NONEXISTENCE OF GLOBAL SOLUTIONS OF SOME NONLINEAR SPACE-NONLOCAL EVOLUTION EQUATIONS ON THE HEISENBERG GROUP

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1 Electronic Journal of Differential Equations, Vol (2015, No. 227, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu NONEXISTENCE OF GLOBAL SOLUTIONS OF SOME NONLINEAR SPACE-NONLOCAL EVOLUTION EUATIONS ON THE HEISENBERG GROUP BASHIR AHMAD, AHMED ALSAEDI, MOKHTAR KIRANE Abstract. This article resents necessary conditions for the existence of weak solutions of the following sace-nonlocal evolution equations on H (0, +, where H is the Heisenberg grou: 2 u t 2 + ( H α/2 u m = u, u t + ( H α/2 u m = u, 2 u t 2 + ( H α/2 u m + u t = u, R, > 1, m N. Moreover, the life san for each equation is estimated under some suitable conditions. Our method of roof is based on the test function method. 1. Introduction and reliminaries In this article we are concerned with the nonexistence of global solutions of nonlinear wave equations, of nonlinear wave equations with linear daming, and of nonlinear arabolic equations with nonlocal diffusion osed on the Heisenberg grou. We start with the wave equation 2 u t 2 + ( H α/2 u m = u, (1.1 where R, > 1, m N, osed in R 2N+1 R + = R 2N+1,1 +, sulemented with the initial data u u(η, 0 = u 0 (η, t (η, 0 = u 1(η, η = (x, y, τ. The oerator ( H α/2 (0 < α < 2 accounts for anomalous diffusion (see below for the definition. Following the lines of the aer of Véron and Pohozaev [22] and using a variant of Cordoba-Cordoba s inequality [3] for the Heisenberg grou roved in this aer, we find an exonent for (1.1 similar to Kato s exonent [9] but with the imrovement on the data given in [22] Mathematics Subject Classification. 35A01, 35R03, 35R11. Key words and hrases. Nonlinear hyerbolic equations; nonlinear arabolic equations; nonlocal oerators; daming; blowing-u solutions. c 2015 Texas State University - San Marcos. Submitted August 19, Published Setember 2,

2 2 B. AHMAD, A. ALSAEDI, M. KIRANE EJDE-2015/227 In a second case, we rove a Fujita-Galaktionov tye result for the heat equation u t + ( H α/2 u m = u, (1.2 where R, > 1, osed in R 2N+1,1 +, sulemented with the initial data u(η, 0 = u 0 (η. Next, we resent the critical exonent for the wave equation with linear daming 2 u t 2 + ( H α/2 u m + u t = u, (1.3 where R, > 1, osed in R 2N+1,1 +, sulemented with the initial data u(η, 0 = u 0 (η, u t (η, 0 = u 1(η. For the reader s convenience, let us briefly recall the definition and the basic roerties of the Heisenberg grou. Heisenberg grou. The Heisenberg grou H, whose oints will be denoted by η = (x, y, τ, is the Lie grou (R 2N+1, with the non-commutative grou oeration defined by η η = (x + x, y + ỹ, τ + τ + 2( x, ỹ x, y, where, is the usual inner roduct in R N. The Lalacian H over H is obtained from the vector fields X i = x i + 2y i τ and Y i = y i 2x i τ, as N H = (Xi 2 + Yi 2. (1.4 i=1 Observe that the vector field T = τ does not aear in (1.4. This fact makes us resume a loss of derivative in the variable τ. The comensation comes from the relation [X i, Y j ] = 4T, i, j {1, 2,, N}. (1.5 The relation (1.5 roves that H is a nilotent Lie grou of order 2. Incidently, (1.5 constitutes an abstract version of the canonical relations of commutation of Heisenberg between momentum and ositions. Exlicit comutation gives the exression H = N ( 2 i=1 x 2 i + 2 y 2 i A natural grou of dilatations on H is given by 2 + 4y i x i τ 4x 2 i y i τ + 4(x2 i + yi 2 2 τ 2. (1.6 δ λ (η = (λx, λy, λ 2 τ, λ > 0, whose Jacobian determinant is λ, where = 2N + 2 (1.7 is the homogeneous dimension of H. The natural distance from η to the origin is introduced by Folland and Stein, see [5] ( ( N 2 1/4 η H = τ 2 + (x 2 i + yi 2. (1.8 i=1

