Well posedness under Levi conditions for a degenerate second order Cauchy problem

Well Posedness Under Levi Conditions for a Degenerate Second Order Cauchy Problem.

ALESSIAASCANELLI(*)

ABSTRACT- We consider the Cauchy problem for a second order equation of hy- perbolic type which degenerates both in the sense that it is weakly hyperbolic and it has non Lipschitz continuous in time coefficients. Intersections between the roots of the equationare of a finite orderk, and the first time derivative of the principal part's coefficients present a blow-up phenomenon at the timetˆ0, behaving ast ^q,q1. The mixture of these two situations gives, under an ap- propriate Levi condition,C¹or Gevrey well posedness of the Cauchy problem, depending on the dominant between the two behaviors.

1. Introduction and main results.

Inthis paper we study the Cauchy problem

P(t;x;D_t;D_x)u(t;x)ˆ0; (t;x)2[0;T]Rⁿ; u(0;x)ˆu0(x);

@tu(0;x)ˆu1(x);

8>

<

>: (1:1)

for a second order operator of the form

PˆD²_t a(t;x;Dx)‡b(t;x;Dx)‡c(t;x);

(1:2)

a(t;x;j)ˆ Xⁿ

i;jˆ1

a_ij(t;x)j_ij_j; b(t;x;j)ˆXⁿ

jˆ1

b_j(t;x)j_j; Dˆ 1

p 1@, which is hyperbolic, i.e.

a(t;x;j)0; t2[0;T];x;j2Rⁿ:

(*) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy.

E-mail: [email protected]

We are going to allow (1.2) to degenerate both in the sense that its char- acteristic roots may coincide at some points and that its coefficients are not regular with respect to the time variable. The purpose of the paper is to obtain well posedness results for the degenerate problem (1.1). We remind that (1.1) is said to be well posed inthe spaceXif for everyu₀;u₁2Xthere is a unique solutionu2C¹([0;T];X).

We know that the Cauchy problem for a weakly hyperbolic equation may be not well posed inC¹, and thatC¹well posedness can be achieved by asking the equation to satisfy some condition (the Levi condition) on the first order termb. For example, the Cauchy problem for

PˆD²_t t^2`D²_x‡t^yDx

is well posed in C¹ if and only if y` 1; for y< ` 1, well posedness holds only in Gevrey classes of indexs<(2` y)=(` y 1), [12]. Inthe particular case of aneffectively hyperbolic operator, C¹ well posedness holds without assuming any Levi condition, see [16]; notice that if aˆa(t;j), (1.2) is effectively hyperbolic if P²

jˆ0j@_t^ja(t;j)j 6ˆ0, t2[0;T], jjj ˆ1:

An intermediate situation between effective and non effective hyperbo- licity is introduced in [9] for (1.2) with coefficients depending only on time: if there exists anintegerk2 an d ag2[0;1=2] such that

X^k

jˆ0

j@^j_ta(t;j)j 6ˆ0; jb(t;j)j Ca^g(t;j); t2[0;T];jjj ˆ1;

then(1.1) isC¹ well posed providedg1=2 1=k, and this choice ofgis optimal; otherwise, (1.1) is well posed inGevrey classes of index s<(1 g)=[1=2 …g‡1=k†]: Similar results for (1.2) depending also on space variables are givenin[11], [1], [2].

Strictly hyperbolic equations with non Lipschitz continuous in time coefficients of the principal part have been widely studied starting from [6];

different ways to weaken the Lipschitz regularity produce quite different effects. Here we are interested in the way considered by [7]: for (1.2) with coefficients depending only on time, and supposing the singular behavior

j@ta(t;j)j C

t^q; q1; t2]0;T];jjj ˆ1;

the authors show that (1.1) isC¹ well posed only ifqˆ1; otherwise, well posedness holds in Gevrey classes of indexs<q=(q 1):The index is sharp.

The same results with dependence also on space variables are in [4], [5].

