QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS

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1 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS SPUR FINAL PAPER, SUMMER 08 CALVIN HSU MENTOR: RUOXUAN YANG PROJECT SUGGESTED BY: ANDREW LAWRIE Augus, 08 Absrac. In his paper, we discuss he decay rae for he soluion o semilinear energy-criical wave equaion ha behaves like he free wave. I is known ha free waves in R n have a decay rae of (+) n. Using he Klainerman- Sobolev inequaliy and Sricharz esimaes, we are able o prove non-linear waves ha scaer in R 3 have a decay rae of. Moreover, we generalize he resuls o R n o obain a ( + ) n decay rae, and also discuss some resuls ha migh be helpful o improve our bound.. Inroducion We sudy he decay rae for soluions o he energy-criical semi-linear wave equaion: u := u + u = ±u 5 u[0] = (u(0, ), u(0, )) = (f, g) C c (R 3 ) C c (R 3 ) in R 3 under he scaering assumpion, which means ha here eiss a free wave u L ha solves u L = 0 wih a possibly differen iniial condiion han f and g, and as. Remark. The equaion u[] u L [] H L 0 u = u 5 is ofen called he defocusing equaion, and he equaion is ofen called he focusing equaion. u = u 5 The moivaion behind his problem is ha any free wave u L ha has smooh and compacly suppored iniial daa in R 3 has a L -norm decay rae of. For semi-linear waves ha scaer (i.e. he difference beween he semi-linear wave and he free wave ends o 0), i is emping o hink ha hey behave very similarly as goes o infiniy. Hence, we conjecured u should also have a decay rae of, and he propose of his paper is o give a bound for hese semi-linear waves. The

2 CALVIN HSU mehod we used is similar o he one used in he free wave equaion, and we can prove a / decay rae for semi-linear wave equaions. Noe ha in his paper, we will need our soluion u o eis for all ime. In he defocusing equaion, his follows from he global well-posedness described in Corollary 5. in []. However, in he focusing case, no all soluion u eis for all ime. (i.e. some u blows up in finie ime). The assumpion of small energy on he iniial daa is required in he focusing equaion for u o eis in all ime. In his paper, we will only be discussing soluions ha eis for all ime and doesn blow up in finie ime. Remark. The inuiion behind focusing equaions can blow up in finie ime while defocusing equaions don is due o he plus and minus sign inside he energy funcion. Namely, in he defocusing case, we have he energy funcion E(u) = R u + u + 6 u 6 d 3 being consan. Hence u, u, and u are all conrolled by a consan ha depends on he iniial daa. However, in he focusing case, we have he energy funcion E(u) = R u + u 6 u 6 d 3 being consan. Since here is a minus sign in fron of u 6, we can no bound any erms like we did before and herefore can no preven u from blowing up. Theorem.3 (Decay rae for u(, ) L ) Le u(, ) be a soluion o he following equaion: u = ±u 5, u[0] = (f, g) C0 (R 3 ) C0 (R 3 ) Assume ha here eiss a free wave u L wih a possibly differen iniial condiion hen f and g, and u[] u L [] Ḣ L 0 as goes o infiniy. Then u(, ) L C + for a consan C ha only depends on u and doesn depend on. We will be using he Klainerman-Sobolev inequaliy, Sricharz esimaes, and oher echniques o esablish his bound. Moreover, our resuls can also be easily generalized ino R n for any n 3. Finally, we will also provide some progress ha we made owards o he decay rae.. Preliminaries, Definiions, Klainerman-Sobolev inequaliy, and Sricharz esimaes Throughou his paper, he defaul space we will be working wih will always be R 3. If we don specify, hen all he norms will also be inegraing over R 3. To simplify hings, we will be using he noaion u[] as u[] = (u(, ), u(, )) o shoren some of our equaions, and he operaor here will denoe n = i. i=

