 + Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

1

pt2− |x−ξ|2dξ2

≤CT(1 + t) (2.25)

and

|∂2I3(t, x)|=Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

∂2ψ(ξ)

pt2− |x−ξ|2dξ2+ψ(0,0)arcsin(x1

pt2−x2

2

)

−π

2ψ(x1−qt2−x2

2,0) + qt2−x2

2Zx1

√t2−x2

2

1

∂1ψ(x1−sqt2−x2

2,0)arcsins ds

≤CT(1 + t) (2.26)

On the other hand, analogous computation yields for x1>0, x2>0 and |x|< t ≤T

|I3(t, x)

t| ≤ CTand

1

tZ0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

ξ· ∇ξψ(ξ)

pt2− |x−ξ|2dξ2≤CT.(2.27)

Therefore, ∂tI3(t, x)∈L∞in the domain {(t, x) : x1>0, x2>0,|x|< t ≤T}by (2.24). Similarly, we

can obtain ∂tI1(t, x)∈L∞and ∂tI2(t, x)∈L∞in the related domains, and thus ∂tv(t, x)∈L∞([0, T ]×R2).

These, together with (2.18), yield

u(t, x)∈L∞([0, T ]×R2) and ku(t, ·)kL∞(R2)≤C t

4

X

i=1 kψi(x)kC1.(2.28)

Consequently, we complete the proof of Lemma 2.4.

Remark 2.4. It is not diﬃcult that by the expression (2.21) of v(t, x), one can get v(t, x)∈C∞((0, T ]×

R2\Σ0∪Σ±

1∪Σ±

2), where Σ0={(t, x) : t > 0,|x|=t},Σ±

1={(t, x) : t > 0, x1=±t}and Σ±

2={(t, x) :

11

t > 0, x2=±t}. On the other hand, v(t, x)6∈ C2((0, T ]×R2)since v(t, x)has a strong singularity when the

variables (t, x)go across Σ0∪Σ±

1∪Σ±

2. Indeed, for example, it follows from (2.25) and a direct computation

that for x1>0, x2>0and |x|< t

∂2

12I3(t, x) = Z0

x1−√t2−x2

2

dξ1Z0

x2−√t2−(x1−ξ1)2

∂2

12ψ(ξ)

pt2− |x−ξ|2dξ2+3ψ(0,0)

pt2− |x|2

−π

2∂1ψ(x1−qt2−x2

2,0)+qt2−x2

2Zx1

√t2−x2

2

1

∂2

1ψ(x1−sqt2−x2

2,0)arcsinsds

−π

2∂2ψ(0, x2−qt2−x2

1) + qt2−x2

1Zx2

√t2−x2

1

1

∂2

2ψ(0, x2−sqt2−x2

1)arcsinsds

=3ψ(0,0)

pt2− |x|2+bounded terms.(2.29)

Thus, (2.29) implies ∂2

12I3(t, x)→ ∞ as (t, x)→Σ0since ψ(0) 6= 0 can be assumed without loss of generality

(this is due to the assumption of ψi(0) 6=ψj(0) for some i6=jand 1≤i < j ≤4in (A3)and the diﬀerent

expressions of ψ(x)in the related domains {(t, x) : t > 0,±x1>0,±x2>0}respectively). In addition, by

an analogous computation, we can derive that ∂2

12I1(t, x)and ∂2

12I2(t, x)are bounded for x1>0, x2>0and

|x|< t. Hence ∂2

12v(t, x)→ ∞ as (t, x)→Σ0and further v(t, x)6∈ C2((0, T ]×R2)is proved. However, by

the expression (2.18) and due to the lack of strong Huyghens’ principle for the Tricomi-type equations, we can

show that the solution u(t, x)6∈ C2((0, T ]×R2\Γ0∪Γ±

1∪Γ±

2)of (2.17) holds true in §6below, which implies

an essential diﬀerence between the degenerate equation and the strict hyperbolic equation.