The purpose of the paper is to study the two different kinds of de- generation mixed together. We consider (1.2) with the following structure:

PˆD²_t a(t)Q(t;x;Dx)‡b(t;x;Dx)‡c(t;x);

a2C¹[0;T]; Q(t;x;j)ˆXⁿ

i;jˆ1

q_ij(t;x)j_ij_j; 8>

><

>>

(1:3) :

the weak hyperbolicity condition is expressed by:

a(t)0; t2[0;T];

Q(t;x;j)q₀jjj²; q₀>0; t2[0;T];x;j2Rⁿ: (

(1:4)

We assume for (1.3) that:

i) there exist anintegerk2 an d ag2[0;1=2] such that X^k

iˆ0

ja⁽ⁱ⁾(t)j 6ˆ0; j@_x^bb_j(t;x)j C_ba(t)^g; jˆ1;. . .;n; t2[0;T];x2Rⁿ; b2Zⁿ_‡; 8>

><

>>

(1:5) :

ii) the coefficients ofQsatisfy j@_tq_ij(t;x)j C

t^q; q1; i;jˆ1; :::;n:

(1:6)

We prove that, under these assumptions, (1.1) is:

- C¹well posed ifqˆ1 an dg1=2 1=k, see Theorem 2.1;

- well posed inGevrey classes of index s<q=(q 1) if q>1 and g1=2 1=k, or s<minn q

q 1; 1 g

1=2 (g‡1=k)

o if q1 an d g<1=2 1=k, see Theorem 3.1.

Both results are in line with [7] and [9].

We use anapproach as similar as possible to the proofs of these results, consisting of three steps:

1) factorizationof the principal part ofPby means of regularized characteristic roots~l;

2) reductionof Puˆ0 to anequivalent 22 system LUˆ0 with Lˆ@_t iL(t;x;D_x)‡R(t;x;D_x), iˆ 

p 1

, L(t;x;j) a real diag- onal matrix having ~l as entries, and R(t;x;j) such that either R^t

0 jR(t;x;j)jdtc0‡dloghji, c0;d>0, t2[0;T]; hji ˆ(1‡ jjj²)¹⁼², in

the C¹ case or R^t

0 jR(t;x;j)jdtdhji^1=s, d>0, t2[0;T] inthe Gevrey case;

3) applicationof the energy estimate inSobolev or inGevrey-So- bolev spaces to the operatorL.

2. C¹well posedness.

This Sectionis devoted to a result ofC¹ well posedness for (1.1), (1.3).

We denote by H^s ˆH^s(Rⁿ) the usual Sobolev space, and by k k_s the Sobolev norm. The usual space of symbols on R²ⁿ is denoted by S^mˆS^m(RⁿRⁿ). The spaceB((0;T];S^m) consists of all functions defined in(0;T] and with values inS^m that are bounded as functions of time.

We prove the following:

THEOREM2.1. Consider the Cauchy problem(1.1), (1.3),under condi- tions(1.4), (1.5)forg1=2 1=k, and(1.6)with qˆ1. Then,(1.1)is C¹ well posed.

This result is consistent with these facts:

- takingkˆ2 one can choosegˆ0: indeed no Levi conditions are needed for an effectively hyperbolic operator, [16];

- gˆ1=2 canbe reached for k! 1: under the usual C¹ Levi condition there is no need to assume thataˆaQhas zeros of finite order, [10], [16].

Inthe proof of Theorem 2.1 we are going to make use of the following result [2]:

THEOREM2.2. Consider the operator Lˆ@_t i ~l₁(t;x;D_x) 0

0 ~l2(t;x;Dx)

!

‡R(t;x;D_x) (2:1)

t2[0;T], x2Rⁿ, where~l_j(t;x;j)2R,~l_j2L¹([0;T];S¹)for jˆ1;2;and R is a22matrix satisfying

R2L¹([0;T];S¹);

R(t;x;j)

j j W(t;j); W2L¹([0;T];S¹);

R^T

0j@_j^bW(t;j)jdtd_bhji ^jbjlog (1‡ hji); b0:

8>

>>

><

>>

>>

: (2:2)

Then, there existsd>0such that the energy estimate kU(t)k²_mC_m kU(0)k²_m‡d‡

Z^t

0

kLU(t)k²_m‡ddt 0

@

1 A

holds for all U2C¹([0;T];H^m‡d)\C([0;T];H^m‡d‡1).

Theorem 2.2 implies well posedness, with the loss of dderivatives, in H^‡1ˆ \sH^sandH ¹ ˆ [sH^s of the Cauchy problem forL.

