NONEXISTENCE RESULTS FOR A PSEUDO-HYPERBOLIC EQUATION IN THE HEISENBERG GROUP

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1 Electronic Journal of Differential Euations, Vol (2015, No. 110, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu NONEXISTENCE RESULTS FOR A PSEUDO-YPERBOLIC EQUATION IN TE EISENBERG GROUP MOKTAR KIRANE, LAKDAR RAGOUB Abstract. Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear seudo-hyerbolic euation u tt u tt u = u, (η, t (0,, > 1, is the Kohn-Lalace oerator on the (2N + 1-dimensional eisenberg grou. Then, this result is extended to the case of a 2 2-system of the same tye. Our techniue of roof is based on judicious choices of the test functions in the weak formulation of the sought solutions. 1. Introduction In this article, we are concerned with the nonexistence of weak solutions to the nonlinear seudo-hyerbolic euation under the initial conditions u tt u tt u = u, (η, t (0,, > 1, (1.1 u(η, 0 = u 0 (η, u t (η, 0 = u 1 (η, η, (1.2 is the Kohn-Lalace oerator on the (2N + 1-dimensional eisenberg grou. In the Euclidean case, seudo-hyerbolic euations served as models for the unidirectional roagation of nonlinear disersive long waves [2], cree buckling [5] for examle. For further alications, one is referred to the valuable book [1] a sizeable number of seudo-hyerbolic euations are studied. Our roofs rely on the test function method [8, 12]. For the reader convenience, some background facts used in the seuel are recalled. The (2N +1-dimensional eisenberg grou is the sace R 2N+1 euied with the grou oeration η η = (x + x, y + y, τ + τ + 2(x y x y, for all η = (x, y, τ, η = (x, y, τ R N R N R, denotes the standard scalar roduct in R N. This grou oeration endows with the structure of a Lie grou Mathematics Subject Classification. 47J35, 34A34, 35R03. Key words and hrases. Nonexistence; nonlinear seudo-hyerbolic euation; systems of seudo-hyerbolic euations; eisenberg grou. c 2015 Texas State University - San Marcos. Submitted January 5, Published Aril 22,

2 2 M. KIRANE, L. RAGOUB EJDE-2015/110 On it is natural to define a distance from η = (x, y, τ =: (z, τ to the origin by ( ( N 2 1/4 η = τ 2 + (x 2 i + yi 2 ( = τ 2 + z 4 1/4, i=1 x = (x 1,, x N and y = (y 1,, y N. The Lalacian over can be defined from the vectors fields for i = 1,, N, as follows X i = xi + 2y i τ and Y i = yi 2x i τ, = N (Xi 2 + Yi 2. i=1 A simle comutation gives the exression N ( u = 2 xix i u + y 2 iy i u + 4y i x 2 iτ u 4x i y 2 iτ u + 4(x 2 i + y i 2 ττ 2 u. i=1 The oerator satisfies the following roerties: It is invariant with resect to the left multilication in the grou, i.e., for all η, η, we have (u(η η = u(η η ; It is homogeneous with resect to a dilatation. More recisely, for λ R and (x, y, τ, we have (u(λx, λy, λ 2 τ = λ 2 ( u(λx, λy, λ 2 τ; If u(η = v( η, then ( d 2 v v(ρ = a(η dρ 2 + Q 1 dv, ρ dρ ρ = η, a(η = ρ 2 N i=1 (x2 i + y2 i and Q = 2N + 2 is the homogeneous dimension of. For more details on eisenberg grous, we refer to [4, 7]. In this work, we first rovide a sufficient condition for the nonexistence of weak solutions to the nonlinear roblem (1.1-(1.2, then we extend the result to the case of the 2 2 system u tt u tt u = v, v tt v tt v = u, u(η, 0 = u 0 (η, u 1 (η, 0 = u 1 (η, (η, t (0,, (η, t (0,, η v(η, 0 = v 0 (η, v(η, 0 = v 1 (η, η, (1.3, > 1 are real numbers, for which we rovide a sufficient condition for the nonexistence of weak solutions. 2. Results and roofs Let T = (0, T, = (0,. For R > 0, let U R = {(x, y, τ, t : 0 t 4 + x 4 + y 4 + τ 2 2R 4 }.

