SCATTERING ABOVE ENERGY NORM OF SOLUTIONS OF A FOCUSING LOGLOGLOG ENERGY-SUPERCRITICAL SCHRO¨DINGER EQUATION WITH RADIAL DATA BELOW 7 1 GROUND STATE 0 2 TRISTANROY n a J Abstract. Given n ∈ {3,4} and k > n2, we prove scattering of the ra- 0 dial H˜k := H˙k(Rn)∩H˙1(Rn)− solutions of the logloglog focusing energy- 1 supercriticalSchro¨dingerequationi∂tu+△u=−|u|n−42ulogγlog(log(1010+|u|2)) for a range of positive γs, for energies below that of the ground states, and ] P for potentials below that of the ground states. The proof uses in particular A arguments of[2,9,11,13,14]. . h t a m 1. Introduction [ We shall study the radial solutions of the following Schrödinger equation in 1 dimension n, n∈{3,4}: v 9 8 6 (1) i∂tu+△u =−|u|n−42ug(|u|) 2 with g(|u|):=logγlog(log(1010+|u|2)) and γ >0. 0 . This equation has many connections with the following power-type Schrödinger 1 equation, p>1 0 7 1 (2) i∂ v+△v =−|v|p−1v : t v i (2) has a natural scaling: if v is a solution of (2) with data v(0):=v and if λ∈R X 0 isaparameterthenv (t,x):= 1 u t ,x isalsoasolutionof(2)butwithdata λ 2 λ2 λ r λp−1 a vλ(0,x) := 12 u0 λx . If sp := n2 −(cid:0)p−21 th(cid:1)en the H˙sp norm of the initial data is λp−1 invariantunderthe(cid:0)sca(cid:1)ling: thisiswhy(2)issaidtobeH˙sp-critical. Ifp=1+ 4 n−2 then (2) is H˙1 (or energy) critical. The energy-critical Schrödinger equation (3) i∂tu+△u =−|u|n−42u has received a great deal of attention. Cazenave and Weissler [3] proved the local well-posedness of (3): given any u(0) such that ku(0)k < ∞ there exists, for H˙1 2(n+2) 2(n+2) some t close to zero, a unique u∈C([0,t ],H˙1)∩L n−2 L n−2 ([0,t ]) satisfying 0 0 t x 0 (3) in the sense of distributions (4) u(t) =eit△u(0)+i tei(t−t′)△ |u(t′)|n−42u(t′) dt′· 0 R 1 h i 2 TRISTANROY The next step is to understand the asymptotic behavior of the solutions of (3). It is well-known that (3) has a family of stationary solutions (the ground states) Wλ,θ(x):=eiθ n1−2W λx thatsatisfy△Wλ,θ+|Wλ,θ|n−42Wλ,θ =0withθ ∈[0,2π] λ 2 and W defined by (cid:0) (cid:1) W(x) := 1 · n−2 1+ |x|2 2 (cid:16) n(n−2)(cid:17) The asymptotic behavior of the solutions for energies below that of the ground states has been studied in [7]. In particular global existence and scattering (i.e the linear asymptotic behavior) were proved for potential energies below that of the ground states. The asymptotic behavior of the solutions was studied in [5] for energies equal to that of the ground states and in [9] for energies slightly larger than that of the ground states. If p > 1 + 4 then s > 1 and we are in the energy supercritical regime. n−2 p Since for all ǫ > 0 there exists cǫ > 0 such that |u|n−42u . |u|n−42ug(|u|) ≤ cǫmax(1,||u|n−42+ǫu|)thenthenonlinearityof(1)is(cid:12)(cid:12)saidtob(cid:12)(cid:12)eba(cid:12)(cid:12)relysupercriti(cid:12)(cid:12)cal. In this paper we study the asymptotic behavior(cid:12)of H˜k-so(cid:12)lutio(cid:12)ns of (1) for n(cid:12) ∈ {3,4}. Recall the local-wellposed result: Proposition 1. “Local well-posedness ” [11] Let n∈{3,4} and let k > n. Let 2 M be such that ku k ≤ M. Then there exists ǫ := ǫ(M) > 0 small such that if 0 H˜k T >0 (T := time of local existence) satisfies l l keit△u k ≤ǫ (5) 0 2(n+2) 2(n+2) Ltn−2 Lxn−2 ([0,Tl]) then there exists a unique 2(n+2) 2(n+2) 2(n+2) 2(n+2) (6) u∈C([0,Tl],H˜k)∩Ltn−2 Lxn−2 ([0,Tl])∩D−1Lt n Lx n ([0,Tl]) 2(n+2) 2(n+2) ∩Lt n D−kLx n ([0,Tl]) such that (7) u(t)=eit△u0+i 0tei(t−t′)△ |u(t′)|n−42u(t′)g(|u(t′)|) dt′ is satisfied in the sense of disRtributions.