NONEXISTENCE OF SOLUTIONS TO CAUCHY PROBLEMS FOR FRACTIONAL TIME SEMI-LINEAR PSEUDO-HYPERBOLIC SYSTEMS

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1 Electronic Journal of Differential Equations, Vol (2016), No. 20, pp ISSN: URL: or ftp ejde.math.txstate.edu NONEXISENCE OF SOLUIONS O CAUCHY PROBLEMS FOR FRACIONAL IME SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS SALEM ABDELMALEK, MAHA BAJNEED, KHALED SIOUD Abstract. We study Cauchy problems time fractional semi-linear pseudohyperbolic equations and systems. Using the method of nonlinear capacity, we show that there are no solutions for certain nonlinearities and initial data. Our work complements the work by Aliev and col. [1, 6, 7]. 1. Introduction In this article, we study Cauchy problems for time fractional pseudo-hyperbolic equations and systems. We start by considering the time fractional equation u tt + η( ) k u tt + ( ) l u + ξ( ) r D0 t α u + γdβ 0 tu = f(u), (1.1) for x, t > 0, supplemented with the initial data and with u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x, (1.2) η, ξ, γ 0 for r, k N {0}, l N, 0 < β α 1, (1.3) is the Laplacian and D0 t α is the left-sided Riemann-Liouville fractional derivative of order α. he aim of this paper is to show, using the method of nonlinear capacity proposed by Pokhozhaev in 1997 [18] and developed successfully and jointly with Mitidieri [15, 16, 17], that under certain conditions, there are no solutions to (1.1)-(1.2). For the non fractional case α = β = 1, Lions monograph [13] considered equation (1.1) in the case where η = 0 and f(u) = u p u. A step forward was achieved by [10, 15, 20] where they considered the absence of global solutions for the case where η = ξ = 0 and f(u) = u p 1 u or f(u) = ± u p. Kato [10] showed that for l = 1, ξ = 0, and 1 < p < N, problem (1.1)-(1.2) admits no global solution under a certain condition on the initial data. A further study of John [9] considered the case l = 1, ξ = 0, η = 0, γ = 0 and f(u) = u p for u close to zero. his was generalized to l N, ξ = η = 0, γ > 0, α = 1, by Zhang [20] and Kirane and 2010 Mathematics Subject Classification. 80A23, 65N21, 26A33, 45J05, 34K37, 42A16. Key words and phrases. Semi-linear pseudo-hyperbolic equation; non-existence of solutions; non-linear capacity method. c 2016 exas State University. Submitted June 25, Published January 12,