PROOF OF THEOREM2.1. The characteristic roots ofPare l(t;x;j)ˆ 

a(t)Q(t;x;j) p

with Q2C([0;T];S²), @_tQ2B((0;T];S²): Givena functionr2C₀¹(R), 0r1, R

r(t)dtˆ1, and extended the symbol Q on R_t by setting Q(t;x;j)ˆQ(T;x;j) fortT,Q(t;x;j)ˆQ(0;x;j) fort0, we approx- imatelby defining new symbols~l(t;x;j) as follows:

~l(t;x;j)ˆ



a(t)‡ hji ²

q Z 

Q(t;x;j)

p r(hji(t t))hjidt:

(2:3)

Thenwe use~lto factorize (1.3). Inthe factorizationwe need to deal with the symbols

@t~l(t;x;j)ˆ ia⁰(t) 2 

a(t)‡ hji ²

q I1(t;x;j)‡



a(t)‡ hji ² q

I2(t;x;j) (2:4)

and

(2:5) ~l…t;x;j† l…t;x;j† ˆ



a…t† ‡ hji ² q

I₃…t;x;j†

‡(



a(t)‡ hji ²

q 

pa(t)

) 

Q(t;x;j) p

where

I₁(t;x;j)ˆ

Z 

Q(t;x;j)

p r(hji(t t))hjidt;

I₂(t;x;j)ˆ

Z 

Q(t;x;j)

p r⁰(hji(t t))hji²dt;

I3(t;x;j)ˆ

Z 

Q(t;x;j)

p 

Q(t;x;j)

p

r(hji(t t))hjidt:

The change of variablehji(t t)ˆsclearly givesI₁2L¹([0;T];S¹). Using

Rr⁰(s)dsˆ0 we canrewrite I2(t;x;j)ˆ

Z 

Q(t;x;j)

p 

Q(t;x;j)

p

r⁰(hji(t t))hji²dt;

by the hypothesis onQwe get

I22C([0;T];S²); tI22B((0;T];S¹);

I32C([0;T];S¹); tI32B((0;T];S⁰):

The same arguments give

[l(t;x;Dx)]² [~l(t;x;Dx)]²ˆ 

a(t)‡ hDxi ² q

‡a(t)

I4(t;x;Dx);

(2:6) with

I₄(t;x;j)2C([0;T];S²); tI₄2B((0;T];S¹):

From (2.3), (2.4), (2.6) we come to the factorization

(2:7) Pˆ …Dt ~l…t;x;Dx††…Dt‡~l…t;x;Dx†† ‡ a⁰…t†

a…t† ‡ hD_xi ²

q S1…t;x;Dx†

‡(



a(t)‡ hD_xi ² q

‡a(t))S₂…t;x;D_x† ‡b…t;x;D_x† ‡c…t;x;D_x†;

with

S₁(t;x;j)2L¹([0;T];S¹);

S₂(t;x;j)2C([0;T];S²); tS₂(t;x;j)2B((0;T];S¹):

To perform now the reduction to a first order system, we consider the operatorv(t;Dx) with symbol

v(t;j)ˆ 

a(t)‡ hji ² q

hji;

and we define the vectorV ˆ(v0;v1) as follows:

v₀ˆv(t;D_x)u;

v₁ˆ(D_t‡~l(t;x;D_x))u:

( (2:8)

By (2.8), and after a straightforward diagonalization of matrixM(t;x;Dx), we obtainthat the scalar problem (1.1) is equivalent to

LUˆ0;

U(0;x)ˆU₀; (

(2:9)

whereUˆM(t;x;Dx)V and Lˆ @_t i

~l(t;x;Dx) 0 0 ~l(t;x;Dx) 0

@

1

A‡ a⁰(t)

a(t)‡ hD_xi ²A(t;x;D_x)

‡b(t;x;D_x)hD_xi ¹ a(t)‡ hDxi ²

q B(t;x;D_x)‡C(t;x;D_x)‡E(t;x;D_x);

(2:10)

with 22 matricesA;B;C;Esuch that

A;B;E2L¹([0;T];S⁰); C2C([0;T];S¹); tC 2B((0;T];S⁰):

Operator (2.10) has the structure (2.1) for a remainder R(t;x;j)ˆ a⁰(t)

a(t)‡ hji ²A(t;x;j)‡C(t;x;j)

‡b(t;x;j)hji ¹ a(t)‡ hji ²

q B(t;x;j)‡E(t;x;j):

We only have to check thatRfulfills condition (2.2); then an application of Theorem 2.2 will giveC¹well posedness of (1.1) by usual arguments in the energy method.