(cid:16)Here D−αLr := H˙α,r e(cid:17)ndowed with the norm kfk :=kDαfk . D−αLr Lr This allows to define the notion of maximal time interval of existence I := max (T ,T ), that is the union of all the intervals I containing 0 such that (7) holds in − + 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) theclassC(I,H˜k)∩Ltn−2 Lxn−2 (I)∩D−1Lt n Lx n (I)∩D−kLt n Lx n (I). Recall the following proposition: Proposition 2. “Global well-posedness: criterion” [11] If |I |<∞ then max kuk =∞ (8) 2(n+2) 2(n+2) Ltn−2 Lxn−2 (Imax) FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 3 Remark 1. Proposition 1 and 2 were proved in [11] for solutions of barely supercritical defocusing nonlinearities. The proof can be easily adapted to solutions of (1). With this in mind, global well-posedness follows from an a priori bound of the form kuk ≤ f(T,ku k ) for arbitrarily large time T > 0. In 2(n+2) 2(n+2) 0 H˜k Ltn−2 Lxn−2 ([−T,T]) factforsomedataweshallprovethatthebounddoesnotdependontimeT,which will show scattering. Before stating the main theorem, we recallsome generalnotation. We write a≪b if the value of a is much smaller that that of b, a ≫ b if the value of a is much larger than that of b, and a∼b if a≪b and b≪a are not true. We say that C˜ is the constant determined by a . b if it the smallest constant C such that a ≤ Cb. We write a = o(b) if there exists a constant 0 < c ≪ 1 such that |a| ≤ c|b|. We define b+ = b+ǫ for 0 < ǫ ≪ 1. If b+ appears in a mathematical expression such as a ≤ Cb+, then we ignore the dependance of C on ǫ in order to make our presentation simple. Let 2∗ := 2n . If f ∈H˜k then we define the energy n−2 (9) E(f) := 1 |∇f(x)|2− F(f,f¯)(x)dx, 2 Rn Rn with R R (10) F(z,z¯) := |z|snn−+22g(s)ds· 0 Indeed R (11) F(f,f¯)(x)dx .kfk2∗ g(kfk ).kfk2∗ g(kfk ): Rn L2∗ L∞ H˙1 H˜k this follows(cid:12)fRrom a simple int(cid:12)egration by part (cid:12) (cid:12) (12) F(z,z¯) ∼|z|n2−n2g(|z|), combined with the Sobolev inequality (13) kukL∞t L∞x (J) .kukL∞t H˜k(J)· If f ∈H˙1 then we denote by E˜(f) the following 1 1 (14) E˜(f):= |∇f(x)|2− |f(x)|2∗ dx· 2ZRn 2∗ ZRn Hence E˜(f)=E(f)+X(f) with X(f):= |f(x)|(g(s)−1)s2∗−1 dsdx· Rn 0 We define the functional R R K˜(f):=k∇fk2 −kfk2∗ · L2 L2∗ Let u be an H˜k solution of (1). A simple computation shows that the energy E(u(t))is conserved,or,inotherwords,thatE(u(t))=E(u ). Letχbe a smooth, 0 4 TRISTANROY radial function supported on |x| ≤ 2 such that χ(x) = 1 if |x| ≤ 1. If x ∈ Rn, 0 R>0, then we define the mass within the ball B(x ,R) 0 1 (15) Mass(B(x ,R),u(t)) := |u(t,x)|2dx 2 0 B(x0,R) (cid:16) (cid:17) Recall (see [13]) that R (16) Mass(B(x ,R),u(t)) .R sup k∇u(t′)k 0 t′∈[0,t] L2 and that its derivative satisfies ′ (17) |∂ Mass(u(t),B(x ,R))| . supt′∈[0,t]k∇u(t )kL2 t 0 R The main result of this paper is: Theorem 3. Let δ > 0, n ∈ {3,4}, and u ∈ H˜k := H˙k(Rn)∩H˙1(Rn),k > n, 0 2 radial such that 1 (18) ku0kH˜k &1, E(u0)<(1−2δ)E˜(W), andku0kL2∗ 0 satisfies 1 (19) logγloglogCaCaCaδ2(αα−1)+ ku0k2H˜k−1≪δ   with α := 1 then the solution of (1) with data u(0) := u exists for all time T. γan 0 Moreover there exists a scattering state u ∈H˜k such that 0,+ (20) tl→im∞ku(t)−eit△u0,+kH˜k =0 and there exists C :=C(ku k ,δ) such that kuk ≤C. 0 H˜k 2(n+2) 2(n+2) Ltn−2 Lxn−2 (R) By symmetry an analogous statement holds for negative times. Remark 2. Notice that ku0kL2∗ = kWkL2∗ is impossible. Indeed, recall [1, 10] the sharp Sobolev inequality kfkL2∗ ≤ C∗k∇fkL2 with C∗ := kk∇WWkLk2L∗2. Hence F(y)<(1−δ)F(kWkL2∗) with y :=ku0kL2∗ and 1 1 (21) F(y):= C2y2− y2∗· 2 ∗ 2∗ Remark 3. Observe from (13) and (19) that kg(u )−1k ≪δ. Hence E˜(u )< 0 L∞ 0 (1−δ)E˜(W). Hence 0<δE˜(W)< 1 k∇Wk2 −k∇u k2 . So in particular δ is 2 L2 0 L2 bounded from above by a constant that does not depend on ku k . (cid:0) (cid:1) 0 H˜k Remark 4. For data satisfying (18), Theorem 3 implies global regularity since by the Sobolev embedding kukL∞t L∞x (R) .kukL∞t H˜k(R) for k > n2. 1If we only assume that ku0kH˜k ≪1 then the same conclusion holds. This is a consequence ofthelocaltheory: seeAppendix. FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 5 We recall some standard inequalities. The following Sobolev inequality holds: kuk .kDuk · (22) 2(n+2) 2(n+2) 2(n+2) 2n(n+2) Ltn−2 Lxn−2 (J) Ltn−2 Lxn2+4 (J) Ifuis asolutionofi∂ u+△u=G, u(t=0):=u onJ suchthatu(t)∈H˜k, t∈J, t 0 then the Strichartz estimates (see for example [6]) yield kuk +kDjuk +kDjuk L∞H˙j(J) 2(n+2) 2(n+2) 2(n+2) 2n(n+2) (23) t Lt n Lx n (J) Ltn−2 Lxn2+4 (J) .kDjGk +ku k 2(n+2) 2(n+2) 0 H˙j Ltn+4 Lxn+4 (J) if j ∈{1,k}. We define (24) Q(J,u):=kuk +kDuk +kDkuk +kuk · L∞H˜k(J) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) t Lt n Lx n (J) Lt n Lx n (J) Ltn−2 Lxn−2 (J) We also recall the following proposition: Proposition 4. “A fractional Leibnitz rule”[11] Let 0 ≤ α ≤ 1, k and β integers such that k ≥ 2 and β ≥ 1, (r,r ,r ) ∈ (1,∞)3, r ∈ (1,∞] be such that 1 2 3 1 = β + 1 + 1 . Let F :R+ →R be a Ck- function and let G:=C×C→C be a r r1 r2 r3 Ck- function such that F[i](x) =O F(x) , τ ∈[0,1]: |F[j](τx+(1−τ)y)|.|F[j](x)|+|F[j](y)| xi and (cid:16) (cid:17) O(|x|β+1−i), i≤β+1 (25) |G[i](x,x¯)|= 0, i>β+1 (cid:26) for 0≤i≤k and 1≤j ≤k. Then (26) Dk−1+α(G(f,f¯)F(|f|) .kfkβ kDk−1+αfk kF(|f|)k Lr Lr1 Lr2 Lr3 Here F[i] a(cid:13)nd G[i] denote the ith- d(cid:13)erivatives of F and G respectively. (cid:13) (cid:13) Now we explain how this paper is organized. In Section 2 we prove the main result of this paper, i.e Theorem 3. The proof relies upon the following bound of kuk on an arbitrarily long time interval 2(n+2) 2(n+2) Ltn−2 Lxn−2 2(n+2) 2(n+2) Proposition 5. “ Bound of L2(n−2)L2(n−2) norm ” Let u be an H˜k solution of t x (1) such that (18) holds. Let J :=[t ,t ] be an interval. Assume that for all t∈J 1 2 (27) g(u(t))−1≪δ· ThenthereexistsaconstantC ≫1suchthatifkuk ≤M forsomeM ≫1, 0 L∞H˜k(J) t then 2(n+2) Cδ−21gan+(M) (28) kuk n−2 ≤C 0 2(n+2) 2(n+2) 0 Ltn−2 Lxn−2 (J) 6 TRISTANROY ThisboundprovedonanarbitrarytimeintervalJ,combinedwithalocalinduc- tionontime ofsome Strichartzestimates, allowsto controla posteriori the L∞H˜k t norm of the solution and some other norms at H˜k regularity on J, and to show a posteriori thatthecondition(27)holdsonJ,assumingthatg growsslowlyenough, in the sense of (19): see [11, 12] for a similar argument. Global well-posedness and scattering of H˜k-solutions of (1) follow easily from the finiteness of these bounds. In Section 3, we prove Proposition 5. We mention the main differences between this paper and [11]. First one has to assume the condition (27). This condition, combined with the energy conservation law and the variational properties of the groundstates,assurethat somerelevantnorms(suchas the kinetic energyandthe potentialenergy)areboundedonJ,sothatwecanapplythetechniquesofconcen- tration (see e.g [2, 13]) in