3 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 3 We will be using X Y o denoe ha X CY for some absolue consan C. The consan C can vary from line o line. We will sar by recalling he homogeneous and normal Sobolev spaces, which are spaces ha we will be using when sudying wave equaions: Definiion. (Sobolev space) The Sobolev space, denoed as H s, consiss of all funcions f such ha f H s <, wih norm f H s = R n ˆf(ξ) ( + ξ ) s dξ. Definiion. (Homogeneous Sobolev space) The homogeneous Sobolev space, denoed as H s, is very similar o he normal Sobolev spaces. However, he (+ ξ ) erm here will be replaced by jus ξ insead. Namely, H s consiss of all funcions f such ha f H <, s wih norm f H s = R n ˆf(ξ) ξ s dξ. Now we recall he free wave and he energy criical wave equaions: Definiion.3 (Free wave) A free wave u L is a funcion ha saisfies he wave equaion u = 0, u[0] = (f, g) C c (R 3 ) C c (R 3 ) for some iniial daa (f, g). Definiion.4 (Energy criical wave equaion) The energy criical wave equaion in R n has he following form: u = ±u n+ n () for all n 3. Remark.5 These wave equaions are called energy criical since hey saisfy he scaling invarian propery, which means ha whenever u(, ) is a soluion o (), hen u λ = u( λ, λ ) λ n will be a soluion as well. Moreover, we also have u λ Ḣ L = u Ḣ L as well, which means ha he energy of u is consan no maer wha λ is. Remark.6 Since we will be mainly working in R 3 here, he energy criical wave equaion will hen be u = ±u 5. Definiion.7 (Scaering condiion) We say an energy criical wave u scaers if here eiss a free wave u L wih possibly differen iniial condiions ha saisfy as goes o infiniy. u[] u L [] H L 0

4 4 CALVIN HSU Remark.8 Noice ha he equaion we are focusing on is u = ±u 5. However, he plus or minus sign doesn maer much in our heorems and proofs since we will always be aking absolue values around u 5. Hence from now on, we will only discuss he equaion u = u 5 unless oherwise saed. Throughou his paper, u will solve he above equaion unless oherwise saed. We will sar off wih a fundamenal propery for general wave equaions, which is he finie speed of propagaion [7]. Finie speed of propagaion is an imporan propery ha allows us o conrol a lo of norms since we will only have o inegrae over a finie volume when he iniial daa has compac suppor. Theorem.9 (Finie Speed of Propagaion) Le u be a soluion o he equaion u = F (u), u[0] = (f, g) C c C c, wih F being smooh and F (0) = 0. Moreover, if f, g are suppored in a ball of radius R cenered a he origin, hen u(, ) = 0 when > + R. Corollary.0 Le u be a soluion for u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose ha f, g are suppored in a ball of radius R cenered a he origin, hen we always have u(, ) = 0 when > + R. Proof. Since F (u) = ±u 5 saisfies F (0) = 0, Theorem.9 proves his immediaely. Therefore from now on, we will only need o consider when R +, since we will always have u(, ) = 0 oherwise. Now we inroduce one of an essenial heorem ha will help us obain a bound for u(, ), which is he Klainerman-Sobolev inequaliy [3]. However, we will need o define some vecor fields before we can sae he Klainerman-Sobolev inequaliy. Definiion. (Invarian vecor fields) Le Γ = {, i, Ω ij := i j j i, S := + n i i, Ω 0i := i + i } be a se of vecor fields. These are he generaors of he linear ransformaions ha commues wih in he equaion u = 0. Moreover, all he vecor fields in Γ acually commues direcly wih, wih he ecepion of S. To be clear, we have i= (Su) = S(u) + u, and so S commues wih in he free wave equaion. Remark. Noice ha in R 3, here are 5 ypes of differen invarian vecor fields up o symmery, and a oal of 4 of hem. Now we are ready o sae he Klainerman-Sobolev inequaliy.

