A FUJITA-TYPE THEOREM FOR A MULTITIME EVOLUTIONARY p-laplace INEQUALITY IN THE HEISENBERG GROUP
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1 Electronic Journal of Differential Equations, Vol (2016), No. 303, pp ISSN: URL: or A FUJITA-TYPE TEOREM FOR A MULTITIME EVOLUTIONARY p-laplace INEQUALITY IN TE EISENBERG GROUP MOAMED JLELI, MOKTAR KIRANE, BESSEM SAMET Abstract. A nonexistence result of global nontrivial positive weak solutions to a multitime evolutionary p-laplace differential inequality in the eisenberg group is obtained. Our technique of proof is based on the test function method. 1. Introduction The standard time-dependent partial differential equations of mathematical physics involve evolution in one-dimensional time. Space can be multidimensional, but time stayed one dimensional. The term multitime was introduced in physics by Dirac, Fock and Podolsky, in 1932, considering multi-temporal wave-functions via m-time evolution equation. It was used in mathematics by Friedman and Littman (1962, 1963). Multitime evolution equations arise for example in Brownian motion (diffusion process with inertia) [2], transport theory (Fokker-Planck-type equations) [21], biology (age-structured population dynamics) [10], wave and Maxwell s equations [5, 9], mechanics, physics and cosmology [20, 25]. Some interesting multitime developments of classical, single-time theories and principles from different fields of mathematical research, appeared in the last years (see [1, 4, 8, 12, 15, 24, 26] and references therein). The study of nonexistence of global solutions to multitime evolutionary problems has begun recently (see [11, 13]). This paper deals with the nonexistence of global (nontrivial) positive solutions to a multitime evolutionary p-laplace differential inequality in the eisenberg group. More precisely, we consider the multitime evolutionary p-laplace problem i=k ( u ti div u p 2 u ) u q, in, u 0, a.e. in, u ti =0 = u i, in E, (1.1) where is the (2N + 1)-dimensional eisenberg group, k is a positive integer (k 1), p, q > 1, = (0, ) k, E = (0, ) k 1, and u i L 1 loc (E), i = 2010 Mathematics Subject Classification. 47J35, 35R03. Key words and phrases. Nonexistence; global solution; multitime; differential inequality; eisenberg group. c 2016 Texas State University. Submitted June 12, Published November 25,
2 2 M. JLELI, M. KIRANE, B. SAMET EJDE-2016/303 1, 2..., k. Using a duality argument [16, 17], we provide a sufficient condition for the nonexistence of global nontrivial positive weak solutions to the above problem. 2. Preliminaries For the reader s convenience, we recall some background facts used here. The (2N + 1)-dimensional eisenberg group is the space R 2N+1 equipped with the group operation ϑ ϑ = (x + x, y + y, τ + τ + 2(x y x y)), for all ϑ = (x, y, τ), ϑ = (x, y, τ ) R N R N R, where denotes the standard scalar product in R N. This group operation endows with the structure of a Lie group. The distance from un element ϑ = (x, y, τ) to the origin is given by ϑ = ( τ 2 + ( N ) 2 ) 1/4, x 2 i + yi 2 where x = (x 1,..., x N ) and y = (y 1,..., y N ). The Gradient over is where for i = 1,..., N, Let = (X 1,..., X N, Y 1,..., Y N ), X i = xi + 2y i τ and Y i = yi 2x i τ. A = ( ) IN 0 2y, 0 I N 2x where I N is the identity matrix of size N N, then = A R 2N+1. A simple computation gives the expression u 2 = 4( x 2 + y 2 )( τ u) 2 + The divergence operator in is N ( ) ( xi u) 2 + ( yi u) τ u(y i xi u x i yi u). div (u) = div R 2N+1(Au). For more details on eisenberg groups and partial differential equations in eisenberg groups, we refer to [3, 7, 14, 22, 23] and references therein. In the proof of our main result, the following inequality will be used several times. Lemma 2.1 (ε-young inequality). Let a, b, ε > 0. Then ab εa p + c ε b p, where p > 1, p is its corresponding conjugate exponent, i.e., 1 p + 1 p c ε = ( 1 εp) p /p 1 p. = 1; and
