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On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

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Abstract

In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation $\p_t^2u-t^m\Delta u=f(t,x,u)$ with typical discontinuous initial data $(u(0,x), \p_tu(0,x))=(0, \vp(x))$; here $m\in\Bbb N$, $x=(x_1, ..., x_n)$, $n\ge 2$, and $f(t,x,u)$ is $C^{\infty}$ smooth in its arguments. When the initial data $\vp(x)$ is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane ${t=x_1=0}$, it is shown that the local solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^n)$ exists and is $C^{\infty}$ away from the forward cuspidal cone $\Gamma_0=\bigl{(t,x)\colon t>0, |x|^2=\ds\f{4t^{m+2}}{(m+2)^2}\bigr}$ and the characteristic cuspidal wedge $\G_1^{\pm}=\bigl{(t,x)\colon t>0, x_1=\pm \ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$, respectively. On the other hand, for $n=2$ and piecewise smooth initial data $\vp(x)$ singular along the two straight lines ${t=x_1=0}$ and ${t=x_2=0}$, we establish the local existence of a solution $u(t,x)\in L^{\infty}([0,T]\times\Bbb R^2)\cap C([0, T], H^{\f{m+6}{2(m+2)}-}(\Bbb R^2))$ and show further that $u(t,x)\not\in C^2((0,T]\times\Bbb R^2\setminus(\G_0\cup\G_1^{\pm}\cup\G_2^{\pm}))$ in general due to the degenerate character of the equation under study; here $\G_2^{\pm}=\bigl{(t,x)\colon t>0, x_2=\pm\ds\f{2t^{\f{m}{2}+1}}{m+2}\bigr}$. This is an essential difference to the well-known result for solutions $v(t,x)\in C^{\infty}(\Bbb R^+\times\Bbb R^2\setminus (\Sigma_0\cup\Sigma_1^{\pm}\cup \Sigma_2^{\pm}))$ to the 2-D semilinear wave equation $\p_t^2v-\Delta v=f(t,x,v)$ with $(v(0,x), \p_tv(0,x))=(0, \vp(x))$, where $\Sigma_0={t=|x|}$, $\Sigma_1^{\pm}={t=\pm x_1}$, and $\Sigma_2^{\pm}={t=\pm x_2}$.
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
tutmu=f(t, x, u) with typical discontinuous initial
data (u(0, x), ∂tu(0, x)) = (0, ϕ(x)); here mN,x= (x1, ..., xn), n2, and f(t, x, u) is Csmooth 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 Caway 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\0Γ±
1Γ±
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\0Σ±
1Σ±
2)) to the 2-D semilinear wave equation 2
tvv=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
tutmu=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 mN,x= (x1, ..., xn), n2, ∆ =
n
X
i=1
2
i,f(t, x, u) is Csmooth 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 i4) 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
2m
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+
1Γ
1), here Γ±
1={(t, x) :
t > 0, x1=±2tm
2+1
m+ 2 }.
(iii) For n= 2, under the condition (A3), if m9, 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\Γ0Γ±
1Γ±
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 Cproperty 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 m9in (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
twtmw=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 0s < 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
Zφ(t)φ(τ)
0
dr1rZB(x,r)
g(τ, y )
pr2− |xy|2dyr=r1
(r1+φ(t) + φ(τ))γ
×(φ(τ)r1+φ(t))γFγ, γ ; 1; (r1+φ(t)φ(τ))(r1φ(t) + φ(τ))
(r1+φ(t) + φ(τ))(r1φ(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
rRB(x,r)
g(τ, y )
pr2− |xy|2dy=rZr
0Z2π
0
g(τ, x1+pr2q2cosθ, x2+pr2q2sinθ)dqdθholds and thus
the Lproperty 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\Γ0Γ±
1Γ±
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\Σ0Σ±
1Σ±
2)will hold
true if v(t, x)is a solution to the 2-D semilinear wave equation 2
tvv=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
tutu=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 uC([0, T ], H s(Rn))
C1([0, T ], H s5
6(Rn)) C2([0, T ], H s11
6(Rn)) for some T > 0 under the crucial assumption that the support
of f(t, x, u) on the variable tlies in {t0}. 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 Lproperty of solution v(t, x) to the linear
problem 2
tvtmv=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 Lestimates, 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, Π1i,jn(xij)kij ϕ(x)Hn
2(Rn) for any kij N∪ {0}in the case of (A1), and
(x11)k1Π2inki
iϕ(x)H1
2(Rn) for any kiN∪ {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
ttm, 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\Γ0Γ±
1Γ±
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 Lproperty 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)Hsδ(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 Lproperty of solution to the linear problem 2
tutmu=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 i4), 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
2δˆϕ(ξ)L2({|ξ| ≤ 1}).