5 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 5 Theorem.3 (Klainerman-Sobolev inequaliy) For any ineger n, le u(, ) C (R n ) vanish when is large for all. Then here eiss a C n > 0 such ha ( + + ) n ( + ) u(, ) Cn Γ α u(, ) L (R n ) α n + holds for all 0. Noice ha his heorem gives us bounds for any smooh funcion u wih compac suppor in space in erms of and, while sacrificing he summaion of invarian vecor fields applying o u. Our goal is o bound all he invarian vecor fields erms o ge a bound for boh u and u, and his will be done in Secion 3. Lasly, we will inroduce anoher essenial ool ha will help us obain bounds for u, which is called he Sricharz esimaes [5]. The Sricharz esimaes gives us an inequaliy ha can be used in a lo of non-linear wave equaions, and we will use i o bound he vecor field erms coming from he Klainerman-Sobolev inequaliy (Theorem.3). Firs, we will have o define wha does a pair of eponens means o be wave admissible. Definiion.4 (Wave admissible) In he R n space, a pair of number (p, q) is called wave admissible if () p, q () p + n q n (3) (p, q, n) (,, 3) all holds. Remark.5 The Hölder conjugae of a is denoed as a, which is defined as a + a =. Now we can sae he Sricharz esimaes. Theorem.6 (Sricharz esimae) Suppose ha () n () (p, q) and (a, b) are boh wave admissible (3) q, b (4) p + n q = n γ = a + n b all holds. Moreover, if u solves he equaion u = F, u[0] = (f, g), hen for ime 0 < T < he following inequaliy holds for some consan C: u L p ([0,T ];Lq (R n )) + u(t, ) Ḣγ (R n ) + u(t, ) Ḣγ (R n ) C( u[0] Ḣγ (R n ) Ḣγ (R n ) + F L a Corollary.7 In R 3, suppose ha () (p, q) and (a, b) are boh wave admissible () q, b (3) p + 3 q = = a + 3 b, ([0,T ];Lb (Rn )) ).

6 6 CALVIN HSU all holds. Moreover, if u solves he equaion hen for ime 0 < T < we have: u = F, u[0] = (f, g), u L p ([0,T ];Lq ) + u(t, ) L + u(t, ) L for some consan C. C( u[0] Ḣ L + F L a ([0,T ];Lb Proof. Plug in γ = and n = 3 in Theorem.6 and we are done. )) Remark.8 Noe ha he gradien operaor in his paper will always be space gradien (i.e. wihou he ime derivaive) unless oherwise saed. Hence is L norm will be comparable o he sum of he L norms of all derivaives in. Therefore we can freely change beween u L and 3 i u L. i= Now ha we have all he ools, we are ready o prove our main resuls in he ne secion. 3. Quaniaive decay for energy criical wave equaions Before proving he decay rae for energy criical wave equaions, we will firs prove he decay rae for free wave equaions. Since he main ideas for proving he energy criical case follows from he free wave case, i will be helpful o undersand how o prove he decay rae for free waves. Theorem 3. (Decay Rae for free wave equaions) Le n be an odd ineger, and u be a soluion o he following equaion: Then he following inequaliy holds: u = 0, u[0] = (f, g) C c (R n ) C c (R n ). u(, ) L. n Remark 3. The sraegy here is o use he Klainerman-Sobolev inequaliy o firs prove a bound for u(, ) L, and hen use he fundamenal heorem of calculus o ge a bound for u(, ) L. The reason ha we don use Klainerman-Sobolev inequaliy direcly on u is ha he erm u(, ) L is no conrollable. We will also follow his idea when i comes o he energy criical wave equaion.

7 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 7 Proof. To simplify noaions, in his proof will denoe boh space and ime derivaive. Apply he Klainerman-Sobolev inequaliy o u(, ), and we obain ha u(, ) C n Now since Γ commues wih, we have α n+ Γ α u(, ) L ( + + ) n ( + ) Γ α u = 0.. () Combine ha wih commuing wo Γ will only resul in a linear combinaion of more Γ and i Γ, we know from () ha u(, ) = C n α n+ Γ α u(, ) L ( + + ) n ( + ) C n α n+ The las equaliy is he energy equaliy. Noice ha since we assume f, g C c for any Γ. Therefore α n+ Γ α u(0, ) L ( + + ) n ( + ) Γ α f, Γ α g C c Hence from equaion (3) and (4), we have u(, ), we also have. (3) Γ α u(0, ) L <. (4) C n ( + + ) n ( + ) for some consan C n. Since n is odd, by Chaper, Theorem. in [5] we know ha u(, ) = 0 if > R, where f, g are suppored in a ball cenered a he origin wih radius R (his is he srong Huygens principle). Therefore for any pair of (, ), here eiss a y such ha u(, y) = 0 and y R. Now by he fundamenal heorem of calculus, we now ha as desired. u(, ) u(, y) + R ( + ) n C n ( + ) n