3 EJDE-2016/303 A FUJITA-TYPE TEOREM FOR A MULTITIME 3 In this paper, we use the notation: 3. Main result d = dt 1... dt k dϑ, de 1 = dt 2... dt k dϑ, de i = dt 1... dt i 1 dt i+1... dt k dϑ (i 1). Moreover, for a given function ϕ : R, we denote ϕ i = ϕ ti=0, i = 1, 2,..., k. For r > 1, we denote by r its corresponding conjugate exponent. Now, let us define the class of solutions under consideration. Definition 3.1. Let u W 1,p loc (; R +) L q loc (; R +) and u i L 1 loc (E; R +), i = 1, 2,..., k, with p = p p 1. We say that u is a global weak solution to problem (1.1) if the following conditions are satisfied: (i) u p 2 u L p loc (; R2N+1 ); (ii) For any ϕ W 1,p loc (; R +) with compact support, u q ϕ d u p 2 u ϕ d uϕ ti d E u i ϕ i de i. (3.1) Observe that all the integrals in (3.1) are well defined. Our main result is given in the following theorem. Theorem 3.2. Let p > 1. If max{1, p 1} < q pk + Q(p 1) Q + (k 1)p, (3.2) where Q = 2N + 2 is the homogeneous dimension of, then (1.1) has no global nontrivial weak solutions. To prove Theorem 3.2, we need the following lemma, which provides a preliminary estimate of possible solutions. Lemma 3.3. Let p > 1, q > max{1, p 1} and α (δ, 0), where δ = max { 1, 1 p, }. Let u be a global weak solution to (1.1). Then for any ϕ W 1, (; R + ) with a compact support, we have u q+α ϕ d + u p u α 1 ϕ d + u α+1 i ϕ i de i E ( ( ϕ ti r ) 1 ) C d + ϕ 1 ps ϕ ps d, ϕ (3.3) for some constant C > 0, where r = q+α 1+α, s = q+α p+α 1 and s is the conjugate exponent of s.
4 4 M. JLELI, M. KIRANE, B. SAMET EJDE-2016/303 Proof. Let ε > 0 be fixed and α (δ, 0). Suppose that u is a global weak solution to (1.1). Let Define ϕ ε as u ε (ϑ, t 1,..., t k ) = u(ϑ, t 1,..., t k ) + ε, (ϑ, t 1,..., t k ). ϕ ε (ϑ, t 1,..., t k ) = u α ε (ϑ, t 1,..., t k )ϕ(ϑ, t 1,..., t k ), where ϕ W 1, (; R + ) has a compact support. Observe that ϕ ε belongs to the set of admissible test functions in the sense of Definition 3.1. By (3.1), we have u q u α ε ϕ d + α u p u α 1 ε ϕ d + 1 (u i + ε) α+1 ϕ i de i E u p 1 u α ε ϕ d + 1 u α+1 ε ϕ ti d. (3.4) Now, using Lemma 2.1, we will estimate the individual terms on the right-hand side of (3.4). For some ε 1 > 0, Lemma 2.1 with parameters r = q+α 1+α and r = q+α q 1 yields ( u α+1 ε ϕ ti d ε 1 u q+α ϕti r ) 1 ε ϕ d + c ε1 d, ϕ from which follows 1 u α+1 ε ϕ ti d kε 1 u q+α ε ϕ d + c (3.5) ( ε 1 ϕti r ) 1 d. ϕ For some ε 2 > 0, applying Lemma 2.1 with parameters p and p = p p 1, we obtain u p 1 u α ε ϕ d (3.6) ε 2 u p u α 1 ε ϕ d + c ε2 u p+α 1 ε ϕ p ϕ 1 p d. Again, for some ε 3 > 0, Lemma 2.1 with parameters s = q+α p+α 1 and s = q+α q p+1, yields u p+α 1 ε ϕ p ϕ 1 p d ε 3 u q+α ε ϕ d + c ε3 ϕ 1 ps ϕ ps d. (3.7) Combining (3.6) with (3.7), we obtain u p 1 u α ε ϕ d ε 2 u p u α 1 ε ϕ d + c ε2 ε 3 + c ε2 c ε3 ϕ 1 ps ϕ ps d. u ε q+α ϕ d (3.8)
5 EJDE-2016/303 A FUJITA-TYPE TEOREM FOR A MULTITIME 5 Furthermore, substituting estimates (3.5) and (3.8) into (3.4), we obtain u q u α ε ϕ d + ( α ε 2 ) u p u α 1 ε ϕ d + 1 (u i + ε) α+1 ϕ i de i E ( kε 1 + c ) ε 2 ε 3 u q+α ε ϕ d + c ε 1 ( ϕ ti r ) 1 d ϕ + c ε2 c ε3 ϕ 1 ps ϕ ps d. Passing to the limit inferior as ε 0 in the above inequality, and applying the Fatou and Lebesgue theorems, we obtain ( kε 1 1 c ) ε 2 ε 3 u q+α ϕ d + ( α ε 2 ) u p u α 1 ϕ d + 1 u α+1 i ϕ i de i c ε 1 E ( ϕti r ϕ ) 1 d + c ε2 c ε3 For ε 1, ε 2, ε 3 sufficiently small, we obtain u q+α ϕ d + u p u α 1 ϕ d + ( C ϕ 1 ps ϕ ps d. E u i α+1 ϕ i de i ( ϕti r ) 1 ) d + ϕ 1 ps ϕ ps d, ϕ for some constant C > 0, which is the desired result. Now we are ready to prove our main result given by Theorem 3.2. Proof of Theorem 3.2. Suppose that u is a nontrivial global weak solution to (1.1). Let us consider the test function ϕ R (ϑ, t) = ϕ R (x, y, τ, t) = φ ω( t 2θ t 2θ1 k + x 4θ2 + y 4θ2 + τ ) 2θ2, R > 0, ω 1, R 4θ2 where φ C0 (R + ) is a decreasing function satisfying { 1 if 0 z 1 φ(z) = 0 if z 2, and θ j, j = 1, 2, are positive parameters, whose exact values will be specified later. Let ρ = t2θ t 2θ1 k + x 4θ2 + y 4θ2 + τ 2θ2. R 4θ2 Clearly ϕ R has support in Ω R = {(ϑ, t) : 0 ρ 2},
6 6 M. JLELI, M. KIRANE, B. SAMET EJDE-2016/303 while (ϕ R ) ti (i = 1, 2,..., k) and ϕ R have support in A simple computation yields Θ R = {(ϑ, t) : 1 ρ 2}. ti ϕ R (tϑ) = 2θ 1 ωt 2θ1 1 i R 4θ2 φ ω 1 (ρ)φ (ρ), i = 1, 2,..., k while ( ϕ R (t, ϑ) 2 = 16θ2ω 2 2 R 8θ2 (φ (ρ)) 2 φ 2ω 2 (ρ) ( x 2 + y 2 )τ 4θ2 2 + ( x 8θ2 2 + y 8θ2 2 ) + 2τ 2θ2 1 N Then, for all (t, ϑ) Ω R and i = 1, 2,..., k, we have ) x i y i ( x 4θ2 2 y 4θ2 2 ). R ϕ R + R 2θ2/θ1 ti ϕ R C φ (ρ) φ ω 1 (ρ). (3.9) For simplicity, in the sequel, we will write ϕ instead of ϕ R. Let us consider now the change of variables where (t 1,..., t k, x, y, τ) = (t, ϑ) ( t 1,..., t k, x, ỹ, τ) = ( t, ϑ), t = R 2θ2/θ1 t, x = R 1 x, ỹ = R 1 y, τ = R 2 τ, d = d t dϑ. In the same way, let ρ = t 1 2θ t k 2θ 1 + x 4θ 2 + ỹ 4θ2 + τ 2θ2, Ω = {( x, ỹ, τ, t) : 0 ρ 2}, Θ = {( x, ỹ, τ, t) : 1 ρ 2}. Using the above change of variables and (3.9), we obtain ( ϕti r ) 1 d CR Q+2 θ 2 θ1 (k r ) φ ω r φ r d, (3.10) ϕ ϕ 1 ps ϕ ps d CR Q+2 θ 2 θ1 k ps φ ω ps φ ps d. (3.11) Setting we have Q + 2 θ 2 θ 1 (k θ 2 = ps (r 1), θ 1 2r r r 1 ) = Q + 2θ 2 θ 1 k ps = Q p(q + α) p(q 1)k + q p + 1 q p + 1. (3.12) Using (3.3), (3.10) (3.12), we obtain u q+α p(q+α) Q ϕ d CR q p+1 + p(q 1)k q p+1. (3.13) Furthermore, noting that for Q p(q + α) p(q 1)k + q p + 1 q p + 1 < 0 q < pk + Q(p 1) Q + (k 1)p
7 EJDE-2016/303 A FUJITA-TYPE TEOREM FOR A MULTITIME 7 and some α (δ, 0) sufficiently small. Under the above condition, letting R in (3.13) and using the monotone convergence theorem, we obtain u q+α d 0, which contradicts our assumption about u. Finally, the limit case pk + Q(p 1) q = Q + (k 1)p can be treated by the same way as in [18]. We now consider some examples where we can apply Theorem 3.2. Applying Theorem 3.2 with p = 2 and k = 1, we obtain the following eisenberg version of Fujita exponent [19]. Corollary 3.4. If 1 < q Q, then the problem u t u u q in, u 0, a.e. in, u t=0 = u 0 0, in, where = (0, ) and u 0 L 1 loc (), has no nontrivial global weak solution. Next, applying Theorem 3.2 with p = 2 and k = 2, we obtain the following result, which is an extension of [11, Theorem 2.1] in the case α = 2, m = 1, s = l = r = 0, to the eisenberg group. Corollary 3.5. If 1 < q Q+2, then the problem u t1 + u t2 u u q in, u 0, a.e. in, u ti =0 = u i 0, in E, where = (0, ) 2, E = (0, ), and u i L 1 loc (E), i = 1, 2, has no nontrivial global weak solution. Acknowledgements. Bessem Samet extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia). References [1] G. A. Abdikalikova; Multiperiodic solution of one boundary-value problem for a parabolic equation with multidimensional time, Abstracts of the Internat. Sci. Conf. of Students, Magistrates, and Young Scientists Lomonosov-2012, Kazakhstan Branch of the Lomonosov Moscow State University, Astana (2012), Part I, pp [2] D. R. Akhmetov, M. M. Lavrentiev Jr.; R. Spigler; Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations, Electron. J. Differential Equations, 24 (2002), [3] L. D Ambrosio; Critical degenerate inequalities on the eisenberg group, Manuscripta Math., 106 (4) (2001), [4] G. A. Anastassiou, G. R. Goldestein, J. A. Goldstein; Uniqueness for evolution in multidimensional time, Nonlinear Anal., 64 (1) (2006), [5] J. C. Baez, I. E. Segal, W. F. Zohn; The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Anal., 93 (1990), [6] A. Benrabah, F. Rebbani, N. Boussetila; A study of the multitime evolution equation with time-nonlocal conditions, Balkan J. Geom. Appl., 16 2 (2011),
