Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001

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Trace identities

and

universal estimates

for

eigenvalues

of linear pencils*

Michael Levitin

Department of Mathematics, Heriot-Watt University

Riccarton

Edinburgh EH14 4AS, U. K.

Leonid Parnovski

Department of Mathematics, University College London

Gower Street,

London WC1E 6BT, U. K.

Email : M.Levitin@ma.hw.ac.uk ; Leonid@math.ucl.ac.uk

Abstract

We describe

the

method

constructing

the

spectral trace identities

and the

estimates

eigenvalue gaps

for the

linear self-adjoint operator pencils

—

XB.

1 Introduction

Since 1950's, mathematicians have devoted

a lot of

efforts

to the

construction

of the

so-called universal eigenvalue estimates

for

elliptic boundary value problems.

As an

illus-

tration, we recall

the

original result

Payne, Polya

and

Weinberger [5], who have shown

1956

that

{A.,}

is the set of

(positive) eigenvalues

of the

Dirichlet boundary value

problem

for the

Laplacian

in a

domain

fi C

, then

—

< — y \j, (i)

ran '—'

3=1

for each

m = 1,2,.... The

term universal estimate

used

this context since

(1)

does

not involve

any

information

on the

domain

apart from

the

dimension

The Payne-Polya-Weinberger inequality

has

been significantly improved over

the

course

several decades; similar universal estimates have been also obtained

in the

spec-

tral problems

for

operators other then

the

Euclidean Dirichlet Laplacian

(or

Schrodinger

"The research

M.L.

was

supported

by the

EPSRC grant GR/M20990. Both authors were also

partially supported

by the

EPSRC Spectral Theory Network

160

161

operator), e.g. higher order differential operators in R", operators on manifolds, systems

like Lame system of elasticity, etc. We refer the reader to the important paper [3] and

the recent survey [1] for more details.

In the recent paper [4], we have shown that the majority of the universal eigenvalue

estimates known so far can be easily obtained from a certain operator trace identity which

is valid not only for boundary value problems for PDEs but, under minimal conditions,

for a general self-adjoint operator acting in a Hilbert space. Namely, we have proved the

following

Theorem 1.1 Let H and G be self-adjoint operators such that G(D

) C D

. Let \j and

4>j

be eigenvalues and eigenvectors of H. Then for each j :

KIM&M£ ,

[WM

(2)

= 5>*-Ai)|(Gfc,&>|

This result implies

Theorem 1.2 Under conditions of Theorem 1.1,

m m

-(A

m+1

- A

) £([[#, G],G]^) < 2^||[tf,G]^||

. (3)

3=1 3=1

The inequalities (3) may be viewed as a family of estimates of the spectral gap A

—

for the operator H, with different estimates being obtained for each choice of the

operator G. The particular choice of G depends, of course, on the problem, and cannot

be prescribed. In [4], we give numerous examples of concrete estimates for boundary

value problems for elliptic PDEs (and systems of PDEs) obtained using Theorem 1.2.

In particular, the classical estimate (1) follows from Theorem 1.2 by taking G to be an

operator of multiplication by the coordinate

X[,

summing the resulting equalities over I,

and using some elementary bounds. The aim of this paper is to extend Theorems 1.1

and 1.2 to the spectra of linear self-adjoint operator pencils of the type A

—

\B. The

statement of the problem and the main results are collected in Section 2; the proofs are

in Section 3.

2 Statement of the problem and main results

We consider a linear operator pencil

V{\) = A-XB, (4)

acting in a Hilbert space

equipped with the scalar product (.,.) and the corresponding

norm ||.||. Here, A and B are self-adjoint operators such that D

C D

- We identify

the domain of the pencil

D-p

with DA- By the spectrum of V we understand the set of

162

complex values A

for

which

V(X) is not

invertible. Throughout this paper

assume,

for simplicity, that

the

spectrum consists

isolated real eigenvalues

Ai < A

< ...

(counted with multiplicity) which

may

accumulate only

to +00, and

that

the

system

corresponding eigenfunctions Uj such that

V(\J)UJ

= 0, is

complete

in H (the

case

of a

pencil with

continuous spectrum

can be

treated

well,

cf.