8 8 CALVIN HSU Now we will sar proving he decay rae for energy criical waves. However, before proving ha all he vecor field bounds are finie, we need he following wo lemmas: Lemma 3.3 (Trapping Lemma) Suppose ha here eiss a family of coninuous funcions K ϵ : R + 0 R ha saisfy boh () K ϵ (0) = 0 () K ϵ () C + ϵk ϵ () m for some fied m >, C > 0. for all ϵ > 0, hen holds for some ϵ > 0. K ϵ ( ) L (R + 0 ) < Remark 3.4 This lemma will be used laer in he scaering lemma (lemma 3.5) o show ha once a funcion is small enough a ime = 0, hen i will always be bounded. Proof. If C >, choose ϵ = (C) m. Then sup{ ϵ m 0} C (C)m = C > C. (C) m If C, hen choose ϵ = 4 m and we know ha Hence by coninuiy sup{ ϵ m 0} 4 = 3 > C. K ϵ () [0, Q m ], where Q m is he smalles posiive roo for he equaion ϵ m = C. Therefore we can conclude ha K ϵ ( ) L (R + 0 ) < holds for some ϵ > 0 as desired. Noice ha one of our assumpion o he energy criical wave is ha u scaers, which means ha u[] u L [] Ḣ L 0. This assumpion isn necessarily easy o use, bu we can ge an equivalen saemen ha conrols some norms of u. Lemma 3.5 (Scaering lemma) () If holds for some (a, b) such ha u L a L < b a + 3 b = and a, b, hen here eiss a linear wave u L such ha as goes o infiniy. u[] u L [] Ḣ L 0

9 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 9 () If here eiss a linear wave u L such ha as goes o infiniy, hen holds for all (a, b) such ha a + 3 b = u[] u L [] Ḣ L 0 u L a L < b and a, b. Proof. () follows from Theorem 6. in [], so we will only be proving () here. Firs le p = 5 4, q = 30 7, hen by he Sricharz esimaes and Hölder s inequaliy we know ha u L 5 L 0 C( f Ḣ + g L + u 5 L p ) Lq = C(K + (u u L + u L ) 5 L p ) Lq C(K + (u u L ) 5 L p + Lq u5 L L p ) Lq C(K + (u u L ) L L 6 (u u L) 4 L + u 5 L 5 L0 L p ). Lq Now again by Sricharz esimaes we know ha u 5 L L p Lq = u L 5 L 5p L5q is bounded by iniial daa and herefore finie. The Sobolev inequaliy [3] also ells us ha u u L L L 6 u u L L Ḣ. Noice ha for any ϵ > 0, we can choose a T big enough such ha u u L L ([T, );Ḣ ) ϵ since he Ḣ norm goes o 0 as goes o infiniy. Combining hese wo inequaliies while seing T as he new iniial saring ime and we have u L 5 L 0 C(K + ϵ (u u L) 4 L ) 5 L0 C(K + ϵ u 4 L ). 5 L0 By he rapping lemma (lemma 3.3) we know ha here eiss an ϵ > 0 such ha u L 5 ([T, );L 0 ) is finie. (Here T depends on ϵ). Moreover, by he finie speed of propagaion, we know ha u(, ) L 0 is jus inegraing u over a finie space, and u L 5 ([0,T );L 0 ) is jus inegraing u(, ) L 0 over a finie inerval, so i mus be finie. Hence is finie. u L 5 L 0 = u L 5 ([0,T );L0 ) + u L 5 ([T, );L 0 )

