On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations

ATHEMATICS AND COMPUTERS N SIMULATION ELSEVIER

Mathematics and Computers in Simulation 37 (1994) 265-277

On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations Jerry

Bona

a,b,*, Keith

Promislow

a, Gene

Wayne

a

a Department of Mathematics, The Pennsylvania State Uniuersity, University Park, PA 16802, USA b Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802, USA

Abstract The renormalization techniques for determining the long-time asymptotics of nonlinear parabolic equations pioneered by Bricmont, Kupiainen and Lin are shown to be effective in analyzing nonlinear wave equations featuring both dissipation and dispersion. These methods allow us to recover recent results of Dix in a way which is both transparent and has interesting prospects for generalization.

1. Motivation

and statement

This study is focussed equation u, -Mu

+ u,,,

of the main results

on the long-time

+ L&l, = 0,

behavior

of solutions

(x E R, t > l),

of the damped,

non-linear

wave

(1.1)

with initial data

U(‘, 1) =f”(*),

(XE q.

(l-2)

Here M is a Fourier-multiplier operator, which in Fourier transformed variables has the form u, where i

2. The initial-value problem (1.1)~(1.2) is always %?~(k) = - I k I 2pA locally well posed, but for larger values of p and large initial data, it appears that it may not be globally well posed (see Bona et al. [1,2]). If attention is restricted to small initial data, then (l.l)-(1.2) is globally well posed and it is not difficult to see in this case that solutions decay to the zero function as t becomes unboundedly large. It is our purpose here to determine the detailed structure of this evanescence.

* Corresponding

author.

037%4754/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0378-4754(94)00019-G

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This program has already been carried out in the elegant paper of Dix [4] (see also Naumkin and Shishmarev [5] for the case p = 1). We will show how some of Dix’s results follow readily from a suitable adaptation of the renormalization techniques put forward by Bricmont et al. [3] in the context of nonlinear parabolic equations. While the results obtained in this way are no more telling than those of Dix, they are here obtained in a transparent and appealing way. Furthermore, this method appears to be a promising avenue of approach for the derivation of more refined aspects of the long-time asymptotics of solutions. The principle result derived herein is easily motivated by consideration of the linear initial-value problem u, - MU + MU,,, = 0, u(.G 1) =f&), with parameter

(1.3)

77,which is formally solved by the formula

U(X, t) = S(t)fO = S,(t)fO = /_,, e-(IklZPfi’Jk3Xt-‘) ePikX&(k) dk,

(1.4)

where &(k) = jY,eikxf,(x) d x is the Fourier transform of fO. For large t, the kernel in (1.4) is very small except where I k I is near 0. For such values of k, I k I 2p x=- 77I k I 3, and hence if & is sufficiently regular, m

u=

,-lklZP(f-I)

/

e-ikx

(&O)+ k&O)+ @I k2I)

dk

--to

+-&)/m

e-ikx

e-lklZa(f-l)

dk.

(1.5)

-m

Defining f*@)

= /* e-ikx ,-IkIZp(f-l) dk, --m and making the change of variables k’ = kt’/2p u(x, t) =

(1.6) in (1.5) yields

$gf*(x,tl/‘P)

(l-7)

as t -+ co. Our main result, here stated informally to provide the reader with a concrete goal, shows both that the approximations made to reach (1.7) are justified, and that they remain so even in the context of the non-linear initial-value problem (1.G(1.2). Theorem 1. For a sufficiently small and smooth initial condition E > 0, there exist constants A, C,,

C, > 0 depending

fo, and

for t < p < 1, p > 2, and only on the giuen data such that

(1.8a) (1.8b) where f * is given in (1.6).