Remark 2.5

[4]).

refer

[2] for a

general discussion

of the

spectral theory

operator pencils.

An important property, which

shall

use

later

on, is the

fact that, under

technical

condition

(BUJ,UJ)

^ 0

(which

satisfied automatically

if, e.g., B > 0), the

eigenfunc-

tions

can be

assumed

to be

normalised

by the

relations

(Buj,u

)

= S

. (5)

Indeed, multiplying

the

equality V{\j)Uj

= 0 by u

in H, and

using

the

fact that

and

B are

self-adjoint,

we get

{Auj,u

)

(uj,Au

)

= \j

(Buj,u

)

which implies

(5).

By completeness,

any

element

/ e K can be

expanded

in a

convergent series

^{Bf,u

and

for any

f,g£H,

{Bf,g)

Y,{Bf:U

)-{u

,Bg).

Thus,

by the

density argument,

(f,g) = J2(f>

*)-(

9)- (

)

In what follows,

[H, G]

denotes

the

standard commutator

—

GH.

Our main result

is the

following spectral trace identities

for

'P(A).

Theorem

2.1 Let V(X) = A

—

\B be a

self-adjoint linear operator pencil with discrete

spectrum

\j and

eigenfunctions

Uj.

Let

auxiliary self-adjoint operator with domain

such that G(D-p)

D-p

DQ. Then

for

each

^=-lm^G

hG]

]}

(7)

and

£(A*

- \,)\

(BG

)

= -i

<[[7>(A,-),G] ,<?]«,•,«,•>

•

(8)

163

Remark 2.2 Instead of the condition G(D(A)) C D(A) we can impose weaker conditions

GUJ

e D(A),

e D(A), j = 1,.... Moreover, the latter condition can be dropped if

the double commutator is understood in the weak sense, i.e., if the right-hand side of (7)

and (8) is replaced by

([V(X),G]UJ,GUJ)

(see (13) below).

The trace identities (8) imply the following universal estimate, which generalises the

Payne-Polya-Weinberger inequality for linear operator pencils.

Theorem 2.3 Under conditions of Theorem 2.1

771

-(A

i-A

)^([p(A

),G],GK,

,)<2^||B-

[P(A

),GK.||

. (9)

J=l

3=1

Proof of the main results

We start with establishing (8). Obviously, we have

[A - XjB,

G) UJ

= (A - \jB)Guj. (10)

Therefore,

([A - XjB, G]

)

= {(A - XjB)

GUJ,GUJ)

. (11)

Since A, B and G are self-adjoint, we have, using (6)

{{A - XjB)

GUJ,GUJ)

= ]T ({A - XjB)

GUj,u

)

,BGUJ)

Y,^

BGu

(12)

= ^(Afc-A^IBGuj.u*!

Using the fact that [A - XjB,G] is skew-adjoint, the left-hand side of (11) can be

rewritten as

(G [A - XjB,

Uj,

) = -

{{[A

- XjB, G],

Uj,

) + {[A - XjB, G]

,Uj)

= -

({{A

XjB,G],G]UJ,UJ)

(UJ,

[A - XjB,G]Uj),

(G[A -

XJB,G]UJ,UJ)

= --

{[[A~XjB,G],G]uj,Uj)

(13)

(notice that (G [A - XjB,

Uj,Uj) is real, see (11) and (12)). This proves (8).

Since (10) implies

{[A - XjB,G]Uj,u

) =

(GUJ,

[A - XjB,G]u

) = {X

- Xj) {BGuj,u

this also proves (7).

164

We now proceed to the proof of (9). Let us sum the equations (7) over j = 1,

...,m.

Then we have

^ g w^rf...£

r(1)Wi%i

„,

(14)

Parceval's equality implies that the left-hand side of (14) is not greater than

"m+1

—

This proves (9).

References

[1] M.S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral

theory and geometry (Edinburgh, 1998), E.B. Davies and Yu Safarov, eds., London

Math. Soc. Lecture Notes, vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp.

95-139.