3 EJDE-2015/227 NONEXISTENCE OF GLOBAL SOLUTIONS 3 Fractional owers of sub-ellitic Lalacians. Here, we recall a result on fractional owers of sub-lalacian in the Heisenberg grou taken from [6]. Let N (t, x be the fundamental solution of H + t. For all 0 < β < 4, the integral R β (x = 1 Γ( β t β 2 1 N (t, x dt converges absolutely for x 0. If β < 0, β / {0, 2, 4,... }, then R β (x = β 2 Γ( β t β 2 1 N (t, x dt defines a smooth function in H \ {0}, since t N (t, x vanishes of infinite order as t 0 if x 0. In addition, R β is ositive and H-homogeneous of degree β 4. Theorem 1.1. For every v S(H, ( H s v L 2 (H and ( H s v(x = (v(x y v(x χ(y H v(x, y R 2s (y dy H = P.V (v(y v(x R 2s (y 1 x dy, H where χ is the characteristic function of the unit ball B ρ (0, 1, (ρ(x = R 1 2+α 2 α (x, 0 < α < 2, ρ is an H-homogeneous norm in H smooth outside the origin. Before we resent our results, let us dwell a while some literature concerning nonexistence or blowing-u solutions to wave equations and arabolic equations osed in the Heisenberg grou. A lot has been said on the non-existence or blowingu solutions to the wave equation (1.1, with α = 2 and = 1, with a final result in [23], for the arabolic equation [7] (for the ioneering aer and [1] (for the alternative roof that will be used here, for the wave equation with a linear daming [10, 21, 26] where the critical has been decided mainly in [21], when the equations are osed in the Euclidian sace. Concerning (1.1, (1.2 and (1.3 with α = 2 and = 1, osed on the Heisenberg grou, a few aers aeared; we may cite [4, 8, 15, 16, 17, 22, 24, 25, 26]. As far as we know, no blowing-u solutions or non-existence results for (1.1, (1.2 and (1.3 have been ublished. Evolution equation with fractional ower of the Lalacian osed on the Heisenberg sace are mentioned, for examle, in [11, 12, 13, 19, 20]. Our method of roof relies on a method due to Baras and Pierre [1]; it had been remained dormant until Zhang [24, 25, 26] revived it. Later, this method has been successfully alied in a great number of situations by Mitidieri and Pohozaev [14]. 2. Main results Proosition 2.1. Consider a convex function F C 2 (R. C0 (R 2N+1. Then Assume that ϕ F (ϕ( H α/2 ϕ ( H α/2 F (ϕ (2.1 holds oint-wise. In articular, if F (0 = 0 and ϕ C 0 (R 2N+1, then R 2N+1 F (ϕ( H α/2 ϕ dη 0. (2.2

4 4 B. AHMAD, A. ALSAEDI, M. KIRANE EJDE-2015/227 Proof. We have F (θ F (ϱ F (ϱ(θ ϱ 0 for all θ, ϱ. We substitute then θ = ϕ(x y, ϱ = ϕ(x and integrate against R α (y 1 x dy (recall that R α (y 1 x 0. Let us mention that hereafter we will use inequality (2.1 for F (ϕ = ϕ σ, σ 1, ϕ 0; in this case it reads σϕ σ 1 ( H α/2 ϕ ( H α/2 ϕ σ. (2.3 We need the following Lemma taken from [18]. Lemma 2.2 ([18, Lemma 3.1]. Let f L 1 (R 2N+1 and fdη 0. Then R 2N+1 there exists a test function 0 ϕ 1 such that fϕ dη 0. (2.4 R 2N+1 Let us set T = T 0 R 2N+1, = 0 R 2N Wave equation (1.1 Definition 3.1. A locally integrable function u L max{,m} loc (R 2N+1 (0, T is called a local weak solution of (1.1 in R 2N+1 (0, T subject to the initial data u 0, u 1 L 1 loc (R2N+1 if the equality T = ( u 2 ϕ t 2 + u m ( H α/2 ϕ dη dt u ϕ dη dt u 0 (η ϕ (η, 0 dη + u 1 (ηϕ(η, 0 dη R t 2N+1 R 2N+1 T is satisfied for any regular function ϕ C 2 ((0, T ]; H α (R 2N+1 C 1 ([0, T ]; H α (R 2N+1, (3.1 with ϕ(, T = 0, ϕ 0, where H α (R 2N+1 is the homogeneous Sobolev sace of order α. The solution is called global if T = +. Theorem 3.2. Let 1 < m < < m,α := 2m+α 2 α, and R 2N+1 u 1 (η dη > 0. Then, (1.1 does not have a nontrivial weak solution. Proof. The roof is by contradiction. For that, let u be a solution and ϕ be a smooth nonnegative test function such that ( ϕ 2 2 ϕ 1 2 A(ϕ := σ(σ 1 + σϕ (σ t t 2 ϕ 1 dη dt <, (3.2 B(ϕ := ( H α/2 ϕ m ϕ (σ m dη dt <.