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267

Remark. A careful look at the proof in Section 3 will convince the reader that the theory could

be formulated in a sharper way, in which for a given p, the decay results in (1.8) are valid for any p 2 2p. In applications of (1.1) to physical problems, the dependent variable is real-valued and consequently non-integer values of p are seen to be somewhat artificial. If non-integer p are contemplated, a complex-valued version of the theory for (l.l)-(1.2) would be required. In consequence, we have eschewed the extra precision that might be possible in favor of the simpler development that is available for integer values of p. The plan of the paper is straightforward. In Section 2 detailed consideration is given to the linear initial-value problem (1.3). This discussion leads naturally to the introduction of a particular weighted norm that turns out to be very useful in the analysis. The theory for the linear problem is both instructive and useful when attention is turned to the nonlinear problem in Sections 3 and 4. The paper concludes with a short summary and suggestions for further inquiry.

2. Norms and the linear problem The approach taken is to apply the renormalization group ideas of Bricmont et al. [3] to the equation (1.1). It is convenient to start with the linear equation (1.3) since the theory for this simple situation already deviates from that of Bricmont et al. [3], because the main ideas appear in bold relief, avoiding the technicalities associated with the nonlinear term, and because use will be made of the linear theory in studying the nonlinear initial-yalue problem. Let 9’ be the Banach space of functions f such that their Fourier transform f lies in E”‘(R), and for which the norm IIf II = SUP (1 + I k I “) I f(k)

I+

SYP I f”‘(k)

I

(2.1)

k

is finite. Note that this norm controls both the L,- and L,-norms of f. Note also that f belongs to 9 if, for example, f lies in W3~‘(R>and xf is a member of L,(Iw). The renormalization group map is now introduced. Taking the solution u given by (1.4), define U&, t) = Lu(Lx,

L%)

(2.2)

for L 2 1. The (linear) renormalization f is (R,,,f)(~)

group map is denoted R,,,

and its action on a function

= %(X7 l),

(2.3)

where u is the solution of (1.4) with initial data f, uLis as in (2.21, and the second subscript on in (1.3). Direct calculation shows that uL satisfies

R connotes the coefficient of the dispersive term u,,,

(2.4) Thus the resealing accomplished

in the definition

of RL_ diminishes

the dispersive term if

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/3 < 3/2 and L is large. Moreover, equation, it transpires that R”

and Computers in Simulation 37 (1994) 265-277

because of the semi-group

property

of the evolution

=R L,a”-~0 RL,a”-~ 0 * * * 0 R,,, 0 RL,I,

(2.5)

where “,‘l LzpP3. Observe that the putative similarity function f* defined in (1.61, which according to the intuition provided by (1.5) captures the long-term asymptotics of solutions of (1.31, is a fixed point

RL,Of*=f*

P-6)

of the renormalization group map R, 0 without dispersion. Assuming that the cy in (2.5) is less than one, which will be true if p <‘5 and L > 1, it is a reasonable conjecture that R, an converges to R,,, as y1 becomes unboundedly large. It is possible then that successive applications of RL+” for it large drive one toward f *. Thus for a given initial datum f, the right-hand side of (2.5) applied to f may, in the right circumstances, converge to f *. This in turn means that R,,,, f + f * and this result, properly interpreted, is exactly what we are after. The latter observation leads to a search for conditions under which R,,, is a contractive mapping. Lemma 1. Let the power p in the dissipative operator be positive and suppose g ~9’ satisfies i(O) = 0. Then th ere exists a positive constant C = C(p) such that

IIR,,,g

II < C(P)L-‘II

g II.

The constant C(p) is independent

W) of 7 E [O, 11.

Proof. The solution of (1.3) with initial condition

D(k, t) = e-

g, written in Fourier-transformed

variables, is

(Ik12P+ik’Xr-l)g+(k),

and therefore

Since $X0) = 0 and g^E C’(R), the Mean-Value Theorem implies that for any k, there is a point 5 = tk with I tk I < k/L such that

In consequence, s;p(l+

we have that

Ik13)IG(k)l

G -&,(I+

G g(P)

I~13)l~l~~~k~2Bl~‘(5~)l II g II.