8 8 M. JLELI, M. KIRANE, B. SAMET EJDE-2016/303 [7] I. Brindelli, I. Capuzzo Dolchetta, A. Cutri; Liouville theorems for semilinear equations on the eisenberg group, Ann. Inst.. Poincaé, 14 (3) (1997), [8] W. Craig, S. Weinstein; On determinism and well-posedness in multiple time dimensions, Proc. R. Soc. A., 465 (8) (2009), [9] P. illion; The Goursat problem for Maxwell s equations, J. Math. Phys., 31 (1990), [10] M. Iannelli; Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori, Pisa, [11] M. Jleli, M. Kirane, B. Samet; Nonexistence of global solutions for a class of two-time nonlinear evolution equations, Comput. Math. Appl., 68 (2014), [12] K. K. Kenzhebaev, G. A. Abdikalikova, A. B. Berzhanov; Multiperiodic solution of a boundary-value problem for one class of parabolic equations with multidimensional time, Ukrainian Math. J., 66.5 (2014), [13] S. Kerbal, M. Kirane; Nonexistence results for the Cauchy problem for nonlinear ultraparabolic equations, Abstr. Appl. Anal., Vol 2011 (2011), Article ID , 10 pages. [14] E. Lanconelli, F. Uguzzoni; Asymptotic behaviour and non existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the eisenberg group, Boll. Un. Mat. Ital., 1 (1) (1998), [15] L. Matei, C. Udrişte; Multitime sine-gordon solitons via geometric characteristics, Balkan J. Geom. Appl., 16 (2) (2011), [16] E. Mitidieri, S. I. Pohozaev; Nonexistence of positive solutions for quasilinear elliptic problems on R N, Proc. Steklov Inst. Math., 227 (1999), [17] E. Mitidieri, S.I. Pohozaev; A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), [18] E. Mitidieri, S. I. Pohozaev; Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan Journal of Mathematics, 72(1) (2004), [19] S. I. Pohozaev, L. Véron; Nonexistence results of solutions of semilinear differential inequalities on the eisenberg group, manuscripta math, 102 (2000), [20] A.D. Rendall; The Characteristic Initial Value Problem of the Einstein Equations, Pitman Res. Notes Math. Ser., 253, [21]. Risken; The Fokker-Planck equation, methods of solution and applications, vol. 18 of Springer, Series in Synergetics, Springer, Berlin, Germany, [22] C. Romero; Potential theory for the Kohn Laplacian on the eisenberg group. Diss. University of Minnesota, [23] S. Thangavelu; armonic analysis on the eisenberg group. Vol Springer Science & Business Media, [24] C. Udrişte; Minimal submanifolds and harmonic maps through multitime maximum principle, Balkan J. Geom. Appl., 18 (2) (2013), [25] J. Uglum; Quantum cosmology of open R S 2 S1, Physical Review. D, (3) 46 (1992), [26] F. Zouyed, F. Rebbani, N. Boussetila; On a class of multitime evolution equations with nonlocal initial conditions, Abstr. Appl. Anal., Volume 2007 (2007), Article ID 16938, 26 pages. Mohamed Jleli Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia address: jleli@ksu.edu.sa Mokhtar Kirane Laboratoire LaSIE, Pôle Sciences et Technologies, Université de La Rochelle, Avenue M. Crépeau, La Rochelle, France address: mkirane@univ-lr.fr Bessem Samet Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia address: bsamet@ksu.edu.sa
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