Asymptotic decay for a one-dimensional nonlinear wave equation
aa r X i v : . [ m a t h . A P ] M a y ASYMPTOTIC DECAY FOR A ONE-DIMENSIONAL NONLINEARWAVE EQUATION
HANS LINDBLAD AND TERENCE TAO
Abstract.
We consider the asymptotic behaviour of finite energy solutions to theone-dimensional defocusing nonlinear wave equation − u tt + u xx = | u | p − u , where p >
1. Standard energy methods guarantee global existence, but do not directlysay much about the behaviour of u ( t ) as t → ∞ . Note that in contrast to higher-dimensional settings, solutions to the linear equation − u tt + u xx = 0 do not exhibitdecay, thus apparently ruling out perturbative methods for understanding such so-lutions. Nevertheless, we will show that solutions for the nonlinear equation behavedifferently from the linear equation, and more specifically that we have the average L ∞ decay lim T → + ∞ T R T k u ( t ) k L ∞ x ( R ) dt = 0, in sharp contrast to the linear case.An unusual ingredient in our arguments is the classical Radamacher differentiationtheorem that asserts that Lipschitz functions are almost everywhere differentiable. Introduction
Fix p >
1. We consider solutions u : R × R → R to the one-dimensional defocusingnonlinear wave equation − u tt + u xx = | u | p − u (1)with the finite energy initial condition k u (0) k H x ( R ) + k u t (0) k L x ( R ) < ∞ . Standard energy methods (using the Sobolev embedding H x ⊂ L ∞ x ) show that theinitial value problem is locally well-posed in this energy class. Furthermore, by usingthe conservation of energy E [ u ] = E [ u ( t )] := Z R T ( t, x ) dx (2)where T is the energy density T := 12 u t + 12 u x + 1 p + 1 | u | p +1 it is easy to show that the H x × L x norm of u ( t ) does not blow up in finite time, andthat the solution to (1) can be continued globally in time. HL is supported by NSF grant DMS-0801120 and TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. We thank DavidStr¨utt, Jason Murphy, Qing Tian Zhang and the anonymous referee for corrections. In order to justify energy conservation for solutions which are in the energy class, one can use stan-dard local well-posedness theory to approximate such solutions by classical (i.e. smooth and compactlysupported) solutions (regularising the nonlinearity | u | p − u if necessary), derive energy conservation forthe classical solutions, and then take strong limits. We omit the standard details. More generally,we shall perform manipulations such as integration by parts on finite energy solutions as if they wereclassical without any further comment. In this paper we study the asymptotic behaviour of finite energy solutions u to (1)as t → ±∞ . Of course, from the conservation of energy (2) we know that u ( t ) staysbounded in ˙ H x ( R ) ∩ L p +1 x ( R ), and thus (by the Gagliardo-Nirenberg inequality) boundedin L ∞ x ( R ) for all time, but this does not settle the question of whether k u ( t ) k L ∞ x ( R ) exhibits any decay as t → ±∞ .For the linear equation − u tt + u xx = 0, the solutions are of course travelling waves u ( t, x ) = f ( x + t ) + g ( x − t ), which do not decay along light rays x = x ± t . Inparticular, for any non-trivial linear solution, k u ( t ) k L ∞ x ( R ) stays bounded away fromzero. It is thus natural to ask whether the same behaviour occurs for solutions to thenonlinear equation (1). However, an easy energy argument shows that the behaviourmust be slightly different. Indeed, if we introduce the momentum density (or energycurrent ) T = T := u t u x and the momentum current T := 12 u t + 12 u x − p + 1 | u | p +1 we observe the conservation laws ∂ t T = ∂ x T (3) ∂ t T = ∂ x T . (4)From (3) and the fundamental theorem of calculus we have ∂ t Z x
1) are easily ruled out by finite speed of propagation(or by a modification of the arguments used to derive (5), (6)), but concentration ontimelike worldlines (in which | x ′ ( t ) | <
1) are not so obviously ruled out. Nevertheless,we are able to rule out this scenario by the following theorem, which is the main resultof this paper.