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Similarly, one determines

and Computers in Simulation 37 (1994) 265-277

that

;(~i(k)=(-(WI 2p-1 + 3iL2SP3qk2)(l

and the result follows.

- L-“P)z(k/L)

q

It will also be useful to understand the action of R,,, large. To this end, we state and prove another lemma.

on the fixed point f* as L becomes

Lemma 2. Suppose that i

L,, one has 11&nf*

-f*

269

and a constant C = C(p)

11
(2.8)

for n = 1, 2,. . . , where CY= L2p-3 and f * is defined in (1.6). Proof. First notice that

(Rz)(k)

=

e-jkj2pL-2~,

e-(Ik~28+i~“k3X1-L-ZP)

from which is follows that (R2)(k) Attention

_flh(k)

=

e-lk128(e-ia”k’(l-~-28)

_ 1).

is now turned to estimating the quantity

SUP(~+

lk13)@>)(k)-f?X(k)l.

k

The estimate is made in two parts. Suppose first that k, L and n are such that I k ( 3 < CY-“~ for some fixed positive ,X < 1. Then it is easily seen that le- ia”k3(1--L-“)- 1 I < Can(l--~~),

while

(1 + I k I “) e-l’lZP

< C(p)

for all k. If on the other hand k, L and y1are such that I k ) 3 2 a-‘+‘, I e-ia”k”(l

-I!-‘~)

-II<2

and, therefore, ,,,;;:

(l+

Ik~3)e-~k~‘B~e-(1/2)a~2e”“3s~p(l+

-w G C(p)


1 L”

e-

(1/q~(3-2mwnP/3

jk13)e-1/21k12”

then

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for L large enough since p, ,U > 0 and 3 - 2/3 > 0. If Al.is restricted to lie in the interval (0, (2 - 2P)/(3 - 2/?)l, then 1 n (Yn(l-c~)= L(2P-3Wl-cL)nG _ iL 1 * In consequence of the above inequalities, depending only on p such that

it is seen that there

are constants

C and L,

n

for L 2 L,. A similar set of inequalities shows that

for L large, where C again depends only on p. q The lemma is thereby established. Remark. If p = 1, the foregoing reasoning leads to an estimate of the form

II&of

* -f*

II < C

valid for any E > 0, where C depends on E as well as p. These two lemmas provide the tools needed to show convergence of the composition of the renormalization group maps. More precisely, suppose f = f0 ~9 to be given and define f, for 122 1 by the formula f,, = RL,oln0 RL,anm~0 . . . 0 R,,, fo.

(2.9)

We aim to show that {f,}~=, is a convergent sequence in 9 and to obtain the rate of convergence. Turning to the just mentioned task, for each IZ= 1, 2,. . . , write fn =A,f*

+g,

where z(O) = 0 so that A, = E(O). Straightforward f n+l

=

computations

RL+n+, f,, =A,,RL,an+lf * + RL,an+lg,,

=A,f*

+A,(RL,an+lf*

-f*)

+R,,,n+lgn.

It is easy to verify ;hat Gf (0) = fi0) ( see Proposition one has A, = A = f,,(O) for all n and g n+l

=

show that

A( RL,an+,f * -f *) +R,,,,+lgn.

Applying Lemmas 1 and 2 leads to the estimate IIg,+l II GA&

+ +

2.1 in Bona et al. [2]). Consequently,

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valid for all n 2 1, where C, which is larger than 1 without loss of generality, depends only on p. A simple induction then shows that n+l

n+l

II

fo II>

from which is follows at once that for all n, n

II fo II.