For |ξ|>1, we decompose ˆϕinto two parts
ˆϕ(ξ) = Z|x|<1
|ξ|
eix·ξϕ(x)dx +Z1
|ξ|≤|x|≤1
eix·ξϕ(x)dx
=I+II =I+
n
X
=1
χ(ξ)II, (2.2)
where {χ}n
=1 is a Cconic decomposition of unity corresponding to the domain {ξRn:|ξ| ≥ 1}, moreover
ξ6= 0 in suppχ.
Obviously, the term Ican be dominated by the multiplier of |ξ|n.
On the other hand, for any 1 n,
χ(ξ)II =χ(ξ)
|ξ|nZ1≤|x|≤|ξ|
eix·ξ
|ξ|g(x
|ξ|,x
|x|)dx
=χ(ξ)
|ξ|n
1
iξ
|ξ|Z1≤|x|≤|ξ|
eix·ξ
|ξ|g(x
|ξ|,x
|x|)dx
+χ(ξ)
|ξ|n
1
iξ
|ξ|Z|x|=1
eix·ξ
|ξ|g(x
|ξ|,x
|x|)cos(~n, x)dS
=χ(ξ)
|ξ|n
1
(iξ
|ξ|)mZ1≤|x|≤|ξ|
eix·ξ
|ξ|m
lg(x
|ξ|,x
|x|)dx
+
m1
X
k=0
χ(ξ)
|ξ|n
1
(iξ
|ξ|)k
1
(iξ
|ξ|)Z|x|=1
eix·ξ
|ξ|k
g(x
|ξ|,x
|x|)cos(~n, x)dS
=III +IV. (2.3)
Due to
|k
g(x
|ξ|,x
|x|)| ≤
k
X
j=0 X
|α|≤kj
Cαj |(j
α
yg)( x
|ξ|,x
|x|)| |x|(kj)|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 mnand |ξ| ≥ 1, we have |ˆϕ(ξ)| ≤ C(1 + ln|ξ|)
|ξ|nby (2.2)-(2.4), which derives (1 +
|ξ|)n
2δˆϕ(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 i4).
Since
|ˆϕ(ξ)| ≡Z1
0Z1
0
ψ1(x)eix·ξdx +Z0
1Z1
0
ψ2(x)eix·ξdx
+Z0
1Z0
1
ψ3(x)eix·ξdx +Z1
0Z0
1
ψ4(x)eix·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|)1δ|ˆϕ(ξ)|2
C
2
Y
i=1 Z
1
(1 + |ξi|)1δ
|ξi|2i+C
2
X
i=1 Z
1
(1 + |ξi|)1δ
|ξi|2i+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})n1, we can also have that α
xϕ(x)
H1
2δ(Rn)for any δ > 0small, here x= (x2, ..., xn). Thus, (1 + |ξ|)1
2δ(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
tutmu= 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
xψ(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)eix·ξdx with ξRnand 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
2+m
(m+ 2)τ
dv
+v= 0.(2.7)
6
As in [25], taking z2=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 t0 and ξRn
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|) = ez
2Φ( m
2(m+2) ,m
m+2 ;z),
V2(t, |ξ|) = tez
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
2δ(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)|C t ZRn
e2i
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)
Ctm+ 2
2tm+2
2nZRn
Φ( m+ 4
2(m+ 2) ,m+ 4
m+ 2 ; 2i|η|)ϕ((m+ 2)η
2tm+2
2
)
Ctm+ 2
2tm+2
2nZRn
1
(1 + |η|2)m+4
4(m+2) |ϕ((m+ 2)η
2tm+2
2
)|(by (2.11))
Cαtm+ 2
2tm+2
2n
2ZRn
1
(1 + |η|2)m+4
2(m+2)
1
(1 + |η1|
tm+2
2
)12δ
1
(1 + |η|
tm+2
2
)2|α|1
2
(by (2.12))
CαtZRn
1
(1 + tm+2|η|2)m+4
2(m+2)
1
(1 + |η1|)12δ
1
(1 + |η|)2|α|1
2
(choosing |α|= [ n
2] + 1 >n
2)
CTt13(m+2)δ
4ZR
1
(1 + |η1|)1+δ11
2
(choosing δ < m+ 4
3(m+ 2) )
CT(choosing δ < 4
3(m+ 2) ).