[2] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nons elf adjoint Oper-

ators. Transl. Math. Monographs, vol. 18, AMS, Providence, 1969.

[3] E.M. Harrell II, J. Stubbe, On trace identities and universal eigenvalue estimates for

some partial differential operators. Trans. Amer. Math. Soc. 349 (1997), 2037-2055.

[4] M. Levitin, L. Parnovski, Commutators, spectral trace identities, and uni-

versal estimates for eigenvalues. J. Funct. Anal, (to appear), preprint at

http://www.ma.hw.ac.uk/~levitin/research.html.

[5] L.E. Payne, G. Polya, H.F. Weinberger, On the ratio of consecutive eigenvalues. J.

Math. Phys. 35(1956), 289-298.

Universal estimates

for the

blow-up rate

in a

semilinear heat equation

Julia MATOS«, Philippe SOUPLET (

)

W Departement de Mathematiques,

Universite d'Evry Val d'Essonne

Boulevard des Coquibus, 91025 Evry Cedex, France

(2)

Departement de Mathematiques, INSSET,

Universite de Picardie, 02109 St-Quentin, France

(3)

Laboratoire de Mathematiques Appliquees, UMR CNRS 7641,

Universite de Versailles, 45 avenue des

Etats-Unis, 78302 Versailles, France

Email : jmatos@maths.univ-evry.fr ; souplet@math.uvsq.fr

1 Introduction

We consider

the

semilinear heat equation

= Au +

\u\P~\ 0<(<T,

ieR" , .

u (x,

= u

(x)

i£R",

(

where

p > 1 and u

L°°(R

Let u be the

unique classical solution

of (1)

which

denned

on a

maximal time interval

[0,

T). This solution

u is

bounded

[0,

T"]

x R^, for

all

0 < T" < T.

When

T <

oo, we have ||u(t)||oo

—>

oo as i

—>

where

•

denotes

the

norm

L°°(H

and we say

that

blows

up in

finite time

Denote

= ^| if N > 3 and p

= oo if N = 1,2. It is

well-known that

if p < p

and

u is a

nonnegative solution

(1) which blows

up in

finite time

then there exists

constant

C > 0,

depending

on u,

such that

||«(t)lloo<c(r-0"^, o<t<r.

(2)

Giga

and

Kohn [8] obtained this upper bound

on the

blow-up rate

of u(t) and

proved

that

(2)

still holds

for

sign-changing solutions provided (3N-A)p

3N+8. Furthermore,

with

restriction

on p > 1, the

same estimate holds

(cf.

Bricher [3]

and

Matos [14])

for

any nonnegative solution

of (1)

which

nondecreasing

time

and

such that

its

blow-up

set

compact. (Recall that

a G R

is in the

blow-up

set if

u(t

) —•

oo for

some

—»

T and x

—>

a.)

However,

it is

known that

one may

have

faster blow-up rate

in dimensions

N > 11 for

sufficiently large values

of p > ps (see

Herrero

and

Velazquez

[12])-

165

166

On the other hand, for

global

positive solutions of (1) on a smoothly bounded domain

fi C R

, with homogeneous Dirichlet boundary conditions, a universal a priori bound of

the form

suplluWHoc^C^.p.r), forallr>0, (3)

t>T

was recently established by Fila, Souplet and Weissler [4] under the assumption (N— l)p <

N + 1, and by Quittner [19] provided iV < 3 and p < p$.

The first goal of this note, motivated by (2) and (3), is to establish a global a priori

bound for the blow-up rate of nonnegative solutions of (1) which is universal, that is

independent of the solution u.

2 Main results

Theorem 2.1 Let p > 1 and e € (0,1). Let w

G L°°(R

), u

> 0, and assume that one

of the following conditions holds:

(i)p<l + %,

(ii)

is radially symmetric and nonincreasing as a function of r = \x\, and

(N-2)p<N + 2 ifN<3,

{N-2)p<N ifN>3.

{

There exists a constant C = C(p, N, e) > 0, independent ofu, such that if the classical

solution u of (1) exists on [0,T) x H

, then

lh(t)||oo<C(T-t)-A,

eT<t<T.