order to prove (28). Roughly speaking, we divide J into 2(n+2) 2(n+2) subintervals(J ) suchthatthe L n−2 L n−2 normofuconcentrates,i.e itis l 1≤l≤L t x smallbutalsosubstantial. Ourgoalistoestimatethenumberofthesesubintervals. It is already known that the mass on a ball centered at the origin concentrates for alltime ofeachof these subintervals. In [11], a Morawetz-typeestimate (combined withthemassconcentration)wasusedtoprovethatthefollowingstatementholds: one of these subintervals is large compare with J. In this paper we use the virial identity and we adapt arguments in [9, 14] to prove a decay at some time of the potential energy on a ball centered at the origin, which leads to a contradiction unless the statement above holds. With the use of this statement one can show that there exists a significantnumber of subintervals (in comparisonwith the total numberofsubintervals)thatconcentratearoundsometimeandsuchthatthemass concentrates around the origin, which yields an estimate of the number of all the subintervals. The process involves several estimates. One has to understand how they depend on g(M) and δ since this will play an important role in the choice of γ (see(19))forwhichwe haveglobalwell-posednessandscatteringofH˜k-solutions of (1). 2. Proof of Theorem 3 The proof is made of two steps: • finite bound of kuk , kuk , kDuk L∞H˜k(R) 2(n+2) 2(n+2) 2(n+2) 2(n+2) t Ltn−2 Lxn−2 (R) Lt n Lx n (R) and kDkuk . By time reversal symmetry 2 and by mono- 2(n+2) 2(n+2) Lt n Lx n (R) tone convergence it is enough to find, for all T ≥ 0, a finite bound of all these norms restricted to [0,T] and the bound should not depend on T. (29) F := T ∈[0,∞):sup Q([0,t],u)≤M ; g(u(t))−1≪δ t∈[0,T] 0 n o We claim that F = [0,∞) for M , a large constant (to be chosen later) 0 depending only on ku k and δ. Indeed 0 H˜k – 0∈F. – F is closed by continuity 2i.eift→u(t,x)isasolutionof(1)thent→u¯(−t,x)isalsoasolutionof(1) FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 7 – F is open. Indeed let T ∈F. Then, by continuity there exists δ′ > 0 such that for T′ ∈ [0,T +δ′] we have Q([0,T′]) ≤ 2M . In view of 0 (28), this implies, in particular, that 2(n+2) Cδ−12gan+(2M0) (30) kuk n−2 ≤C 0 2(n+2) 2(n+2) 0 Ltn−2 Lxn−2 ([0,T′]) LetJ :=[0,a]be aninterval. We getfrom(23), (13), andProposition 4 (31) 4 Q(J,u).ku k + kDuk +kDkuk kukn−2 0 H˜k 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) (cid:18) Lt n Lx n (J) Lt n Lx n (J)(cid:19) Ltn−2 Lxn−2 (J) g kuk L∞H˜k(J) t .(cid:16)ku k +Q((cid:17)J,u)kukn−42 g(Q(J,u)) 0 H˜k 2(n+2) 2(n+2) Ltn−2 Lxn−2 (J) LetC be the constantdetermined by . inthe secondline of(31). We may assume WLOG that C ≫1. Let 0<ǫ≪1. Notice that if J satisfies kuk = ǫ then a simple continuity 2(n+2) 2(n+2) n−2 Ltn−2 Lxn−2 (J) g 4 (2Cku0kH˜k) argument shows that (32) Q(J,u) ≤2Cku k · 0 H˜k Wedivide[0,T′]intosubintervals(J ) suchthatkuk = i 1≤i≤I 2(n+2) 2(n+2) ǫ ,1≤i 0 such that (39) holds for T′ ∈ [t ,T +β] with δ′ substituted for 2δ′. Hence in view of the 1 assumptions, the Sobolev embedding, and the conservation of the energy (40) E˜(u(t))=E(u)+X(u(t))<(1−2δ)E˜(W)+δ <(1−δ)E˜(W) Let us define (a): ku(t)kL2∗

Scattering above energy norm of solutions of a focusing logloglog energy-supercritical Schrodinger equation with radial data below ground state

Similar Scattering above energy norm of solutions of a focusing logloglog energy-supercritical Schrodinger equation with radial data below ground state