10 0 CALVIN HSU Now ha we have u L 5 L is finie, we can again use he Sricharz esimaes 0 o show ha for all (a, b) such ha we have as desired. u L a L b a + 3 b = and a, b, C( f Ḣ + g L + u 5 L ) < 5 L0 Wih his lemma, we can use he scaering assumpion o conrol some of he norms for u and use hem in he Sricharz esimaes once again. However, his ime we will be puing hem on he righ-hand side of he inequaliy and ry o bound he lef-hand side, which consiss of he invarian vecor fields applied o u. Lemma 3.6 (Bounds for Γ α u(, ) L ) Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha u[] u L [] Ḣ L 0 as, hen Γ α u(, ) L is bounded by a consan for every α 0 ha is independen of. Proof. Le s firs prove he case α = 0, which means ha we will have o bound by a consan. Since u(, ) L u = u 5, his is jus a sraighforward applicaion of he Sricharz esimaes (Corollary.7): u(t, ) L C( u[0] Ḣ L + u 5 L ([0,T ];L ) ) = C( u[0] Ḣ L + u 5 L 5 ([0,T )) ];L0 C( u[0] Ḣ L + u 5 L 5 )). ([0, );L0 The las epression above is a consan independen of T since u L 5 ([0, );L 0 ) is bounded by Lemma 3.5. Now for α, he idea is ha we apply Γ α on boh sides of he original equaion u = u 5, which will give us Γ α u = Γ α u 5. Noice ha all Γ ecep for S commues wih. Moreover, even when Γ = S we have S(u) = (Su) u = (Su) u 5. When here are wo S we have S (u) = S((Su) u 5 ) = (S u) (Su) Su 5,

11 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS and so on. Combine ha wih all he Γ being linear; we know ha afer applying Γ α o he original equaion, i will have he form of Where P is homogeneous, (Γ α u) = P (Γ α u, Γ α u,..., Γu, u). (5) P Z[Γ α u, Γ α u,..., Γu, u], deg(p ) = 5, and Γ α m u means here are m less Γ applied o he equaion. Moreover, he erm u 4 Γ α u will always have a nonzero coefficien in P. By he Sricharz esimaes applying on (5) we know ha Now by Lemma 3.5 we know ha Γ α u L 5 ([0,T ];L 0 ) + Γ α (T, )u L C( Γ α u[0] Ḣ L + P L ([0,T ];L ) ). u L 5 ([0, );L 0 ) <. Therefore for any ϵ > 0, we can find a series of number 0 = T < T < T 3 < < T M = such ha u L 5 ([T i,t i+ ];L 0 ) < ϵ for all i M. Noice ha we can make any T i T M be he iniial ime, hen from he above inequaliy we know ha for all T [T i, T i+ ], he following inequaliy holds: Γ α u L 5 ([T i,t ];L 0 ) + Γ α (T, )u L C( Γ α u[t i ] Ḣ L + P L ([T i,t ];L ) ). By Hölder s inequaliy and here is a erm in P ha consiss of boh Γ α u and u, we can coninue he above inequaliy: C( Γ α u[t i ] Ḣ L + Γ α u L 5 ([T i,t ];L 0 ) u L 5 ([T i,t ];L 0 ) P 5/3 L ([T i,t ];L 0/3 ) + P L ([T i,t ];L ) ). Here P, P Z[Γ α u,..., Γu, u], deg(p ) = 3, deg(p ) = 5. Choose ϵ small enough such ha he consan for (6) Γ α u L 5 ([T i,t ];L 0 ) is less han for all i in he righ hand side of (6), hen we have Γα u L 5 ([T i,t ];L 0 ) + Γ α (T, )u L C( Γ α u[t i ] Ḣ L + P L ([T i,t ];L ) ). By he finie speed of propagaion we know ha he righ hand side of he above equaion is always finie. Le Q = ma{c( Γ α u[t i ] Ḣ L + P L ([T i,t ];L ) )} i M } <, hen we have Γ α (T, )u L Q for some consan Q independen of T as desired.

12 CALVIN HSU The above lemma is very close o wha we waned, he only hing ha is missing is ha we successfully bounded Γ α u(, ) L, bu wha we have in he Klainerman Sobolev inequaliy is Γ α u(, ) L. To fi his, we will need one more lemma: Lemma 3.7 (Bounds for Γ α u(, ) L ) Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha u[] u L [] Ḣ L 0 as, hen Γ α u(, ) L is bounded by a consan for every α 0 ha is independen of. Proof. By Lemma 3.6 we know ha Γ α u(, ) L is bounded by a consan for every α 0 ha is independen of. Now we know from remark.8 ha we can change ino he sums of i. Moreover, commuing i and Γ will resul in a linear summaion of muliple j erms ( j 3). Hence every Γ α u(, ) L can be wrien as a linear combinaion of which means hey are also bounded. Γ α u(, ) L, Wih he above lemmas, we can now easily prove he decay rae for u by applying he Klainerman-Sobolev inequaliy o u. However, he Klainerman- Sobolev inequaliy requires he funcion being applied on he be smooh in space and has compac space suppor for all > 0, and we will need o show ha u does have hese characerisics. The compac space suppor par is a direc corollary of he finie speed of propagaion, and we will show ha u is smooh here. Proposiion 3.8 (Persisence of regulariy) Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha as, hen u[] u L [] Ḣ L 0 u(, ) C (R 3 )