(5)

It is unknown whether the upper blow-up rate (2) can fail for all N > 3 and all

P > Ps- However it is interesting to note that when N > 3 and p > p

, the universal

global blow-up rate (5) can be true for no value of e £ (0,1), even if one restricts to

radially symmetric nonincreasing initial data. (Indeed this would imply that Theorem 2.2

below holds for such p - see after that Theorem.)

Let us note that Theorem 2.1 holds whenever u exists on [0,T) x R" and that we

do not need to assume that u blows up at t = T. Actually, Theorem 2.1 has several

interesting consequences, not only related to blowing-up solutions. The first one concerns

the decay rate in large time of

global

nonnegative solutions of (1).

Theorem 2.2 Letp > 1 satisfy (4)- Assume thatuo € L°°, u

> 0, is radially symmetric

and nonincreasing as

function ofr = \x\, and that the classical solution u of (1) is

global

(i.e., T = oo,). Then there exists a constant C = C(p,N) > 0, independent ofu, such

that

\\u(t)

||oo

<Cf^, t>0. (6)

167

When p < ps and u

, Vu

satisfy a fast decay condition (namely, / \f(x)\

2/4

< oo), the estimate (6) was obtained by Kavian [13], with a constant C depending on

u. The result of [13] relied on the method of forward self-similar variables. Interestingly,

although a result on global solutions, Theorem 2.2 relies on the use of

backward

self-similar

variables (and on some new ideas - see section 3). Later in [21], the second author proved

that when p < p

, any global nonnegative solution of (1) (with w

6 L

nL°°) must satisfy

lim

||w(t)||

= 0.

On the other hand, we see that (6) cannot be true unless p < p$. Indeed, when

N > 3 and p > Ps, there exist positive, classical stationary solutions of (1), which are

radially symmetric nonincreasing (see e.g. Haraux and Weissler [10]). Also, note that

the self-similar solutions to (1) (constructed by [10] for p >

4- 2/JV) decay precisely like

||«(*)lloo =Cr^.

For other sufficient conditions ensuring global existence and decay of positive solutions

of (1), we refer to the recent paper by Gui, Ni and Wang [9] and to the references therein.

In particular it is observed in [9] (see p. 590) that all previous results concerning the decay

in time provide rates no slower than t~p^ when p < ps- (Some solutions with slower decay

rates are constructed in [9] for N > 11 and sufficiently large values of p > p§.) From the

works [13, 10, 9] and the references in [9], it thus seems natural to make the following

conjecture:

Conjecture 2.3 When p < p$, all

global

nonnegative classical solutions of (1) decay at

least like i"

/^"

' as t -> oo.

Theorem 2.2 proves this conjecture in dimensions N < 3 in the radial symmetric case.

Moreover, the estimate is global on (0, oo), with a universal constant.

As a consequence of Theorem 2.2, we obtain a new kind of parabolic Liouville type

Theorem, concerning solutions of (1) that are defined globally on (—00,00).

Corollary 2.4 Let p > 1 and let u be a

global

nonnegative classical solution of

= Au + u

, — 00 < t < 00, x € R

Assume that (4) holds and that for all t, u(t,.) is radially symmetric and nonincreasing

as a function of r

—

\x\. Then u = 0.

Furthermore, we partially improve a parabolic Liouville Theorem of Merle and Zaag,

concerning solutions of (1) that are defined globally in the past (see Corollary 1.6 in [16]).

Corollary 2.5 Assume p > 1, T < 00 and let u be a

global

nonnegative solution of

= Au + u

, -00

<t<T,

i6R

with u(t,.) £ L°° for all t < T. Moreover, assume that one of the following conditions

holds:

(i) p <

+ |,

168

(ii) for all t <T, u(t,.) is radially symmetric and nonincreasing as

function ofr = \x\,

and (4) holds.

Then u = 0 or there exists T

> T such that u(t,x) = K(T

—

where

= (p-l)-i/(p-D.