For anyN2, a⁰(t)

a(t)‡ hji ²A(t;x;j)ˆ a⁰(t)

(a(t)‡ hji ²)^{1 1=N} A(t;x;j) (a(t)‡ hji ²)^1=N; we know from Lemma 1 in [10] that iff 2C^N[0;T] is real valued and non negative, thenf^1=Nis absolutely continuous on [0;T]; we apply the Lemma tof(t)ˆa(t)‡ hji ²and we obtain

a⁰(t)A(t;x;j)= a(t)‡ hji ²

2L¹([0;T];S^N2);N2:

Using the Levi condition, we split b(t;x;j)hji ¹B(t;x;j)

a(t)‡ hji ²₁₌₂ ˆ b(t;x;j)hji ¹

(a(t)‡ hji ²)^g B(t;x;j) (a(t)‡ hji ²)^{1=2 g}; and get

b(t;x;j)hji ¹B(t;x;j)a(t)‡ hji ² ₁₌₂

2L¹([0;T];S^{1 2g}):

The first condition in (2.2) holds true.

Now we have to constructWas in(2.2). To this aim we take a smooth function c, 0c1, c(y)ˆ1 forjyj 1, c(y)ˆ0 for jyj 2, and we introduce the function

~W(t;j)ˆc…thji†dhji ‡d

t…1 c(thji)†; d>0:

We canchoose d so large that jC(t;x;j)j ~W(t;j); for all (t;x;j). We define

W(t;j)ˆK a⁰(t)

a(t)‡ hji ²‡ 1

(a(t)‡ hji ²)^{1=2 g}‡~W(t;j)

!

; K>0 and large enough to havejR(t;x;j)j W(t;j) for all (t;x;j).

We get, from Lemma 1 and Lemma 2 of [9], that Z^T

0

ja⁰(t)j

a(t)‡ hji ²dtc₀‡dloghji;

Z^T

0

1

(a(t)‡ hji ²)^{1=2 g}dtc₀‡dloghji;

8>

>>

>>

>>

><

>>

>>

>>

>>

:

for somec₀>0;d>0, and thanks tog1=2 1=k. As to~W, we have Z^T

0

~

W(t;j)dtd Z

2hji¹

0

hjidt‡d Z^T

2hji ¹

(1=t)dtc₀‡dloghji:

Condition (2.2) is fulfilled. Theorem (2.1) is proved.

3. Gevrey well posedness.

Inthis Sectionwe prove a result of Gevrey well posedness for (1.1), (1.3). Fors1, we denote by G^sˆG^s(Rⁿ) the Gevrey class of indexs, consisting of all functionsf such that

j@_x^bf(x)j CA^jbjb!^s; C;A>0; 8b2Zⁿ_‡:

Inthe proof we use Gevrey-Sobolev spacesH^s;e;sˆH^s;e;s(Rⁿ), defined for e0 an ds1 by

H^s;e;s(Rⁿ)ˆ fu(x);e^ehDxi1=su2H^s(Rⁿ)g;

there the norm iskuk_s;e;sˆ ke^ehDxi1=suk_s:Notice thatH^s;e;sG^s; e>0:

A class of bounded pseudodifferential operators in Sobolev-Gevrey spaces is the class S^m;sˆS^m;s(RⁿRⁿ) of Gevrey symbols, defined for s1 as the space of all functionsa(x;j) satisfying

j@^a_j@_x^ba(x;j)j C_aA^jbjb!^shji^{m jaj}; a;b2Zⁿ_‡; A;c_a>0;

which is the limit space S^m;s:ˆlim

`!‡1

S^m;s_{`</sub> ; S<sup>m;s</sup><sub>`} :ˆ lim

A!‡1!