13 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 3 Proof. By Sobolev embedding, we only need o prove ha u(, ) H s for all s. Now o prove u is in H s, we will need o show ha α u(, ) L is finie for all 0 α s. (There is one era since our original funcion already has ). Moreover, noice ha can be pariioned ino 3 differen i, and all of hem are one of he invarian vecor fields described in Definiion.. Hence by Lemma 3.6, we know ha α u(, ) L Γ α u(, ) L is finie for all α 0 Now we are all se o show he decay rae for u: Theorem 3.9 Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha as, hen u(, ) u[] u L [] Ḣ L 0 ( + + )( + ) Proof. Apply he Klainerman-Sobolev inequaliy o u(, ), and we obain ha Γ α u(, ) L Now by Lemma 3.7 we know ha C 3 α u(, ) ( + + )( +. ) α Γ α u(, ) L is bounded by a consan independen of. Hence we have C u(, ) ( + + )( +, ) where C is a consan only depending on u. Corollary 3.0 Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha as, hen u[] u L [] Ḣ L 0 u(, ) L +..

14 4 CALVIN HSU Proof. This is a sraighforward applicaion of Theorem 3.9. Since u(, ) ( + + )( + ) we know ha for any fied he following holds: u(, ) +., Now we are finally ready o prove Theorem.3, which is u(, ) L / : decays like Proof. By he fundamenal heorem of calculus, we know ha u(, ) u(, u(, (R + )) + (R + )) + R+ R+ 0 u(, y) dy u(, y) dy. Combine his wih finie speed of propagaion (Theorem.9) and he bounds for u (Theorem 3.9), we obain Now u(, ) 0 R+ 0 0 dy ( + + y) + y dy = 0 ( + + y) + y + ( dy ( + + y) + y R+ ) + R +. dy ( + + y) + y dy ( + + y) + y = ln( + (+) + ) ln(3 ). ( + ) Since ln(3 ) is a consan, we will only need o ry o bound ln(+ (+) + ). Noice ha 3 + ( + ) + holds for all 0, which means ha Therefore we can conclude ha as desired. ln(3 ) ln( + ( + ) ) 0. + u(, ) ln(3 ) + +

15 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 5 4. Discussion In his secion, we will show some resuls we have ha could lead us o prove he + decay rae. From Theorem 3.9 we have a bound for u ha is approimaely (+) when.5 is away from. By he fundamenal heorem of calculus i is sraighforward o guess ha he bound for u using u will be +. However, if we resric ino some smaller area, hen we will be able o obain a beer bound han +. Proposiion 4. Le u(, ) be a soluion o he following equaion: u = ±u 5, u[0] = (f, g) C 0 (R 3 ) C 0 (R 3 ) Assume ha here eiss a free wave u L wih a possibly differen iniial condiion hen f and g, and u[] u L [] Ḣ L 0 as goes o infiniy. Then for any β <, here eiss an ϵ = β.5 C u(, ) L ( β ) ( + ) 0.5+ϵ holds for a consan C ha only depends on u. Proof. From Theorem 3.9 we know ha u(, ) ( + + )( +. ) The finie speed of propagaion ells us ha we only need o consider R +. Hence by he fundamenal heorem of calculus, we know ha when β R +, he following inequaliy holds Now choose We know ha u(, ) u(, (R + )) + β u(, y) dy + a = 0.5 ϵ, a i+ = a i R+ R ϵ. u(, y) dy u(, y) dy. lim a n = ϵ > β, n since ϵ = β.5. Hence we have N a i u(, ) u(, y) dy + u(, y) dy + a i= a i+ R+ > 0 such ha u(, y) dy for some N large enough such ha a N > β. Now each erm from he above inequaliy can be bounded as follow: a u(, y) dy a ( + ) ( + ) 0.5+ϵ