5 EJDE-2015/227 NONEXISTENCE OF GLOBAL SOLUTIONS 5 Then, taking ϕ σ, σ 1 instead of ϕ in (3.1 and using inequality (2.3, we obtain u ϕ σ dη dt + u 1 (ηϕ σ σ 1 ϕ (η, 0 dη σ u 0 (ηϕ (η, 0 dη R 2N+1 R t 2N+1 ( u(σ(σ 1ϕ σ 2( ϕ 2 + σϕ σ 1 2 ϕ t t 2 dη dt (3.3 + σ u m ϕ σ 1 ( H α/2 ϕ dη dt 1 u ϕ σ dη dt + C ( A(ϕ + B(ϕ 2 by means of the ε-young s inequality ab εa + C(εb, + =, a 0, b 0. Choosing ϕ such that ϕ (η, 0 = 0, (3.4 t from (3.3, (3.4, and using Lemma 2.2, we obtain u ϕ σ dη dt C ( A(ϕ + B(ϕ. (3.5 Set ϕ(η, t = ϕ 1 (ηϕ 2 (t = Φ ( τ 2 + x 4 + y 4 R 4 Φ ( t 2 R 2ρ, ρ = α( 1 2( m, where R > 0, and Φ D([0, + [ is the standard cut-off function 1, 0 r 1, Φ(r =, 1 r 2, 0, r 2. Set Ω 1 = { η H; 0 τ 2 + x 4 + ỹ 4 2 }, Ω 2 = { t; 0 t 2 2 }. Note that ϕ ( τ 2 t (η, t = + x 4 + y 4 2tR 2 Φ R 4 Φ ( t 2 ϕ = R 2ρ (η, 0 = 0; t so equality (3.4 is satisfied as required. Moreover, using the scaled variables we obtain the estimates τ = R 2 τ, x = R 1 x, ỹ = R 1 y, t = R ρ t, A(ϕ C 1 R ϑ, B(ϕ C 2 R ϑ, (3.6 with ϑ = α m + + ρ; the constants C 1, C 2 are A(ϕ and B(ϕ evaluated on Ω 1 Ω 2. Now, if α m + + ρ < 0 < α,m, by letting R in (3.5, we obtain u dη dt = 0 = u 0; this is a contradiction.