3 EJDE-2015/110 NONEXISTENCE OF SOLUTIONS IN TE EISENBERG GROUP Case of a single euation. The definition of solutions we adot for (1.1- (1.2 is: We say that u is a local weak solution to (1.1-(1.2 on with initial data u(0, = u 0 L 1 loc (, if u L loc ( and satisfies u ϕ dϑ dt + u 1 (ϑϕ(ϑ, 0 dϑ + u 1 (ϑ ϕ(ϑ, 0 dϑ = uϕ tt dϑ dt + u ϕ tt dt dϑ u ϕ dt dϑ, for any test function ϕ, ϕ(, t = 0, ϕ t (, t = 0, t T. The solution u is said global if it exists on (0,. Our first main result is given by the following theorem. Theorem 2.1. Let u 1 L 1 (. Suose that u 0 dϑ > 0. (2.1 If 1 < Q 1, then any weak solution to (1.1-(1.2 blows-u in a finite time. Proof. Suose that u is a weak solution to (1.1-(1.2. Then for any regular test function ϕ, we have u ϕ dϑ dt + u 1 (ϑϕ(ϑ, 0 dϑ u ϕ tt dϑ dt + u ϕ tt dt dϑ (2.2 + u ϕ dt dϑ + u 1 (ϑ ϕ(ϑ, 0 dϑ. Using the ε-young ineuality ab εa + C ε b, a, b, ε, C ε > 0, 1 <,, + =, with arameters and = /( 1, we obtain u ϕ tt dϑ dt = u ϕ 1/ ϕtt dϑ dt ε u ϕ dϑ dt + c ε for some ositive constant c ε. Similarly, we have u ϕ tt dt dϑ ε u ϕ dϑ dt + c ε u ϕ dt dϑ ε u ϕ dϑ dt + c ε 1 ϕtt 1 dϑ dt, (2.3 1 ϕ tt 1 dt dϑ, (2.4 P 1 ϕ P 1 dt dϑ. (2.5

4 4 M. KIRANE, L. RAGOUB EJDE-2015/110 Using (2.2, (2.3, (2.4 and (2.5, for ε > 0 small enough, we obtain u ϕ dϑ dt + u 1 (ϑϕ(ϑ, 0 dϑ ( C A (ϕ + B (ϕ + C (ϕ + u 1 (ϑ ϕ(ϑ, 0 dϑ, A (ϕ = B (ϕ = C (ϕ = 1 ϕtt (2.6 1 dϑ dt, (2.7 1 ϕ tt 1 dϑ dt, (2.8 1 ϕ 1 dϑ dt. (2.9 Now, let us consider the test function ϕ R (t, ϑ = φ ω( t 4 + x 4 + y 4 + τ 2 R 4, R > 0, ω 1, (2.10 φ C0 (R + is a decreasing function satisfying { 1 if 0 r 1, φ(r = 0 if r 2. Observe that su(ϕ R is a subset of U R, while su(ϕ Rtt, su( ϕ R and su( (ϕ R tt are subsets of Let Then we have Θ R = {(t, x, y, τ : R 4 t 4 + x 4 + y 4 + τ 2 2R 4 }. ρ = t4 + x 4 + y 4 + τ 2 R 4. ϕ R (t, ϑ 4ω(N + 4 ( = R 4 x 2 + y 2 φ (ρφ ω 1 (ρ 16ω(ω 1 ( + R 8 ( x 6 + y 6 + 2τ( x 2 y 2 x y + τ 2 ( x 2 + y 2 φ 2 (ρφ ω 2 (ρ + 16ω ( R 8 ( x 6 + y 6 + 2τ( x 2 y 2 x y + τ 2 ( x 2 + y 2 φ (ρφ ω 1 (ρ for examle. Observe first that (ϕ R t (ϑ, 0 = 0 as reuired in the definition. It follows that there is a ositive constant C > 0, indeendent of R, such that for all (t, ϑ R, we have ϕ R (t, ϑ CR 2 φ ω 2 (ρχ(ρ, (2.11 and χ(ρ = φ (ρ φ(ρ + φ 2 (ρ + φ (ρ φ(ρ, ( ϕ R t (t, ϑ CR 3, (2.12 (ϕ R tt (t, ϑ CR 4. (2.13

5 EJDE-2015/110 NONEXISTENCE OF SOLUTIONS IN TE EISENBERG GROUP 5 Using (2.11 and (2.12, we obtain Let us consider now the change of variables Let 2 Q+1 A (ϕ R CR 1, ( Q+1 B (ϕ R CR 1, ( Q+1 C (ϕ R CR 1. (2.16 (t, x, y, τ = (t, ϑ ( t, ṽ = ( t, x, ỹ, τ, (2.17 t = R 1 t, τ = R 2 τ, x = R 1 x, ỹ = R 1 y. ρ = t 4 + x 4 + ỹ 4 + τ 2, C R = {( t, x, ỹ, τ : 1 ρ 2}, C R = {(x, y, τ : R 4 x 4 + y 4 + τ 2 2R 4 }. Using (2.6, (2.15 and (2.16, we obtain u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ ( C R ϑ1 + R ϑ2 + u 1 (ϑ ϕ R (ϑ, 0 dϑ C R, ϑ 1 = Q and ϑ 2 = Q On the other hand, we have lim inf u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ lim inf u ϕ R dϑ dt + lim inf u 1 (ϑϕ R (ϑ, 0 dϑ. Using the monotone convergence theorem, we obtain lim inf u ϕ R dϑ dt = u dϑ dt. Since u 1 L 1 (, by the dominated convergence theorem, we have lim inf u 1 (ϑϕ R (ϑ, 0 dϑ = u 1 (ϑ dϑ. Now, we have ( lim inf from (2.1, u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ u dϑ dt + l, l = u 1 (ϑ dϑ > 0. (2.18 By the definition of the limit inferior, for every ε > 0, there exists R 0 > 0 such that u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ

6 6 M. KIRANE, L. RAGOUB EJDE-2015/110 ( > lim inf u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ ε u dϑ dt + l ε, for every R R 0. Taking ε = l/2, we obtain u ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ u dϑ dt + l 2, for every R R 0. Then from (2.18, we have u dϑ dt + l (R 2 C ϑ1 + R ϑ2 + u 0 (ϑ ϕ R (ϑ, 0 dϑ, (2.19 C R for R large enough. Now, we reuire that ϑ 1 = max{ϑ 1, ϑ 2 } 0, which is euivalent to 1 < Q 1. We distinguish two cases. Case 1: 1 < < Q 1. In this case, letting R in (2.19 and using the dominated convergence theorem, we obtain u dϑ dt + l 2 0, which is a contradiction as l > 0. Case 2: = Q 1. From (2.19, we obtain u dϑ dt C < lim u ϕ R dϑ dt = 0. (2.20 C R Using the ölder ineuality with arameters and /( 1, from (2.2, we obtain u ϕ R dϑ dt + l ( 1/. 2 C u ϕ R dϑ dt Θ R Letting R in the above ineuality and using (2.20, we obtain u dϑ dt + l 2 = 0. A contradiction; the roof of the theorem is comlete The case of system (1.3. The definition of solutions we adot for (1.3 is: We say that the air (u, v is a local weak solution to (1.3 on with initial data (u(0,, v(0, = (u 0, v 0 L 1 loc ( L1 loc (, if (u, v L loc ( L loc ( and satisfies v ϕ dϑ dt + u 1 (ϑϕ(ϑ, 0 dϑ = uϕ tt dϑ dt + u( ϕ tt dt dϑ u ϕ dt dϑ + u 1 (ϑ ϕ(ϑ, 0 dϑ and u ϕ dϑ dt + v 1 (ϑϕ(ϑ, 0 dϑ = vϕ tt dϑ dt + v( ϕ tt dt dϑ v ϕ dt dϑ + v 1 (ϑ ϕ(ϑ, 0 dϑ,

7 EJDE-2015/110 NONEXISTENCE OF SOLUTIONS IN TE EISENBERG GROUP 7 for any test function ϕ, ϕ(, t = 0, ϕ t (, t = 0, t T. The solution is said global if it exists for T = +. Our second main result is given by the following theorem. Theorem 2.2. Let (u 1, v 1 L 1 ( L 1 (. Suose that u 1 dϑ > 0 and v 1 dϑ > 0. If 1 < (, ( = max{ + 1, + 1}, Q 1 then there exists no nontrivial weak solution to (1.3. Proof. Suose that (u, v is a nontrivial weak solution to (1.3. regular test function ϕ, we have v ϕ dϑ dt + u 1 (ϑϕ(ϑ, 0 dϑ u ϕ tt dϑ dt + u ( ϕ tt dt dϑ + u ϕ dt dϑ + u 1 (ϑ ϕ(ϑ, 0 dϑ and u ϕ dϑ dt + v 1 (ϑϕ(ϑ, 0 dϑ v ϕ tt dϑ dt + v ( ϕ tt dt dϑ + v ϕ dt dϑ + v 1 (ϑ ϕ(ϑ, 0 dϑ. Then for any Taking ϕ = ϕ R, the test function given by (2.10, and using the ölder ineuality with arameters and /( 1, we obtain v ϕ R dϑ dt + u 1 (ϑϕ R (ϑ, 0 dϑ u 1 (ϑ ϕ R (ϑ, 0 dϑ ( (A 1/, (ϕ R 1 + B (ϕ R 1 + C (ϕ R 1 u ϕ R dϑ dt A (ϕ, B (ϕ and C (ϕ are given resectively by (2.7, (2.8 and (2.9. Similarly, by the ölder ineuality with arameters and /( 1, we get u ϕ R dϑ dt + v 1 (ϑϕ R (ϑ, 0 dϑ v 1 (ϑ ϕ R (ϑ, 0 dϑ ( (A 1/. (ϕ R 1 + B (ϕ R 1 + C (ϕ R 1 v ϕ R dϑ dt Without restriction of the generality, we may assume that for R large enough, we have u 1 (ϑϕ R (ϑ, 0 dϑ u 1 (ϑ ϕ R (ϑ, 0 dϑ 0, v 1 (ϑϕ R (ϑ, 0 dϑ v 1 (ϑ ϕ R (ϑ, 0 dϑ 0.