§3. Some regularity estimates on the solutions to linear generalized Tricomi equations

At ﬁrst, we list some results on the conﬂuent hypergeometric functions for our computations later on.

The conﬂuent hypergeometric equation is

zw′′ (z) + (c−z)w′(z)−aw(z) = 0,(3.1)

where z∈C,aand care constants. The solution of (3.1) is called the conﬂuent hypergeometric function.

When cis not an integer, (3.1) has two linearly independent solutions:

w1(z) = Φ(a, c;z), w2(z) = z1−cΦ(a−c+ 1,2−c;z).

Below are some crucial properties of the conﬂuent hypergeometric functions.

Lemma 3.1.

(i) (pages 278 of [13])). For −π < argz < π and large |z|, then

Φ(a, c;z) = Γ(c)

Γ(c−a)(eiπǫz−1)a

M

X

n=0

(a)n(a−c+ 1)n

n!(−z)−n+O|z|−a−M−1

+Γ(c)

Γ(a)ezza−c

N

X

n=0

(c−a)n(1 −a)n

n!z−n+O|ezza−c−N−1|,(3.2)

where ǫ= 1 if Imz > 0,ǫ=−1if I mz < 0,(a)0≡1,(a)n≡a(a+ 1) · · · (a+n−1), and M, N = 0,1,2,3....

(ii) (page 253 of [13]). Φ(a, c;z) = ezΦ(c−a, c;−z).

(iii) (page 254 of [13]).

dn

dznΦ(a, c;z) = (a)n

(c)n

Φ(a+n, c +n;z) (3.3)

12

and

d

dz Φ(a, c;z) = 1−c

zΦ(a, c;z)−Φ(a, c −1; z).(3.4)

For such a problem

∂2

tu−tm∆u= 0,(t, x)∈[0,+∞)×Rn,

u(0, x) = φ1(x), ∂tu(0, x) = φ2(x),(3.5)

by the results in [23], one has for t≥0

u∧(t, ξ) = V1(t, |ξ|)φ∧

1(ξ) + V2(t, |ξ|)φ∧

2(ξ),(3.6)

where the expressions of V1(t, |ξ|) and V2(t, |ξ|) have been given in (2.10).

In order to analyze the regularities of u∧(t, ξ) in (3.6) under some restrictions on φi(x)(i= 1,2), we require

to establish the following estimates:

Lemma 3.2. For 0≤s1≤m

2(m+2) ,0≤s2≤m+4

2(m+2) and some ﬁxed positive constant T, if g(x)∈Hs(Rn)

with s∈R, then we have for 0< t ≤T

(i) (kV1(t, |ξ|)g∧(ξ)∨kHs+s1≤Ct−s1(m+2)

2kgkHs,

kV2(t, |ξ|)g∧(ξ)∨kHs+s2≤Ct1−s2(m+2)

2kgkHs.

(3.7)

(ii)

k∂tV1(t, |ξ|)g∧(ξ)∨kHs−m+4

2(m+2) ≤CkgkHs,

k∂tV2(t, |ξ|)g∧(ξ)∨kHs−m

2(m+2) ≤CkgkHs

(3.8)

Proof. (i) First, we ﬁx t= ( m+2

2)2

m+2 to show (3.7) (in this case, the corresponding variable zin (2.10)

becomes z= 2i|ξ|). Subsequently, for the variable t, as in [25] and so on, we can use the scaling technique to

derive (3.7).

Since Φ(a, c;z) is an analytic function of z, then Φ( m

2(m+2) ,m

m+2 ; 2i|ξ|) and Φ( m+4

2(m+2) ,m+4

m+2 ; 2i|ξ|) are bounded

for |ξ| ≤ C. On the other hand, it follows from (3.2) that for large |ξ|

|Φ( m

2(m+ 2) ,m

m+ 2; 2i|ξ|)| ≤ C(1 + |ξ|2)−m

4(m+2)

and

|Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2; 2i|ξ|)| ≤ C(1 + |ξ|2)−m+4

4(m+2) .