6 6 B. AHMAD, A. ALSAEDI, M. KIRANE EJDE-2015/227 Remark 3.3. For the equation the found exonent is α,1 2,1 = +1 1, the one obtained in [22]. 2 u t 2 + ( H α/2 u = u, := 2+α 2 α which, in the limiting case α = 2, gives Remark 3.4. For u 0, u 1 C 0 (R 2N+1, the solution exists locally and is very regular as it can be roved following the lines of [27]. So in this case, the nonexistence result we have roved is in fact a blow-u result, that is, the solutions blows-u at a finite time. 4. Parabolic equation (1.2 Definition 4.1. A locally integrable function u L loc (R2N+1 (0, T is called a local weak solution of the differential equation (1.2 in R 2N+1 (0, T subject to the initial data u 0 L 1 loc (R2N+1 if the equality u ϕ dη dt + u 0 (ηϕ(η, 0 dη T R ( 2N+1 = u ϕ (4.1 t + u m ( H α/2 ϕ dη dt T is satisfied for any regular function ϕ C 1 ((0, T ]; H α (R 2N+1 C([0, T ]; H α (R 2N+1, with ϕ(., T = 0, ϕ 0. The solution is called global if T = +. Theorem 4.2. Let 1 < < m + α and u R 2N+1 0 (η 0. Then, (1.2 does not have a nontrivial global weak solution. Proof. The roof is similar to the revious one. Let u be a global weak solution and ϕ be a smooth nonnegative test function such that ( ϕ E(ϕ = 1 ϕ σ 1 + ( H α/2 ϕ m ϕ σ m dη dt <. (4.2 t Then, using inequality (2.3 in (4.1 with ϕ σ as a test function and using ε-young s inequality, we obtain u ϕ σ dη dt + u 0 (ηϕ σ (η, 0 dη R ( 2N+1 = u ϕσ + u m ( H α/2 ϕ σ dη dt t ( σ 1 ϕ σ uϕ t + u m ϕ σ 1 ( H α/2 ϕ dη dt (4.3 1 u ϕ dη dt + C ϕ 1 ϕ σ 1 dη dt 2 t + C ( H α/2 ϕ m ϕ σ m dη dt,

7 EJDE-2015/227 NONEXISTENCE OF GLOBAL SOLUTIONS 7 so, using Lemma 2.2, we obtain u ϕ σ dη dt C ϕ 1 ϕ σ 1 dη dt t + C ( H α/2 ϕ (4.4 m ϕ σ m dη dt, for some constant C > 0. Setting ( τ 2 + x 4 + y 4 ϕ(η, t = Φ R 4 Φ ( t α( 1, ρ = R ρ m and using the change of variables, we obtain the estimates ϕ t ( H α/2 (ϕ τ = R 2 τ, x = R 1 x, ỹ = R 1 y, t = R ρ t, 1 ϕ σ 1 dη dt CR ρ m ϕ σ m dη dt CR α 1 ++ρ, (4.5 The constraint 1 < < m + α allows us to obtain the contradiction u dη dt = lim u ϕ σ dη dt = 0 u 0. R m ++ρ. (4.6 Remark 4.3. For m = 1, as in the Euclidian case, we can obtain analogous estimates for the semi-grou generated by the linear art of (1.2 via estimates of the semi-grou generated by the one of linear art with α = 2. Whereuon, the existence of a local mild solution to (1.2 can be obtained by simle alication of the Banach fixed oint theorem (see [17] for the case α = 2. This suggests that the limiting exonent for (1.2 is in fact a Fujita s exonent. 5. Wave equation with linear daming (1.3 Now, we consider the wave equation with a linear daming. Definition 5.1. A locally integrable function u L max{,m} loc (R 2N+1 (0, T is called a local weak solution of the differential equation (1.3 in R 2N+1 (0, T subject to the initial data u(η, 0 = u 0 (η, u t (η, 0 = u 1(η, u 0, u 1 L 1 loc (R2N+1 if the equality u ϕ dη dt + T = T R 2N+1 ( u 0 (ηϕ(η, 0 + u 0 (η ϕ ( u 2 ϕ t 2 u ϕ t + u m ( H α/2 ϕ is satisfied for any regular function t (η, 0 + u 1(ηϕ(η, 0 dη (5.1 dη dt ϕ C 2 ((0, T ]; H α (R 2N+1 C 1 ([0, T ]; H α (R 2N+1, with ϕ(., T = 0, ϕ 0. The solution is called global if T = +. Theorem 5.2. Let 1 < < m + α, and R 2N+1 (u 1 (η + u 0 (η dη 0. Then (5.1 does not admit a nontrivial weak solution.