16 6 CALVIN HSU a i a i+ u(, y) dy a i+ ( + ) + ai a i+ ( + )( + ) a i ( + ) 0.5+ϵ R+ u(, y) dy R + +. Since N is finie, we can sum up he hree inequaliies above and proved ha u(, ) for some ϵ > 0 when β. Therefore holds. u(, ) L ( β ) ( + ) 0.5+ϵ C ( + ) 0.5+ϵ Remark 4. The above proposiion ells us ha he decay rae for energy criical waves is beer han + when is near. However, he volume where β is sill grows like 3 no maer how close β is o. Theorem.3 gives us he + bound, which is no quie he + we epeced. The above proposiion gave us a slighly beer (+) bound for cerain areas. 0.5+ϵ Now we show ha he ϵ we improved can acually be very beneficial for us o improve he bound owards +. Namely, he global (+) decay will give us 0.5+ϵ he global + decay immediaely: Proposiion 4.3 Le u(, ) be a soluion o he following equaion: u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Assume ha here eiss a linear wave u L () such ha as goes o infiniy, and for some ϵ > 0, hen u() u L () Ḣ L 0 u(, ) L ( + ) +ϵ u(, ) L +. Remark 4.4 Assume ha we already have he bound improved o (+) +ϵ, hen we can use his resul o improve he bound o +. This implies ha if he decay rae for u(, ) L is a polynomial in, hen i can only be + or +. Proof. Suppose ha we already have u(, ) L ( + ) +ϵ

17 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 7 holds for a fied ϵ > 0, hen by he Duhamel s formula [] we know ha u(, ) u L (, ) iπ sin(( s) ξ ) e û R ξ 5 (s, ξ)dξ ds 3 = u 5 (s, )ds( )ds. 4π(s ) = = s The equaliy above comes from equaion (7) from [6]. Now from our assumpion for he decay rae of u, we know ha u 5 (s, )ds( )ds. 4π(s ) Hence 4π(s ) 4π(s ) = s ds = ( + s).5+5ϵ u(, ) u L (, ) (s ) ( + s).5+5ϵ ds ( + s) 0.5+5ϵ. ( + ) 0.5+5ϵ. We already know from Theorem 3. ha he free wave u L decays like +, which implies ha u(, ) L has a decay rae of (+). Do he above progress for 0.5+5ϵ K imes and we will be able o prove ha u(, ) L if we ake K large enough. ( + ) min{0.5+5k ϵ,} = + The above argumen gave us he + bound bu required some addiional assumpions. On he oher hand, we can also prove he desired + decay rae wihou any addiional condiions inside any compac space using Hardy s inequaliy [4]. Theorem 4.5 (Hardy s inequaliy) Le f be a smooh funcion wih compac suppor, hen he following inequaliy holds: f() p n p f L p (R n ). Lp (R n ) Theorem 4.6 (Decay rae wihin compac suppor) Le u(, ) be a soluion o he following equaion: u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Assume ha here eiss a linear wave u L wih possibly differen iniial condiions hen f and g, and u() u L () Ḣ L 0 as goes o infiniy. Le K be any compac space, hen u(, ) L (K) +. Proof. Since K is a compac se, we know ha here eiss a real number R such ha K B(0, R).

