Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 303, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu 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, 2016. Published November 25, 2016. 1
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) 2 + 4 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
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 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)
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 2θ1 1 + + 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 = t2θ1 1 + + 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 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 1 + + 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
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 1 + 2 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 1 + 2 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. 7 9. [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), 1 17. [3] L. D Ambrosio; Critical degenerate inequalities on the eisenberg group, Manuscripta Math., 106 (4) (2001), 519 536. [4] G. A. Anastassiou, G. R. Goldestein, J. A. Goldstein; Uniqueness for evolution in multidimensional time, Nonlinear Anal., 64 (1) (2006), 33 41. [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), 239 269. [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), 13 24.
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