(2.10)

Recalling from the definition (2.3), that f,(x) = LnU(Lnx, L2pn),

where u is the solution of (1.3) with initial value u(., 1) = fo, and setting t = L2@“, we see that f,(x) = t”%(Xwfi,

t)

and that nL-” < C(log t)t-(‘m

We have thus established the following proposition. Proposition 1. For i < /3< 1 and any initial condition f. ES’, there exists a constant C > 0 and a time to > 0 such that for t > to, the solution v of (1.3) with 17= 1 satisfies IIv(*WP,

t) --At-“2fif

*(*)

II GC(t-“P

log t) II f. II,

(2.11)

where A = f&O) and f * is defined in (1.6).

for the sequence of times t, = L2@. However, the result is self improving if one notes that as in Bricmont et al. [3], the analysis is unchanged if L2@ is replaced by 7L2@ throughout for 1 G 7 < L 2p. One thereby infers (2.11) for all t > L2P. The above analysis may be adapted to the case 0 < j3 G + in the linear case. One has to compensate for the non-differentiability of ( k 1” for p in this range by modifying the norm defining the space 9, replacing the Cl-norm on f^ by an appropriate Holder norm. Remarks. The proof only showed convergence

3. The nonlinear

problem

Attention is returned to the nonlinear problem (1.1)~(1.2) with the aim of showing that the term upu,, p 2 2, does not affect the results of Proposition 1 provided IIf. )I is sufficiently small. Let u be a solution of (1.1)~(1.2) and consider the renormalized version UL(X, t) = Lu(Lx, L%) of u. The renormalized

(34

solution uL satisfies the evolution equation

$LIL = Mu, - L2+%;u,

- L2~-p-1Ll~aXu,.

(34

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272

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Introduce the notation (Y= L20P3 as in Section 2, y = L2P-p-1, and let R,-,ll,p denote the renormalization map (3.1) corresponding to the semi-group for equation (1.1) with coefficient 7 for the dispersive term and p for the nonlinear term. Thus if f ~9 is given, then ( RL,7,pf)( x) is u,(x, 1) where uL is as in (3.1) and u is the solution of the initial-value problem u, + pupu, + ruxnx -Mu = 0, u(x, 0) =f(x),

(3.3)

for x E R and t > 0. Note that (RL,l,lfO)(~) = Lu(Lx, L2P) where u solves (l.l)-(1.2). To proceed with the program that was effective in Section 2 for the linear problem, estimates on the Fourier transform of the nonlinear term in the evolution equation are needed. The following lemma is helpful in this regard. Lemma 3. Let p 2 1 be an integer. Then there is a constant C depending only on p such that for any f EA?and all k E [w, IF(k)1

< l+7k,3

If(k)1 < &If

Ilf II’+‘,

(3.4a)

II’+‘,

(3.4b)

aYlf IIpfl.

l-&x(k)1

Proof. These straightforward

F(k)

+

f”f,(k)

=f^* e.0 *f:(k)

(3.4c)

estimates follow immediately upon writing

. . . z+ f(k) = ,-&

-

(k, + *. . +k,))fjk,)

= lRP{(k - (kl + - -. +k,))fjk,)

. * * &p)

* *. f”(kp_l)ikpfl(kp)

dkl

. ”

dk~,

dk, *a. dk,

and $mx(k)

= l,Pr”(k - (k, + . . . +k,))fjk,)

. . . fn(kp_l)ikpfn(kp)

dk, * *. dk,.

q

A bound on the kernel of the linear propagator will also prove to be useful. Lemma 4. There are constants C, and Cl such that for any17E LO, 11,t t-1

/I

e-s(lkl

*‘+i&)

ds

l+ -s(JklZP+iqk3)

These two inequalities elementary estimates. 0

Proof.

Cot

G

0

> 1, and for

d

(3

lk12P’ s <

C,t(l + k2).

follow by computing

all k,

Sa)

(3Sb) the integrals

in question

and making

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273

A little later in the analysis, Duhamel’s principle will be used and it will then be helpful to have estimates on the functional N = N, defined for u E C(1, T; ~25’)by N(u)@,

t) = /l’,S(, -s + l)P(x,

s)u,(x,

s) ds,

(3 *6)

where {S(r)), > 1 = {S,(r)), > 1 is the semigroup defined in (1.4) associated to the initial-value problem for the linear equation (1.3). By using the estimates in Lemmas 3 and 4, one deduces that if u E C(1, T; ~25’1,then for 1 < t G T, sup(l+

lk13)l$q(k

t)l

k f

yp(l+

(l+


<sup><

-WW&&-(k,

Ikl’)lj;‘e

k

I<~
i

+

’ k I31

‘(l

llullpTtlsup

4

sup

Ikl’)

i

1)u(. , s> )I p+‘)lf ‘-l

e-r,,,2”