8 8 M. KIRANE, L. RAGOUB EJDE-2015/110 Slight modifications yield the roof in the general case (see the roof of Theorem 2.1. Then for R large enough, we have v ϕ R dϑ dt and (A (ϕ R 1 u ϕ R dϑ dt (A (ϕ R 1 + B (ϕ R 1 + B (ϕ R 1 ( + C (ϕ R 1 ( + C (ϕ R 1 1/ (2.21 u ϕ R dϑ dt 1/. (2.22 v ϕ R dϑ dt Using the change of variables (2.17, from (2.21 and (2.22, we obtain ( 1/, v ϕ R dϑ dt CR Q( 1 2 u ϕ R dϑ dt (2.23 ( 1/. u ϕ R dϑ dt CR Q( 1 2 v ϕ R dϑ dt (2.24 Combining (2.23 with (2.24, we obtain ( u ϕ R dϑ dt ( υ 1 = Q( 1 2( v ϕ R dϑ dt and υ 2 = CR υ1, (2.25 CR υ2, (2.26 Q( 1 2( We reuire that υ 1 0 or υ 2 0 which is euivalent to 1 < Q max{ + 1, + 1}. We distinguish two cases. Case 1: 1 < < Q max{ + 1, + 1}. Without loss of the generality, we may suose that 0 <. In this case, letting R in (2.25, we obtain u dϑ dt = 0, which is a contradiction. Case 2: = Q max{ + 1, + 1}. This case can be treated in the same way as in the roof of Theorem 2.1. Remark 2.3. If = and u = v in Theorem 2.2, we obtain the result for a single euation given by Theorem 2.1. References [1] A. B. Al shin, M. O. Korusov, A. G. Sveshnikov; Blow-u in nonlinear Sobolev tye, De Gruyter, [2] T. B. Benjamin, J. L. Bona, J. J. Mahony; Model euations for long waves in nonlinear disersive systems, Philos. Trans. R. Soc. Lond. Ser. A 272 (1220 (1972, [3] Y. Cao, J. Yin, C. Wang; Cauchy roblems of semilinear seudo-arabolic euations, J. Differential Euations. 246 (2009,

9 EJDE-2015/110 NONEXISTENCE OF SOLUTIONS IN TE EISENBERG GROUP 9 [4] G. B. Folland, E. M. Stein; Estimate for the comlex and analysis on the eisenberg grou, Comm. Pure Al. Math. 27 (1974, [5] N. J. off; Cree buckling, Aeron. Quart. 7 (1956, No 1, [6] E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev; The Cauchy roblem for a Sobolev tye euation with ower like nonlinearity, Izv Math. 69 (2005, [7] E. Lanconelli, F. Uguzzoni; Asymtotic behaviour and non existence theorems for semilinear Dirichlet roblems involving critical exonent on unbounded domains of the eisenberg grou, Boll. Un. Mat. Ital. 1(1 (1998, [8] E. Mitidieri, S. I. Pohozaev; A riori estimates and the absence of solutions of nonlinear artial differential euations and ineualities, Tr. Mat. Inst. Steklova. 234 ( [9] S. I. Pokhozhaev; Nonexistence of Global Solutions of Nonlinear Evolution Euations, Differential Euations. 49, No. 5 (2013, [10] S.I. Pohozaev, L. Véron; Nonexistence results of solutions of semilinear differential i neualities on the eisenberg grou, Manuscrita Math. 102 ( [11] S. L. Sobolev; On a new roblem of mathematical hysics, Izv. Akad. Nauk USSR Ser. Math. 18 (1954, [12] Qi S. Zhang; The critical exonent of a reaction diffusion euation on some Lie grous, Math. Z. 228 (1 (1998, Mokhtar Kirane Laboratoire de Mathématiues, Image et Alications, Pôle Sciences et Technologies, Université de La Rochelle, Avenue M. Créeau, La Rochelle, France. NAAM Research Grou, Deartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia address: mkirane@univ-lr.fr Lakhdar Ragoub Al Yamamah University, College of Comuters and Information Systems P.O. Box 45180, Riyadh 11512, Saudi Arabia address: l ragoub@yu.edu.sa