8 8 B. AHMAD, A. ALSAEDI, M. KIRANE EJDE-2015/227 Proof. Choosing ( τ 2 + x 4 + y 4 ϕ(η, t = Φ R 4 Φ ( t 2 α( 1, ρ = R 2ρ m ( ϕ t (η, 0 = 0, and using Lemma 2.2 together with the estimates, as before, we obtain u ϕ σ dη dt C ( A(ϕ + B(ϕ + C(ϕ, (5.2 where C(ϕ := ϕ t After using the scaled the variables 1 ϕ σ 1 dη dt <. τ = R 2 τ, x = R 1 x, ỹ = R 1 y, t = R ρ t, we obtain the estimate u ϕ σ dη dt + u 1 (ηϕ(η, 0 dη + u 0 (ηϕ(η, 0 dη CR α R 2N+1 R 2N+1 The requirement 1 < < m + α allows us to conclude as in the revious cases. m ++ρ. Remark 5.3. We conjecture that the limiting exonent here is the critical exonent. However to show that in fact it is, one needs to obtain similar results as in [21]. 6. Life san of solutions Our first result concerns the wave equation. satisfies the condition We assume that the data u 1 (η u 1 (η η k H, x R2N+1, k >. (6.1 Theorem 6.1. Suose that (6.1 is satisfied, 1 < m < < m,α, and let u be the solution of (1.1 with the initial data u(η, 0 = µu 0 (η, where µ > 0. Denote by [0, T µ the life san of u. Then there exists a constant C > 0 such that where κ := k α(+1 2( m < 0. T µ Cµ 1/κ, (6.2 Proof. We take, for T µ > 0, ϕ(η, t := ϕ 1 (ηϕ 2 (t, where ( τ 2 + x 4 + y 4 ϕ 1 (η := Φ, ϕ 2 (t := ( t 2. T 4 µ We clearly note that: t ϕ(η, 0 = 0. Now, as for the estimate (3.3, we have, with := [0, Tµ] 2 R 2N+1, u ϕ σ (η, t dη dt + µ u 1 (ηϕ(η, 0 dη CA + CB. R 2N+1 (6.3 Using the ositivity of the first term in the left-hand side of (6.3, we have µ u 1 (ηϕ(η, 0 dη CA + CB. R 2N+1 Next, by the assumtion on the data u 1 (η, we get µ η k ϕ(η, 0 dη µ u 1 (ηϕ(η, 0 dη. R 2N+1 R 2N+1 T 2ρ µ

9 EJDE-2015/227 NONEXISTENCE OF GLOBAL SOLUTIONS 9 Thus µ η k ϕ(η, 0 dη CA + CB. R 2N+1 We ass to the new variables t = T ρ µ and η = ( τ, x, ỹ such that τ = T 2 T 1 µ x, ỹ = T 1 µ y, to obtain Whereuon µt k µ R 2N+1 η Φ(ν σ d η CT θ µ. α(+1 k µ CT 2( m = Tµ Cµ 1/κ, where κ := k α(+1 2( m < 0. This comletes the roof of the theorem. µ τ, x = The results concerning the heat equation and the damed wave equation are: For heat equation, we have the estimate T µ Cµ 1/κ, where κ := k α m < 0, with u 0 (η η k H ; For wave equation with linear daming, we have the estimate T µ Cµ 1/κ, where κ := k α m < 0, with u 0(η + u 1 (η η k H, whose roofs are similar to the revious one and hence are omitted. Acknowledgements. The roject was funded by the Deanshi of Scientific Research (DSR, King Abdulaziz University, under grant no. ( RG. The authors, therefore, acknowledge with thanks DSR technical and financial suort. References [1] P. Baras, M. Pierre; Critère d existence de solutions ositives our des équations semilinéaires non monotones, Annales de l Institut H. Poincaré Analyse Non Linéaire, 2 ( [2] Pierre Baras, Robert Kersner; Local and global solvability of a class of semilinear arabolic equations, J. Differential Equations, 68 (1987, no. 2, [3] A. Cordoba, D. Cordoba; A maximum rincile alied to quasi-geostrohic equations, Comunn. Math. Phys, 249 ( [4] A. El Hamidi, M. Kirane; Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg grou, Abstract and Alied Analysis Volume 2004 (2004, Issue 2, Pages [5] G. B. Folland, E. M. Stein; Estimates for the h comlex and analysis on the Heisenberg Grou, Comm. Pure Al. Math. 27 (1974, [6] F. Ferrari, B. Franchi; Harnack inequality for fractional sub-lalacian