Thus, for any s1∈[0,m

2(m+2) ] and s2∈[0,m+4

2(m+2) ], by a direct computation, we arrive at

k(V1(( m+ 2

2)2

m+2 ,|ξ|)g∧(ξ))∨kHs+s1

=k(1 + |ξ|2)s+s1

2e−i|ξ|Φ( m

2(m+ 2) ,m

m+ 2 ; 2i|ξ|)g∧(ξ)kL2

≤k(1 + |ξ|2)s1

2Φ( m

2(m+ 2) ,m

m+ 2 ; 2i|ξ|)kL∞k(1 + |ξ|2)s

2g∧(ξ)kL2

≤CkgkHs(3.9)

and

k(V2(( m+ 2

2)2

m+2 ,|ξ|)g∧(ξ))∨kHs+s2≤CkgkHs.(3.10)

13

Next we treat k(V1(t, |ξ|)g∧(ξ))∨kHs+s1and k(V2(t, |ξ|)g∧(ξ))∨kHs+s2. To this end, we introduce the follow-

ing transformation

η=2

m+ 2tm+2

2ξ,

and then we have

k(V1(t, |ξ|)g∧(ξ))∨kHs+s1

=ZRn

(1 + |ξ|2)s1

2e−2i

m+2 tm+2

2|ξ|Φ( m

2(m+ 2) ,m

m+ 2;4i

m+ 2 tm+2

2|ξ|)(1 + |ξ|2)s

2g∧(ξ)

2

dξ1

2

=m+ 2

2tm+2

2n

2ZRn

(1 + |(m+ 2)η

tm+2

2|2)s1

2Φ( m

2(m+ 2) ,m

m+ 2 ; 2i|η|)G∧(η)

2

dη1

2

(3.11)

and

k(V2(t, |ξ|)g∧(ξ))∨kHs+s2

=tm+ 2

2tm+2

2n

2ZRn

(1 + |(m+ 2)η

2tm+2

2|2)s2

2Φ( m+ 4

2(m+ 2) ,m+ 4

m+ 2 ; 2i|η|)G∧(η)

2

dη1

2

,(3.12)

here and below the notation G∧(η) is deﬁned as

G∧(η) = 1 +

(m+ 2)η

2tm+2

2

2s

2

g∧((m+ 2)η

2tm+2

2

).

It is noted that

kG∧(η)kL2= (ZRn|(1 + |ξ|2)s

2g∧(ξ)|2(2tm+2

2

m+ 2 )ndξ)1

2≤Ct n(m+2 )

4kgkHs.(3.13)

Additionally, for 0 < t ≤Tand α≥0, we have

1 + |(m+ 2)η

2tm+2

2|2α

< Ct−α(m+2) (1 + |η|2)α.(3.14)

Thus, we obtain from (3.11)-(3.14) that for 0 < t ≤T

k(V1(t, |ξ|)g∧(ξ))∨kHs+s1

≤Ct−s1(m+2)

2−n(m+2)

4(ZRn|(1 + |η|2)s1

2Φ( m

2(m+ 2) ,m

m+ 2 ; 2i|η|)G∧(η)|2dη)1

2

=Ct−s1(m+2)

2−n(m+2)

4kG∧(η)kL2

≤Ct−s1(m+2)

2kgkHs(3.15)

and

k(V2(t, |ξ|)g∧(ξ))∨kHs+s2≤Ct1−s2(m+2)

2kgkHs.(3.16)

Consequently, we complete the proof of Lemma 3.2.(i).

(ii). It follows from a direct computation and (3.3)-(3.4) that

∂tV1(t, |ξ|)

=2i(m+ 2

4i)m

m+2 |ξ|2

m+2 zm

m+2 −1

2e−z

2Φ( m

2(m+ 2) ,m

m+ 2, z) +