Theorem 1.1 (Average L ∞ x decay) . Let u be a finite energy solution to (1) , with anupper bound E [ u ] ≤ E on the energy. Then T Z t + Tt − T k u ( t ) k L ∞ x ( R ) dt ≤ c E,p ( T ) for all t ∈ R and T > , where c E,p : R + → R + is a function depending only on theenergy bound E and the exponent p such that c E,p ( t ) → as t → ∞ . In particular, wehave lim T → + ∞ sup t ∈ R T Z t + Tt − T k u ( t ) k L ∞ x ( R ) dt = 0 . The proof of this theorem will use energy estimates combined with a version of theRademacher differentiation theorem (or Lebesgue differentiation theorem), that Lips-chitz functions are almost everywhere differentiable. The basic idea is to observe that if u concentrates on a timelike worldline { ( t, x ( t )) : t ∈ R } , then x should be Lipschitz, andthus mostly differentiable. This implies that u concentrates on certain parallelogramsin spacetime; we will then use energy estimates to rule out such concentration.In principle, the decaying bound c E,p ( T ) could be made explicit, but this wouldrequire a quantitative version of the Radamacher differentiation theorem. Such resultsexist (see [5] or [6, Section 2.4]), but they are fairly weak (involving the inverse towerexponential function log ∗ ). Presumably a more refined argument than the one givenin this paper would give better bounds. For instance, it is plausible to conjecture that k u ( t ) k L ∞ x ( R ) should decay at a polynomial rate in t , at least in the perturbative regimewhen u is small.We remark that our methods do not seem to give any precise asymptotics for thesolution. Of course Theorem 1.1 indicates that the solution will not scatter to a linearsolution, but it is not clear what the solution scatters to instead, even in the perturbativeregime. It may be that techniques from nonlinear geometric optics could be useful tosettle this question, but the extremely weak decay of the solution means that it wouldbe very difficult for these methods to be made rigorous, at least until one can improvethe results of Theorem 1.1 significantly.2. Energy estimates
In this section we derive the basic energy estimates needed to establish Theorem 1.1.Henceforth we fix p and the finite energy solution u . We adopt the notation X . Y or HANS LINDBLAD AND TERENCE TAO X = O ( Y ) to denote the estimate | X | ≤ CY , where C can depend on p and the energybound E . Thus from energy conservation we obtain the bounds Z R | u t | ( t, x ) + | u x | ( t, x ) + | u | p +1 ( t, x ) dx . t . Lemma 2.1 (H¨older continuity) . For all t, x, t ′ , x ′ ∈ R we have the pointwise bound u ( t, x ) = O (1) (8) and the H¨older continuity property u ( t, x ) − u ( t ′ , x ′ ) = O ( | t − t ′ | / + | x − x ′ | / ) . (9) Proof.
The bound (8) follows immediately from (7) and the Gagliardo-Nirenberg in-equality. Using the bound on | u x | in (7) together with the fundamental theorem ofcalculus and the Cauchy-Schwarz inequality, we also have the spatial H¨older continuitybound u ( t, x ) − u ( t, x ′ ) = O ( | x − x ′ | / ) . Thus to prove (9) it will suffice to show that u ( t , x ) − u ( t , x ) = O (( t − t ) / ) (10)for all t > t . In view of (8) we may also assume t = t + O (1).Fix t , t . From (4) and the fundamental theorem of calculus we have ∂ t Z x Proposition 2.2 (Nonlinear energy decay in a parallelogram) . Let T ≥ R ≥ , let x , t ∈ R , and let v ∈ R be a velocity. Then we have Z t + Tt − T Z x + vt + Rx + vt − R | u ( t, x ) | p +1 dxdt . R / T / + TR . (11) Remark 2.3. Energy conservation (7) only gives the bound of O ( T ) for this integral,thus this proposition is non-trivial when T is much larger than R . A key point here isthat the bounds do not blow up in the neighbourhood of the speed of light v = 1 . It maybe possible to improve the right-hand side of (11) , and to also control other componentsof the energy, but the above bound will suffice for our purposes. SYMPTOTIC DECAY FOR A ONE-DIMENSIONAL NONLINEAR WAVE EQUATION 5 Proof. By translation invariance we can set x = t = 0. By reflection symmetry wemay assume that v ≥ χ : R → R be a non-negative bump function supported on [ − , 2] which equals1 on [ − , ψ ( x ) := R y Case 1: (Spacelike case) v ≥ . In this case, we can verify the pointwise bound1 p + 1 | u | p +1 ≤ T + v T and so (11) follows immediately from (13) (note that R = O ( R / T / )). Case 2: (Lightlike case) − R / T / < v < . In this case we have the bound vp + 1 | u | p +1 ≤ ( T + v T ) + O ( R / T / T ) HANS LINDBLAD AND TERENCE TAO and so from (13) and (7) we have vp + 1 Z R Z R χ ( tT ) χ ( x − vtR ) | u ( t, x ) | p +1 dtdx . R + R / T / and (11) follows. Case 3: (Timelike case) ≤ v ≤ − R / T / . Here we use the identity( T + v T ) + v ( T + v T ) + 1 − v Q = ( vu t + u x ) + ( p − − v )2( p + 1) | u | p +1 . Taking the indicated linear combination of (12), (13), (14) and discarding the non-negative quantity ( vu t + u x ) , we conclude that( p − − v )2( p + 1) Z R Z R χ ( tT ) χ ( x − vtR ) | u ( t, x ) | p +1 dtdx . R + 1 − v TR and thus (noting that 1 − v = (1 − v )(1 + v ) is comparable to 1 − v ) Z R Z R χ ( tT ) χ ( x − vtR ) | u ( t, x ) | p +1 dtdx . R − v + TR . Since 1 − v & R / /T / by hypothesis, the claim follows. (cid:3) Proof of Theorem 1.1 We are now ready to prove Theorem 1.1. Suppose that this claim failed for some E, p . Carefully negating the quantifiers, we may thus find a sequence of times T n → ∞ and t n ∈ R , a δ > n , and a family of solutions u n which uniformlyobey the energy bound E [ u n ] ≤ E such that12 T n Z t n + T n t n − T n k u n ( t ) k L ∞ x ( R ) dt ≥ δ. By translating each u n by t n , we may normalise t n = 0.Let n be large. We will now allow our implied constants in the . notation to dependon δ , thus Z T n − T n k u n ( t ) k L ∞ ( R ) dt & T n . From this bound and (8), we now conclude that the set { t ∈ [ − T n , T n ] : k u n ( t ) k L ∞ ( R ) & } has Lebesgue measure & T n (for suitable choices of implied constants). In particular,we can find a finite set ∆ n ⊂ [ − T n , T n ] of times which are 1-separated and of cardinality n & T n such that k u n ( t ) k L ∞ ( R ) & t ∈ ∆ n .For each t ∈ ∆ n , let x n ( t ) ∈ R be a point such that | u n ( t, x n ( t )) | ≥ k u n ( t ) k L ∞ ( R ) .From (15), one has | u n ( t, x n ( t )) | & t ∈ ∆ n . SYMPTOTIC DECAY FOR A ONE-DIMENSIONAL NONLINEAR WAVE EQUATION 7 Let us say that two times t, t ′ ∈ ∆ n are spacelike if we have | x n ( t ′ ) − x n ( t ) | ≥ | t − t ′ | + 1 . There is a limit as to how many spacelike pairs of times can exist: Lemma 3.1 (Finite speed of propagation) . Let n be sufficiently large, and let t , . . . , t m ∈ ∆ n be times which are pairwise spacelike. Then we have m = O (1) .Proof. Without loss of generality we may assume that t < . . . < t m . Consider thespacetime regionΩ := R × R \ [ ≤ j ≤ m { ( t, x ) : t ≥ t j ; | x − x n ( t j ) | ≤ t − t j + 12 } . Standard energy estimates reveal that Z x :( t j ,x ) ∈ Ω T ( t j , x ) dx + Z x : | x − x n ( t j ) |≤ T ( t j , x ) dx ≤ Z x :( t j − ,x ) ∈ Ω T ( t j − , x ) dx for all 1 < j ≤ m , where T = T ,n is the energy density of u n . Iterating this and thenusing (7), we conclude that X Corollary 3.2 (Existence of Lipschitz worldline) . Let ε : (0 , → (0 , be an arbitraryfunction. Then there exists a constant < c = c ( ε ) ≤ with the following property:for all sufficiently large n , there exists c < c < (depending on n ) and a subset ∆ ′ n of ∆ n with ′ n ≥ cT n such that we have the Lipschitz property | x n ( t ′ ) − x n ( t ) | ≤ | t − t ′ | + ε ( c ) T n (17) for all t, t ′ ∈ ∆ ′ n .Proof. Fix ε , and let n be sufficiently large. Define the particle number of a set ∆ tobe the largest integer m for which one can find pairwise spacelike times t , . . . , t m in ∆.By the previous lemma, we see that ∆ n has particle number O (1). The key lemma isthe following: HANS LINDBLAD AND TERENCE TAO Lemma 3.3 (Dichotomy) . Let ∆ ′ ⊂ ∆ n , m = O (1) and c > be such that ′ ≥ cT n and ∆ ′ has particle number at most m . Suppose n is sufficiently large depending on c .Then at least one of the following is true: (i) There exists a subset ∆ ′′ ⊂ ∆ ′ of cardinality at least cT n such that (17) holds forall t, t ′ ∈ ∆ ′′ . (ii) There exists a subset ∆ ′′′ ⊂ ∆ ′ of cardinality at least cε ( c ) T n / with particlenumber at most m − . Iterating this lemma at most O (1) times we obtain the claim.It remains to prove the lemma. We subdivide the interval [ − T n , T n ] into intervals I of length between ε ( c ) T n / ε ( c ) T n / 8. Call an interval sparse if ′ ∩ I ) ≤ cε ( c ) T n / 8, and dense otherwise. Observe that at most cT n elements of ∆ ′ lie in sparseintervals. Thus if we let ∆ ′′ denote the intersection of ∆ ′ with the union of all the denseintervals, then ′′ ≥ cT n .If ∆ ′′ obeys (17) then we are done. Otherwise, we can find t , t ∈ ∆ ′′ such that | x n ( t ) − x n ( t ) | > | t − t | + ε ( c ) T n . The time t must lie in some dense interval I . We split ∆ ′′ ∩ I = ∆ ′′′ ∪ ∆ ′′′ , where∆ ′′′ consists of all t ∈ ∆ ′′ ∩ I with | x n ( t ) − x n ( t ) | ≤ ε ( c ) T n / 2, and ∆ ′′′ consists ofthe remainder of ∆ ′′ ∩ I . Observe from the triangle inequality (if n is sufficiently largedepending on c ) that all times in ∆ ′′′ are spacelike with respect to t , and similarly alltimes in ∆ ′′′ are spacelike with respect to t . Thus each of ∆ ′′′ and ∆ ′′′ can have particlenumber at most m − 1. On the other hand, by the pigeonhole principle, one of ∆ ′′′ and∆ ′′′ must have cardinality at least ′′ ∩ I ), which is at least cε ( c ) T n / 16 since I isdense. The lemma, and hence the corollary, follows. (cid:3) Let ε : (0 , → (0 , 1] to be a function to be chosen later (one should think of ε ( c )as going to zero very rapidly as c → n , let c , c and ∆ ′ n beas in Corollary 3.2.Define the function x ′ n : [ − T n , T n ] → R by x ′ n ( t ) := inf t ′ ∈ ∆ ′ n ( x n ( t ′ ) + | t − t ′ | ) . One easily verifies that x ′ n is Lipschitz with constant at most 1. From (17) we also seethat | x n ( t ) − x ′ n ( t ) | ≤ ε ( c ) T n (18)for all t ∈ ∆ ′ n .We now apply a quantitative version of the Rademacher (or Lebesgue) differentiationtheorem to ensure that x ′ n ( t ) is approximately differentiable on a large interval. Proposition 3.4 (Quantitative Rademacher differentiation theorem) . Let ε : (0 , → (0 , be a function, and let δ > . Then there exists r = r ( ε , δ ) > with the followingproperty: given any Lipschitz function f : [ − , → R with Lipschitz constant at most , there exists r ≤ r ≤ such that the set { x ∈ [ − , 1] : There exists L ∈ R such that | f ( y ) − f ( x ) y − x − L | ≤ δ whenever y ∈ [ − , is such that ε ( r ) ≤ | y − x | ≤ r } SYMPTOTIC DECAY FOR A ONE-DIMENSIONAL NONLINEAR WAVE EQUATION 9 (which, intuitively, is the set where f is approximately differentiable) has Lebesgue mea-sure at least − δ .Proof. We give an indirect “compactness and contradiction” proof. Suppose for contra-diction that the claim failed. Negating the quantifiers carefully, this means that thereexists a function ε : (0 , → (0 , δ > 0, a sequence r n → 0, and a sequence f n : [0 , → R of Lipschitz functions with constant at most 1, such that the sets { x ∈ [ − , 1] : There exists L ∈ R such that | f n ( y ) − f n ( x ) y − x − L | ≤ δ whenever y ∈ [ − , 1] is such that ε ( r ) ≤ | y − x | ≤ r } have Lebesgue measure at most 2 − δ for all n and all r n ≤ r ≤ f n by a constant if necessary, we may normalise f n (0) = 0. TheLipschitz functions then form a bounded equicontinuous family on the compact domain[ − , f n converge uniformly to a limit f . We conclude that theset { x ∈ [ − , 1] : There exists L ∈ R such that | f ( y ) − f ( x ) y − x − L | ≤ δ/ y ∈ [ − , 1] is such that ε ( r ) ≤ | y − x | ≤ r } has Lebesgue measure at most 2 − δ for all 0 < r ≤ 1. On the other hand, f is clearlyLipschitz with constant at most 1, and so by the Lipschitz differentiation theorem, f isdifferentiable almost everywhere. In particular, the set ∞ [ m =1 { x ∈ [ − , 1] : There exists L ∈ R such that | f ( y ) − f ( x ) y − x − L | ≤ δ/ y ∈ [ − , 1] is such that 0 < | y − x | ≤ − m } has full measure in [ − , − δ . But this contradicts theprevious claim. (cid:3) Remark 3.5. It is also possible to give a more direct “martingale” or “multiscaleanalysis” proof of this proposition, which we sketch as follows. For each n ≥ , let f n be the piecewise linear continuous function which agrees with f on multiples of − n , andis linear between such intervals. One easily verifies that the functions f n +1 − f n arepairwise orthogonal in the Hilbert space ˙ H ([ − , , and thus by Bessel’s inequality wehave ∞ X n =1 k f n +1 − f n k H ([ − , ≤ . Now let F : N → N be a function to be chosen later, and let σ > be a small quantityto be chosen later. From the pigeonhole principle, one can find ≤ n ≤ C ( F, σ ) suchthat F ( n ) X n = n k f n +1 − f n k H ([ − , ≤ σ. Indeed, the arguments here are closely related to some classical martingale inequalities of Doob[1]and L´epingle[2]. If one then sets r := σ − n , one can verify all the required claims if σ is chosen suffi-ciently small depending on δ , and F is sufficiently rapidly growing depending on δ , σ ,and ε ; the quantity L can basically be taken to be f ′ n ( x ) . We omit the details, but see [5] for some similar arguments in this spirit. Let δ > c ) to be chosen later, and let ε : (0 , → (0 , 1] be the function ε ( r ) := δr . We let n be sufficiently large, and apply the aboveproposition to the Lipschitz function f = f n : [ − , → R defined by f ( y ) := T n x ′ n ( T n y ).We conclude that there exists r = r ( δ ) and r < r < δ and n ) suchthat the set { t ∈ [ − T n , T n ] :There exists L ∈ R such that | x ′ n ( t ′ ) − x ′ n ( t ) t ′ − t − L | ≤ δ whenever