in Carnot grous, rerint. [7] H. Fujita; On the blowing-u of solutions to the Cauchy roblems for u t = u + u 1+α, J. Fac. Sci. Univ. Tokyo, Sect. IA 13 (1966, [8] J. A. Goldstein, I. Kombe; Nonlinear degenerate arabolic equations on the Heisenberg grou, Int. J. Evol. Equ. (1 1 (2005, 122. [9] T. Kato; Blow-u of solutions of some nonlinear hyerbolic equations in three sace dimensions, Comm. Pure Al. Math., 37 (1984, [10] M. Kirane, M. afsaoui; Fujita s exonent for a semilinear wave equation with linear daming, Advanced Nonlinear Studies, (2 1 (2002, [11] N. Laskin; Fractional uantum Mechanics and Lévy Path Integrals, Physics Letters (268 A (2000, [12] N. Laskin; Fractional Schrödinger equation, Physical Review E 66 ( ages. [13] N. Laskin; Fractional uantum Mechanics, Physical Review E 62 ( [14] E. Mitidieri, S. I. Pohozaev; A riori estimates and blow-u of solutions to nonlinear artial differential equations and inequalities, Proceedings of the Steklov Institute of Mathematics, 234 (2001,

10 10 B. AHMAD, A. ALSAEDI, M. KIRANE EJDE-2015/227 [15] D. Müller, E. M. Stein; L -estimates for the wave equation on the Heisenberg grou, Rev. Mat. Iberoamericana (15 2 (1999, [16] A. I. Nachman; The wave equation on the Heisenberg grou, Comm. Partial Differential Equations (7 6 (1982, [17] A. Pascucci; Semilinear equations on nilotent Lie grous: global existence and blow-u of solutions, Matematiche (53 2 (1998, [18] S. I. Pokhozhaev; Nonexistence of Global Solutions of Nonlinear Evolution Equations, Differential Equations. 49 ( No. 5 (2013, [19] V. E. Tarasov; Fractional dynamics. Nonlinear hysical science, Sringer. ISBN , [20] V. E. Tarasov; Fractional Heisenberg equation, Phys. Lett. A 372 ( [21] G. Todorova, B. Yordanov; Critical exonent for the wave equation with daming, Journal Diff. Eq., 174, (2001, [22] L. Véron, S. I. Pohozaev; Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg grou, Manuscrita Math. 102 ( [23] B. Yordanov, i S. Zhang; Finite time blow u for critical wave equations in high dimensions, J. Functional Analysis, Vol. 231, no2, [24] i S. Zhang; The critical exonent of a reaction diffusion equation on some lie grous, Math. Z. 228 (1998, [25] i S. Zhang; Blow-u results for nonlinear arabolic equations on manifolds, Duke Math. J. 97 (1999, [26] i S. Zhang; A blow u result for a nonlinear wave equation with daming: the critical case, C. R. Acad. Sci. Paris, Vol. 333, No. 2, , [27] C. Zuily; Existence globale de solutions régulières our l équation des ondes non linéaires amortie sur le groue de Heisenberg, (French [Global existence of regular solutions for the damed nonlinear wave equation on the Heisenberg grou], Indiana Uni. Math. J. (62 2 (1993. Bashir Ahmad Nonlinear Analysis and Alied Mathematics (NAAM Research Grou, Deartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia address: bashirahmad qau@yahoo.com Ahmed Alsaedi Nonlinear Analysis and Alied Mathematics (NAAM Research Grou, Deartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia address: aalsaedi@hotmail.com Mokhtar Kirane Laboratoire de Mathématiques, Image et Alications, Université de La Rochelle, Avenue M. Créeau, La Rochelle Cedex, France. Nonlinear Analysis and Alied Mathematics (NAAM Research Grou, Deartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia address: mokhtar.kirane@univ-lr.fr