18 8 CALVIN HSU Le χ R be a funcion ha is consan in and smooh in. Moreover, χ R is inside B(0, R), 0 ouside of B(0, R + ), and is decreasing in beween hem. Then by Theorem 4.5, we know ha R + χ Ru(, ) L Hence we know ha χ R u(, ) L n (χ Ru(, )) L. χ R u(, ) L (χ R u(, )) L. Now since χ R u is a smooh funcion, is derivaive is bounded be a consan. Combine ha wih Theorem.3 and Theorem 3.9 we know ha (χ R u(, )) L = (χ R u(, )) L (B(0,R+)) u(, ) L (B(0,R+)) + ( χ R )u(, ) L (B(0,R+)) ( + ).5 + u(, ) L (B(0,R+)) ( + ) Noe ha we have u(, ) ( + ).5 since we are only considering a compac se, and herefore is far away from when is big enough. Therefore we proved ha χ R u(, ) L (R n ) +. (7) For he oher invarian vecor field erms, we know ha Γ α χ R u(, ) L will be bounded like he one above since is finie, ecep when Γ consiss of he weigh. Noice ha he weigh always pairs up wih or i in Γ. Now when we apply on χ R, i will simply vanish since χ R is consan in. Moreover, when is applied o u, i won change he order of u since we will gain from and back righ afer. For i, when we apply i on χ R, we will gain a ; when we applied o u, i won change he order of u since we will again gain from i and back righ afer. Therefore he wors possible erm is Γ α χ R u(, ) L, where α = and boh Γ consiss of i. Le i = Ω, hen Ω χ R u(, ) L Ωu(, ) L + Ωχ R u(, ) L u(, ) L + u(, ) L + u(, ) L (8)

19 QUANTITATIVE DECAY FOR NONLINEAR WAVE EQUATIONS 9 Since χ R u(, ) is smooh in space and has compac space suppor, we can use he Klainerman-Sobolev inequaliy on χ R u(, ) and obain ha Γ α χ R u(, ) L α χ R u(, ) ( + + ) +. (9) Now by R +, (7), (8), and (9) we know ha Γ α χ R u(, ) L χ R u(, ) α ( + ).5 +. Finally, since χ R u(, ) = u(, ) when K, we proved ha u(, ) L (K) +. For he las par of he paper, I would like o poin ou ha almos all he heorems we proved above can be easily generalized from R 3 ino R n for all n 3. Recall he energy criical equaion in R n is u = u n+ n. Using he same argumen, we have he following wo decay raes for u: Theorem 4.7 Le u be a soluion o he equaion u = u 5, u[0] = (f, g) C c (R 3 ) C c (R 3 ). Suppose here eiss a linear wave u L such ha as, hen u(, ) u[] u L [] Ḣ L 0 Theorem 4.8 (Decay rae for u(, ) L following equaion: ( + + ) n ( + ). in R n ) Le u(, ) be a soluion o he u = u n+ n, u[0] = (f, g) C 0 (R n ) C 0 (R n ) Assume ha here eiss a free wave u L wih a possibly differen iniial condiion hen f and g, and u[] u L [] Ḣ L 0 as goes o infiniy. Then u(, ) L for a consan C ha only depends on u. C ( + ) n The proof for boh heorems is enirely analogous o he R 3 case. The only difference is when using he Klainerman Sobolev inequaliy, we ge more powers of ( + + ) in he denominaor and herefore have a bound ha depends on n.

20 0 CALVIN HSU 5. Acknowledgemens This paper is wrien for he MIT SPUR(Summer Program in Undergraduae Research). I would like o hank professor Andrew Lawrie for suggesing he opic and provide very useful suggesions and insighs hroughou his summer, and professor Casey Rodriguez for helping me sar and wrap up he projec, as well as poining ou some possible approaches o he problem. I will also like o hank my menor RuoXuan Yang for eplaining a lo of he new maerials I sudied, coming up wih new ideas and mehods o ackle he problem, and checking my paper and work carefully. Finally, I would like o hank professor Ankur Moira and Davesh Maulik for giving me grea suggesions during our weekly alk. References [] L. Evans, Parial Differenial Equaions: Second Ediion, American Mahemaical Sociey [] A. Lawrie, Nonlinear Wave Equaions, hp://mah.mi.edu/ alawrie/ [3] J. Luk, Inroducion o Nonlinear Wave Equaions, hp://web.sanford.edu/ jluk/ [4] M. Ruzhansky, D. Suragan, Hardy and Rellich Inequaliies, Ideniies, and Sharp Remainders on Homogeneous Groups, Advances in Mahemaics Volume 37, 7 Sepember 07, Pages [5] C. Sogge, Lecures on Non-Linear Wave Equaions, Second Ediion, Inernaional Press, Boson, 008 [6] T. Tao, Spaceime Bounds for The Energy-Criical Nonlinear Wave Equaion in Three Spaial Dimensions, hps://ariv.org/abs/mah/06064 [7] Q. Wang, Lecures on Nonlinear Wave Equaions, hp://