I

I k I “)

(1 + k2)(1 + 1k I 2p)

k

provided that p 2 i. Here O(k) = I k I 2p + iqk3 and we have introduced IIu II T = II LfII C(l,T;.!a)

SUP

=

l
II q-7 t) II.

t)l

k

e-(t-s)s(k)q(

k, s) ds

d/+ sup /’ e -(r-s)o=-u”ux(k, k

s) ds

1

$ +c

11

u

l)pTtl

’ e-“-d

sup

k < Ct

/

Re e(k)

e -reck)

dr

ds

1

II LfTllpT+l

for 1 G t G T, where C is independent

the space-time norm

(3J)

In a similar vein, it transpires that for u E Ccl, T; 991, sup (N(u)‘(k,

ds}

dr

/o (1 +

< IlullpT+lSUP cc,

e-(‘-s)Re@(k)

1 +k

l+k2

i

s) ,-Js

of both T and u.

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These estimates lead to the following result, which will find use immediately in the attack on the main result. Proposition 2. Suppose i</sup>

where T = L2P with L > 1 given.

Then the following inequalities hold: 11N(u,)I/

T<

CT

II N(u,) -N(4)

iIUIII%+l,

(3.9)

II T
Ib~~%)h1

-U2iiT.

(3.10)

Proof. Inequality

(3.8) shows that (IN(u,)(*, t>II < Ct IIul IIpT+l for 1
Iup+l_ up+11= lu-vllu~+Up-lv+

*+vPl ,
and the same sort of estimates that appear in (3.8), the inequality (3.10) is likewise verified.

0

With these preliminaries in hand, we turn to the primary task of determining the asymptotic behavior of solutions of (l.l)-(1.2). Let f0 in S’ be given. Much as in the linear case, we consider the sequence f, =R L”,l,lffJ, n=l,%..., where L > 0 will be specified later. Because of the semigroup property, (3.11)

f, =RL,Ol”~l,yn-Lo * * - o RL,ol,y o R,,l,lf07 or what is the same, f,, = RL,cx+-lfn- 1, for n = 1, 2,. . , , where LY= L2p-3 and y = L2P-p-1 as above. To analyse the sequence ( fn}zso, it is convenient to consider the initial-value problem a,u, = Mu, - “n-la$$ u,(.,

- yn%l;Ql,, (3.12)

1) =f.

Duhamel’s principle is applied to (3.12) to obtain the formulation u,(,,

t) =

S,(t)f

=uf(*,

+ y”-‘jl’s,(t t) +F’N,(u,)(*,

-s)u$,

s)Q&,

s) ds

t),

(3.13)

where S,(t) = S,.-l(t> is the semigroup defined in (1.4) associated with the linear initial-value problem (1.3) with 77= an-‘. Because of the previously derived results, the terms on the right-hand side of (3.13) can be estimated thusly: Ib:/tTGC2/tf

II and

Yn-111~,(u,)II~~~3TII~,II~+1Yn-1,

for T > 0, where C, and C, are independent of ~1,T, f, (Y and y. Define a mapping T, on functions v E Ccl, T; 9) by T,(u) = u; + yn?Vn(v),

(3.14)

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275

where U: is as in (3.13). Let B,f = {U E Ccl, T; 23’): IIu - uz IIT < IIf II} and presume that f is drawn from B, = (g ~9: 1)g II
C,Tllull$+l~n-l<

Ilfll

for u E B,f, and the latter CJY’( Because

(3.15)

holds if

IIf II + IIu: IIT)~+’ G 11f 11.

of the first inequality

C,Ty”-‘(1

+ C2)‘+’

Since y = L2p-P-1, C,(l

+

in (3.141, this holds provided

IIf 11’ < 1.