7
Similarly,
ZRn|y1(t, ξ)|Cm+ 2
2tm+2
2nZRn
Φ( m
2(m+ 2) ,m
m+ 2 ; 2i|η|)ψ((m+ 2)η
2tm+2
2
)
Cαm+ 2
2tm+2
2n
2ZRn
1
(1 + |η|2)m
2(m+2)
1
(1 + |η1|
tm+2
2
)32δ
1
(1 + |η|
tm+2
2
)2|α|1
2
CαZRn
1
(1 + |η1|)32δ
1
(1 + |η|)2|α|1
2
CT(choosing |α|= [ n
2] + 1 and 0 < δ < 1).
Thus, |u(t, x)| ≤ RRn|y(t, ξ)|RRn|y1(t, ξ)|+RRn|y2(t, ξ)|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
xf(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
tutmu=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)
(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)|+Zt
0|V1(t, |ξ|)V2(τ, |ξ|)f(τ , ξ)|
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
)|
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|
8
and thus
ZRn|I|Cαt1(m+2)n
4Zt
0
ZRn
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|1
2
CαtZt
0
ZRn
1
(1 + |η1|)2s(1 + |η|)2|α|1
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 |C t2.(2.16)
Substituting (2.15) and (2.16) into (2.14) yields
ZRn|u(t, ξ)|Ct2.
Consequently, |u(t, x)| ≤ RRn|u(t, ξ)|C t2, and the proof on Lemma 2.3 is completed.
Finally, we study the Lproperty 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
tutmu= 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 uL([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 m2, 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)((t), x)ds, (2.18)
where Cm= ( 2
m+ 2)m
m+2 22
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
tvv= 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(x1ξ1)2(x2ξ2)2dξ. (2.20)
9
Let vi(t, x) be the smooth solution to the linear wave equation 2
tvivi= 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
t1, x2
t1;
v1(t, x) + I1(t, x) for x1
t1, x2
t1;
v1(t, x) + I2(t, x) for x2
t1, x1
t1;
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)
(ψ2ψ1)(x)
t2r2dθ,
I2(t, x) = 1
2πZt
x2
rdr Zarccos(x2
r)
arccos(x2
r)
(ψ4ψ1)(x)
t2r2dθ,
I3(t, x) = 1
2πZt
|x|
rdr Zarccos(x2
r)
arcsin(x1
r)
(ψ1+ψ3ψ2ψ4)(x)
t2r2dθ,
where ω= (cosθ, sinθ), r=p|x1ξ1|2+|x2ξ2|2and (x1ξ1, x2ξ2) = (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
t1, x2
t1;
tv1(t, x) + tI1(t, x) for x1
t1, x2
t1;
tv1(t, x) + tI2(t, x) for x2
t1, x1
t1;
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+ψ3ψ2ψ4, then it follows from a direct computation that for x1>0, x2>0 and
|x|< t
I3(t, x) = Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
ψ(ξ)
pt2− |xξ|22.(2.23)
Taking the transformations x=ty and ξ=in (2.23) yields
I3(t, ty) = tJ (t, x
t)
where
J(t, z) = Z0
z11z2
2
1Z0
z21(z1η1)2
ψ()
p1− |zη|22for 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
x1t2x2
2
1Z0
x2t2(x1ξ1)2
ξ· ∇ξψ(ξ)
pt2− |xξ|22x· ∇xI(t, x)
t.(2.24)
It is noted that for x1>0, x2>0 and |x|< t,
|1I3(t, x)|=|lim
h0
I3(t, x1+h, x2)I3(t, x1, x2)
h|
=Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
1ψ(ξ)
pt2− |xξ|22Z0
x2t2x2
1
ψ(0, ξ2)
pt2x2
1(x2ξ2)22
=Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
1ψ(ξ)
pt2− |xξ|22+Zx2
t2x2
1
1
ψ(0, x2sqt2x2
1)d(arcsins)
=Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
1ψ(ξ)
pt2− |xξ|22+ψ(0,0)arcsin(x2
pt2x2
1
)
π
2ψ(0, x2qt2x2
1) + qt2x2
1Zx2
t2x2
1
1
2ψ(0, x2sqt2x2
1)arcsins ds
CT1 + Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
1
pt2− |xξ|22
CT(1 + t) (2.25)
and
|2I3(t, x)|=Z0
x1t2x2
2
1Z0
x2t2(x1ξ1)2
2ψ(ξ)
pt2− |xξ|22+ψ(0,0)arcsin(x1
pt2x2
2
)
π
2ψ(x1qt2x2
2,0) + qt2x2
2Zx1
t2x2
2
1
1ψ(x1sqt2x2
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
x1t2x2
2
1Z0
x2t2(x1ξ1)2
ξ· ∇ξψ(ξ)
pt2− |xξ|22CT.(2.27)
Therefore, tI3(t, x)Lin the domain {(t, x) : x1>0, x2>0,|x|< t T}by (2.24). Similarly, we
can obtain tI1(t, x)Land tI2(t, x)Lin 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\Σ0Σ±
1Σ±
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 Σ0Σ±
1Σ±
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
x1t2x2
2
1Z0
x2t2(x1ξ1)2
2
12ψ(ξ)
pt2− |xξ|22+3ψ(0,0)
pt2− |x|2
π
21ψ(x1qt2x2
2,0)+qt2x2
2Zx1
t2x2
2
1
2
1ψ(x1sqt2x2
2,0)arcsinsds
π
22ψ(0, x2qt2x2
1) + qt2x2
1Zx2
t2x2
1
1
2
2ψ(0, x2sqt2x2
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 1i < 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\Γ0Γ±
1Γ±
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) + (cz)w(z)aw(z) = 0,(3.1)
where zC,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) = z1cΦ(ac+ 1,2c;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(ca)(eiπǫz1)a
M
X
n=0
(a)n(ac+ 1)n
n!(z)n+O|z|aM1
+Γ(c)
Γ(a)ezzac
N
X
n=0
(ca)n(1 a)n
n!zn+O|ezzacN1|,(3.2)
where ǫ= 1 if Imz > 0,ǫ=1if I mz < 0,(a)01,(a)na(a+ 1) · · · (a+n1), and M, N = 0,1,2,3....
(ii) (page 253 of [13]). Φ(a, c;z) = ezΦ(ca, 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) = 1c
zΦ(a, c;z)Φ(a, c 1; z).(3.4)
For such a problem
2
tutmu= 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 t0
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 0s1m
2(m+2) ,0s2m+4
2(m+2) and some fixed positive constant T, if g(x)Hs(Rn)
with sR, then we have for 0< t T
(i) (kV1(t, |ξ|)g(ξ)kHs+s1Cts1(m+2)
2kgkHs,
kV2(t, |ξ|)g(ξ)kHs+s2Ct1s2(m+2)
2kgkHs.
(3.7)
(ii)
ktV1(t, |ξ|)g(ξ)kHsm+4
2(m+2) CkgkHs,
ktV2(t, |ξ|)g(ξ)kHsm
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
2ei|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|)kLk(1 + |ξ|2)s
2g(ξ)kL2
CkgkHs(3.9)
and
k(V2(( m+ 2
2)2
m+2 ,|ξ|)g(ξ))kHs+s2CkgkHs.(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
2e2i
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
1
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
1
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
1
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 )n)1
2Ct n(m+2 )
4kgkHs.(3.13)
Additionally, for 0 < t Tand α0, we have
1 + |(m+ 2)η
2tm+2
2|2α
< Ctα(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
Cts1(m+2)
2n(m+2)
4(ZRn|(1 + |η|2)s1
2Φ( m
2(m+ 2) ,m
m+ 2 ; 2i|η|)G(η)|2)1
2
=Cts1(m+2)
2n(m+2)
4kG(η)kL2
Cts1(m+2)
2kgkHs(3.15)
and
k(V2(t, |ξ|)g(ξ))kHs+s2Ct1s2(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
2ez
2Φ( m
2(m+ 2) ,m
m+ 2, z) +