In Corollary 1.6 of [16], the same conclusion was obtained under the additional as-

sumption that there exists C > 0 such that

u(t,x) <C{T-t)-

(

-''-

, -oo

<t<T,

leR",

But instead of (i) or (ii), it was only assumed that (N

—

2)p < N +

We prove Corollary

2.5 as a consequence of Theorem 2.1 and of [16, Corollary 1.6].

Finally, we obtain an estimate for the initial blow-up rates of local solutions to u

Au + u

Corollary 2.6 Assume p > 1, 0 < T < oo and let u be a nonnegative classical solution

= Au + u

0<t<T,

ie R

with u(t,.) e L°° for all t < T. Moreover, assume that (i) or (ii) in Corollary 2.5 is

satisfied. Then

\Ht)\\oo < Ct~&, 0<i<T/2, (7)

where C = C(p, N) > 0.

Results on initial blow-up rates for equation (1) were first derived by Bidaut-Veron [2].

Namely, it was proved there that (7) holds under the condition p < N(N + 2)/(N

—

without radial symmetry restriction. Note that this condition is better than ours when

N > 4, worse when N = 2,3 and identical when N = 1. The method of [2] is completely

different from ours and quite involved (relying on Bernstein type gradient estimates,

Aronson-Serrin Harnack inequalities and multiplier arguments). Also for the Cauchy-

Dirichlet problem on a bounded domain, initial blow-up rates up to the boundary are

studied by Quittner and the second author in [20].

Remark 2.7 a. Theorem 2.1 fails for e = 0.

b. It is well-known that all blowing-up solutions of (1) satisfy the following universal

global lower

bound:

H^lloo^er-i)"^

, o<t<r, (8)

where

K = (p -

1)-V(P-I).

c. Assuming p > 1 and (N

—

2)p < N + 2, and w

£ i?

), Merle and Zaag

[17] derived the following uniform estimates of order one for the blow-up rate of a

169

nonnegative solution

u of (1): for all e > 0,

there exists

= t

(e) € (0,T)

such

that

)

7bl

(

)\^^

+{^

)

l]og{

t)l

forallte[t

,T).

The constant

appearing

in the

term

order

one is

optimal

and t

(e)

depends

N_K

the H

norm

the initial data

UQ-

However this estimate only holds

on a

neighbourhood

of the

blow-up time

ofu(t), whereas estimate

(5) is

global

in the

existence time interval

the solution.

d. It

seems that when

p <

2/iV,

the

estimate

(5)

can be obtained

by the

methods

[1], where, among other things,

the

authors prove upper blow-up rates

for a

reaction-

diffusion system which generalizes equation

(1). The

arguments

[1]

are

rather

involved

and

rely

on a

Moser's iteration scheme.

Our

approach here

completely

different

and

simpler, especially

in the

case

p <

+ 2/N.

Throughout this note,

denotes various positive constants which depend only

on the

indicated arguments.

In the next sections, we present the main tools and ideas used

the proofs

Theorems

2.1

and

2.2.

For the

detailed proofs

the

reader

referred

to [15].

3 Main tools: self-similar variables

and

convolution

Lebesgue spaces

The study

the blow-up rate

of u(t) is

done

using

the

method

backward self-similar

variables, introduced

Giga

and

Kohn [7]

and

Galaktionov

and

Posashkov [5].

For

each

R/^, we

rescale

u by

similarity variables around

{T,a) by

setting

l0g(T

y =

W=t (9)

{s,y)

(T-t)^u(t,x).

This function

w = w

defined

, +oo)

x R

with

—

logT,

and

satisfies

Aw-\yVw +

\w\

w-f^,

s>s

jeR", ,

(so,y) =

<l>{v),

2/eR

where

<f>(y)

T^u

y/Ty).

The

study

of the

boundedness

of (T - t)r-^

IKOIU

when

approaches

T is

equivalent

to the

study

of the

boundedness

of the

global solution

. Theorem 2.1

has

then

the

following equivalent formulation:

Let

8 > 0.

Assume either

(i) or (ii).

Then there exists

constant

C =

C(p,N,8)

> 0

such that

for any

global nonnegative classical solution

w of

(10),

holds

II^MIloo

< C for all s >

+ 6.