T = L2fi and

(I f II G R, (3.15) will hold if

C2)a+1RPL2”P-(P+1Xlt-1)

<

1.

Since p a 2 and p G 1, the exponent is negative for II 2 3. Moreover, the inequality (3.10) in Proposition 2 shows that T, is a contraction on B,f, if II f IIis small enough and thus (3.12) has a unique solution there. Taking f=f,_ 1 in (3.121, then using (3.11) and (3.13) leads to the formula f,(*)

= Lu,_l(*L, =R

L”P) + y”-‘LN(u,)(

L,a~4n-1(9

‘L, L’P)

(3.16)

+Tk)

for f,, where

II w, II GLY”_’

II N(u,)( ‘L, L”P) II

< L3y”_1

IIN(u,)

< cL2p+3y”~’

Writing

f, =A,f*

IIfn_l

+RL,an-lgn-l

+An-l(RL,an-lf*

IIu, Il”t’

IIp+1.

+ g, with a constant

f,,=An--lRL,an-lf* =A,plf*

II,r_6 cL2P+3ynp1

(3.17) A,, chosen

so that

g^,(O) = 0, we see from (3.14) that

+w,, -f*)

+RL,anmlgn-l

+w,.

So A, = A, ~ 1 + GJO> and f*-f*)+R&Ig,_,+w,-Gn(0)f*.

g, =A,_,(R;,,n-I

(3.18)

for the linear case, A,, = A, ~ 1 = A, say, for all IZ. Furthermore,

As observed

<

1A 1;

+ C;

where we used Lemmas

11g,_,

11+CL2p+3y”-’

II f,-1

we see that

II p+l,

1 and 2 from the linear case and the bound

(3.19) on w, from (3.17).

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Assume inductively that for all m < n - 1, we have (i) IIf, II G IIf0 II, IIf0 II G l/L and (ii) IIg, II G CL-"('-") II f. II. Note that 1)g, II G C IIf0 II, and that (3.19) implies II g, II CL-”

II fo II

p+l

+ CL-EL-~--E)

ll f.

11+~~2~+3~(2p-~~l)(n-1)

11 f.

11 p+l

Since p 2 2 and $ G p G 1, if follows that 2p -p - 1 G - 1. Hence, for any F > 0, for all L large enough and R small enough, we have II g,

II =G cL-“(‘-“)

II fo II

where C depends on E but not on n. It follows that IIf,-@*

II
(3.20)

II.

Setting t = L2Pn, we see from (3.11) that IItVLqt’/W

. , t) -Af”(*)

II
(3.21)

II f-0 Il(t-“2p)‘-”

for t large enough and (If0 II small enough. As we remarked in Section 2, this result can be extended to any t 2 L2p. Theorem 1’. Let 3 < p G 1 and p > 2, then for II f. exist constants A and C > 0 such that for all t > t,

II u( .ty

t) -

g&f*(-)

II < c

II ffj

ll

small enough and t, large enough, there

(3.22)

Il(t-“P)‘?

4. L,- and L,-bounds

The error estimates in (3.22) also imply estimates in the L,- ?nd L,-norms on R. For any function f E L2(rW>,Plancherel’s Theorem states that IIf II2 = )If IILzG C IIf II. Observing that IIf(r~)IIL~=Y-1’211f(~)llL~,

(3.22) implies t”4p

IIu(., t) -

Similarly IIf IIcEG II f t1’2p

II up, t) -

) II 2 < c II fo II Ll <

C

II f

Il(t-“2~)‘-E.

(4.1)

II, and so we have

~~*(./r’~2~)ll~~CIlfil

ll(t-“2y.

These are precisely the estimates claimed in Theorem

1.

P-2)

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