S^m;s_{`;A</sub> of the Banach spacesS<sup>m;s</sup><sub>A;`} of all symbols such that

jaj_m;s;A;`:ˆ sup

jaj`;b2Zⁿ‡

supx;j j@_j^a@_x^ba(x;j)jA ^jbjb! ^shji ^m‡jaj<‡1:

Inwhat follows,B((0;T];S^m;s) will denote the space of all functions defined in(0;T] and with values inS^m;sthat are bounded as functions of time.

We are going to prove the following:

THEOREM3.1. Consider the Cauchy problem(1.1), (1.3)under condi- tions(1.4), (1.5)with C_bˆCA^jbjb!^s,(1.6).

Ifg1=2 1=k and q>1, then problem(1.1)is G^swell posed provided 1s< q

q 1: (3:1)

Ifg<1=2 1=kandq1, thenproblem (1.1) isG^swell posed for 1s<s0ˆmin q

q 1; 1 g 1=2 (g‡1=k)

: (3:2)

REMARK 3.2. Inthe proof of Theorem 3.1, we are precisely going to show that:

- 1q< 2k

k‡2, 0g<1 q 1 2‡1

k

)s₀ˆ 1 g 1=2 (g‡1=k); - 1q< 2k

k‡2, 1 q 1 2‡1

k

g1

2)s₀ˆ q q 1; - q 2k

k‡2, 0g1

2)s0ˆ q q 1.

It becomes easy to compare these results with [7], [9] if only we notice that

1 q…1=2‡1=k† 1=2 1=k; 8q1; k2;

1 q(1=2‡1=k)0 forq2k=(k‡2):

Inthe proof of Theorem 3.1 we apply the following result [2]:

THEOREM 3.3. Consider the operator L in (2.2), suppose

~l_j 2L¹([0;T];S^1;s)for jˆ1;2and

jR(t;x;j)j W(t;j); W2L¹([0;T];S¹);

Z^t

0

W(t;j)dt2C([0;T];S^1=s):

8>

>>

<

>>

>: (3:3)

Then, there existl0>0and b(t)2L¹[0;T], b(t)0, such that for w(t;j)ˆexp

Z^t

0

W(t;j)‡b(t)hji^1=s

dt 0

@

1 A

the energy estimate

kw(t;D_x)U(t)k²_m;l;sC_m kU(0)k²_m;l;s‡ Z^t

0

kw(t;D_x)LU(t)k²_m;l;sdt 0

@

1 A;

0tT; w(T;j)e^lhji1=s;

holds for all U2C¹([0;T];H^m;l;s)\C([0;T];H^m‡1;l;s),0<ll0. PROOF OFTHEOREM3.1. We define

~l(t;x;j)ˆ



a(t)‡ hji ^11g

q Z 

Q(t;x;j)

p r(hji(t t))hjidt;

thenwe use the same arguments of the proof of Theorem 2.1 inthe frame of the calculus of Gevrey symbolsS^m;s and we come to the following fac- torizationforP:

Pˆ (D_t ~l(t;x;D_x))(D_t‡~l(t;x;D_x))‡ a⁰(t) a(t)‡ hD_xi ^11g

q S₁(t;x;D_x)

‡



a(t)‡ hDxi ^11g q

‡a(t)

S2(t;x;Dx)‡S3(t;x;Dx)

‡b(t;x;Dx)‡c(t;x;Dx);

where

S₁(t;x;j)2L¹([0;T];S^1;s);S₃2L¹([0;T];S^{2 11g;s});

S2(t;x;j)2C([0;T];S^2;s); t^qS2(t;x;j)2B((0;T];S^1;s):

Defining this time

v(t;j)ˆ



a(t)‡ hji ^11g q

hji;

we retrace the reductionto a first order system of Theorem 2.1, coming to the equivalent Cauchy problem (2.9) for

Lˆ @_t i ~l(t;x;Dx) 0 0 ~l(t;x;Dx)

‡ a⁰(t)

a(t)‡ hDxi ^11gA(t;x;D_x)‡b(t;x;D_x)hD_xi ¹ a(t)‡ hDxi ^11g

q B(t;x;D_x)

‡C(t;x;Dx)‡ D(t;x;Dx) a(t)‡ hD_xi ^11g

q ‡E(t;x;Dx);

(3:4)

where

A;B;E2L¹([0;T];S^0;s); D2L¹([0;T];S^{1 11g;s});