t ′ ∈ [ − T n , T n ] is such that δrT n ≤ | t − t ′ | ≤ rT n } has measure at least (2 − δ ) T n .On the other hand, the set ∆ ′ n has cardinality at least cT n . As in the proof of Lemma3.3, we partition [ − T n , T n ] into intervals I of length between rT n / rT n / 8, and let∆ ′′ n be the portion of ∆ ′ n which are contained inside those intervals I which are dense in the sense that they contain at least crT n / 16 elements of ∆ ′ n . It is easy to see that∆ ′′ n has cardinality at least cT n / 2. Also, ∆ ′′ n is 1-separated.Thus, if we let δ = δ ( c ) be sufficiently small compared to c , we can find t ∗ ∈ [ − T n , T n ]within a distance 1 of ∆ ′′ n and v ∈ R such that (cid:12)(cid:12)(cid:12)(cid:12) x ′ n ( t ′ ) − x ′ n ( t ∗ ) t ′ − t ∗ − v (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ whenever t ′ ∈ [ − T n , T n ]is such that δrT n ≤ | t ∗ − t ′ | ≤ rT n . Let t be an element of ∆ ′′ n within 1 of t ∗ . Applying (18), the triangle inequality, andthe Lipschitz nature of x ′ n , we conclude that x n ( t ) = x n ( t ) + v ( t − t ) + O ( δ | t − t | ) + O ( ε ( c ) T n ) + O (1)whenever t ∈ ∆ ′′ n is such that δT n + 1 ≤ | t − t | ≤ rT n − 1. Applying the Lipschitzproperty again, we conclude that x n ( t ) = x n ( t ) + v ( t − t ) + O ( δrT n ) + O ( ε ( c ) T n ) + O (1)for all t ∈ ∆ ′′ n with | t − t | ≤ rT n − 1. If we set ε ( c ) := δ ( c ) r ( δ ( c )), and assume n issufficiently large depending on all other parameters, we thus have x n ( t ) = x n ( t ) + v ( t − t ) + O ( δrT n )whenever t ∈ ∆ ′′ n and | t − t | ≤ rT n / 4. One should view this as an assertion that x n is approximately differentiable near t .By definition of ∆ ′′ n , we know that t is contained in an interval I of length at most rT n / & crT n elements of ∆ n . We thus see that the parallelogram P := { ( t, x ) : t ∈ I ; | x − x n ( t ) − v ( t − t ) | ≤ R/ } contains at least & crT n points of the form ( t, x n ( t )) with t ∈ ∆ n , where R is a quantityof size ∼ δrT n . On the other hand, by definition of ∆ n , we have | u n ( t, x ( t )) | & t ∈ ∆ n . Applying (9), we conclude that Z P | u n ( t, x ) | p +1 dtdx & crT n . SYMPTOTIC DECAY FOR A ONE-DIMENSIONAL NONLINEAR WAVE EQUATION 11 On the other hand, from Proposition 2.2 we have Z P | u n ( t, x ) | p +1 dtdx . R / ( rT n ) / + rT n R . δ / rT n + δ − . If we set δ to be sufficiently small depending on c , and let n be sufficiently large de-pending on all other parameters, we obtain a contradiction as desired. This completesthe proof of Theorem 1.1. References [1] L. Doob, Stochastic processes, New York, 1953.[2] D. L´epingle, La variation d’ordre p des semi-martingales , Z. Wahrscheinlichkeitstheor. Verw. Geb. (1976), 295-316.[3] H. Lindblad, A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation , Lett. Math. Phys. (2005), no. 3, 249-258.[4] M. Reed, Propagation of singularities for non-linear wave equations in one dimension , Comm.Partial Differential Equations (1978), no. 2, 153-199.[5] T. Tao, A quantitative version of the Besicovitch projection theorem via multiscale analysis , Proc.Lond. Math. Soc. (3) (2009), no. 3, 559-584.[6] T. Tao, Structure and Randomness: pages from year one of a mathematical blog, AmericanMathematical Society, 2008. Department of Mathematics, UCSD, San Diego CA 92013-0112 E-mail address : [email protected] Department of Mathematics, UCLA, Los Angeles CA 90095-1555 E-mail address ::