C2C([0;T];S^1;s); t^qC2B((0;T];S^0;s):

System (3.4) has the structure (2.1) with R(t;x;j)ˆ a⁰(t)

a(t)‡ hji ^11gA(t;x;j)‡ b(t;x;j)hji ¹ a(t)‡ hji ^11g

q B(t;x;j)

‡C(t;x;j)‡ D(t;x;j) a(t)‡ hji ^11g

q ‡E(t;x;j):

We are going to show that there existd>0 an d 0<h<1=s,sas in(3.1) or (3.2), such that

Z^t

0

R(s;x;j)dsdhji^h; t2[0;T]:

(3:5)

Then, an application of Theorem 3.3 withW(j)ˆdhji^hwill give Gevrey well posedness of (1.1) by usual arguments.

By Lemma 1 in[9] we have again Z^T

0

ja⁰(t)j

a(t)‡ hji ^11gdtc0‡dloghji; c0;d>0:

By the Levi condition in (1.5) we split b(t;x;j)hji ¹B(t;x;j)

a(t)‡ hji ^11g₁₌₂ ˆ b(t;x;j)hji ¹

a(t)‡ hji ^11g_g B(t;x;j) a(t)‡ hji ^11g_{1=2 g}; and Lemma 2 in [9] gives

Z^T

0

1

(a(t)‡ hji ^11g)^{1=2 g}dt c1‡dloghji; if g1=2 1=k dhji^{1=2 g 1=k†=(1 g)}; if g<1=2 1=k 8<

: for positive constantsd,c₁.

As to the new termD(t;x;j)(a(t)‡ hji ^11g) ¹⁼², by Lemma 2 we obtain

hji^{1 11g} Z^T

0

1 a(t)‡ hji ^11g

q dtdhji^{(1=2 g 1=k)=(1 g)}; d>0:

Finally, ifqˆ1 we have

jC(t;x;j)j W(t;~ j);

Z^T

0

~

W(t;j)dtc₂‡dloghji; c₂;d>0;

while if q>1 we need to mix the two estimates C2C([0;T];S^1;s), t^qC2B((0;T];S^0;s) to find an intermediate one. Given any e2(0;1), we introduce a separation in the phase space:

- fort^{1 e}hji^{(1 e)=q}1 we useC2C([0;T];S^1;s), - fort^{1 e}hji^{(1 e)=q}1 we uset^qC2B((0;T];S^0;s), and we obtain

j@_j^g@_x^bC(t;x;j)j 1

t^{1 e}hji^{1 (1 e)=q jgj}; g;b2Zⁿ_‡; that is the global estimate

t^{1 e}C2B((0;T];S^{1 (1 e)=q;s});

wich under the integral sign becomes Z^T

0

jC(t;x;j)jdtdhji^{1 (1 e)=q};d>0:

Thus:

- if q>1 an d g1=2 1=k, then(3.5) holds true with hˆ1 1 e

q ; it is possible to choose an e2(0;1) such that h<1=s, thanks to (3.1);

- if qˆ1 an d g<1=2 1=k, then(3.5) is satisfied with hˆ1=2 (g‡1=k)

1 g ; clearlyh<1=s, since we are supposing (3.2);

- ifq>1 an dg<1=2 1=k, thenwe get (3.5) with hˆmax 1 1 e

q ;1=2 (g‡1=k) 1 g

; (3:6)

and condition (3.2) provides h<1=s. To make precise estimate (3.6), we compute the crucial exponent

gˆ1 q 1 e

1 2‡1

k

;

and notice that g0 if q2k=(k‡2): This leads us to consider three cases:

- if 1q<2k=(k‡2) and 0g<1 q…1=2‡1=k†, then hˆ[1=2 (g‡1=k)](1 g), and we gets0ˆ(1 g)=[1=2 (g‡1=k)];

- if 1q<2k=(k‡2) and 1 q…1=2‡1=k† g1=2, then hˆ1 (1 e)=qands₀ˆq=(q 1);

- ifq2k=(k‡2), againhˆ1 (1 e)=q, an ds₀ˆq=(q 1):

The proof of Theorem (3.1) is complete via Theorem 3.3.

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Manoscritto pervenuto in redazione il 3 ottobre 2005