2 2 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 afsaoui [12]. he two studies proved that the critical exponent for this case is in fact p = N. he existence of global solutions of problem (1.1)-(1.3) for the non-fractional case α = β = 1, η, ξ, γ 0, r, k N {0} and l N was achieved by Aliev and Kazymov [5]. Recently, by using the method of the test function Aliev and col. [1, 6, 7] established sufficient conditions for the nonexistence of global solutions of problem (1.1)-(1.3) for the non-fractional case α = β = 1: Aliev and Lichaei [6] considered the case α = β = 1 and η, ξ, γ > 0 for r = k N {0}, l N, and f(u) C u p. Aliev and Kazymov [1] examined the case 1 α = β = 1, k = 0, r = 0, l N and f(u) = (1+ x 2 ) u p. Aliev and Mamedov s [7] treated the non existence of global solutions of a semilinear hyperbolic equation with an anisotopic elliptic part (α = β = 1, k = r = 0), u tt + εu t + N ( 1) l k D 2l k x k u = f(u), f(u) c u p. k=1 Our work will complement the results of [6] for r, k N {0}, η, ξ, γ 0 and l N and extend it to the time-fractional case 0 < β α < 1, using the test function method. In the second part of this paper, we study the Cauchy problem for the timefractional pseudo-hyperbolic system u tt + η 1 ( ) k1 u tt + ( ) l1 u + ξ 1 ( ) r1 D α1 0 t u + γ 1D β1 0 tu = f(v) = v p v tt + η 2 ( ) k2 v tt + ( ) l2 v + ξ 2 ( ) r2 D α2 0 t v + γ 2D β2 0 t v = g(u) = (1.4) u q posed in := (0, ), subject to the initial conditions u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x, v(x, 0) = v 0 (x), v t (x, 0) = v 1 (x)x (1.5) with p, q > 1, r i, k i N {0}, l i N, η i, ξ i, γ i 0 and 0 < β i α i 1 for i = 1, 2. he non-existence of global solutions in the case of a non fractional system of two (or more) equations with α i = 0 or 1 and β i = 0 or 1, is investigated in numerous studies of Aliev and colleagues: Aliev, Mammadzada, and Lichaei [8] considered the case β i = 1, γ i = η i = 1, ξ i = 0, l 1 = 1, l 2 = 2, p = 7 2 and q = 5 2 ; Aliev and Kazymov [4] examined the case β i = 1, γ i = η i = 1, ξ i = 0, l i N, and f i (u, v) C i,1 u pi + C i,2 v qi ; Aliev and Kazymov [2] considered the case β i = 1, γ i = η i = 1, ξ i = 0, l i N, and f(v) C v p and g(u) C u q ; Aliev and Kazymov [3] dealt with a system of three equations that is similar to the case presented in [4]. Our work will complement these papers for the system of two equations in the cases γ i, η i, ξ i > 0, r i, k i N {0}, l i N and extend it to the time-fractional case, using again the test function method. 2. Preliminaries For the convenience of the reader, we start by recalling some basic definitions and properties which will be useful throughout this paper.

3 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 3 Definition 2.1. he left- and right-sided Riemann-Liouville integrals of order 0 < α < 1 for an integrable function are defined as ( I α 0 t f ) (t) := 1 Γ(α) ( I α t f ) (t) := 1 Γ(α) where Γ is the Euler gamma function. t 0 t (t s) α 1 f(s)ds, (2.1) (s t) α 1 f(s)ds, (2.2) Definition 2.2. Let AC[0, ] be the space of functions f which are absolutely continuous on [0, ]. he left and right-handed Riemann-Liouville fractional derivatives of order n 1 < γ < n for a function f AC n [0, ] := {f : [0, ] R, D n 1 f AC[0, ]}, n N is defined as (see [11]) where D is the usual time derivative. D γ 0 t f(t) := Dn (I n γ 0 t f)(t), t > 0, (2.3) D γ t f(t) := ( 1)n D n (I n γ t f)(t), (2.4) Furthermore, for every f, g C([0, ]) such that D0 t α f(t), Dα 0 tg(t) exist and are continuous for all t [0, ], 0 < α < 1, the formula of integration by parts can be given according to Love and Young [14] by 0 g(t)(d α 0 t f)(t)dt = In addition, [19, Lemma 2.2] provides us with the formula 0 f(t)(dt α g)(t)dt. (2.5) or D α t f(t) := 1 Γ(1 α) [ f( ) ( t) α Dt α f(t) := 1 d Γ(1 α) dt t t (t s) α f (s)ds] (2.6) (t s) α f(s)ds. (2.7) 3. Non-existence of global solutions of one equation In this section, we study the non-existence of global solutions for the timefractional semi-linear pseudo-hyperbolic equation (1.1) for certain initial data with f(u) = u p. Before we state our result. let us define the weak solution of problem (1.1)-(1.3). In this article, denotes the set := (0, ), 0 < +. We set f := f(x, t) dx dt, f := f(x, t) dx dt, 0 0 f := f(x, 0)dx.

4 4 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 Definition 3.1. he function u L p loc ( ) is a weak solution of problem (1.1)- (1.3) on with initial data u 0 (x), u 1 (x) L 1 loc (RN ) if it satisfies u p ϕ + u 1 (x)ϕ(x, 0) + η u 1 (x)( ) k ϕ(x, 0) R N = u 0 (x)ϕ t (x, 0) + η u 0 (x)( ) k ϕ t (x, 0) + uϕ tt + η u( ) k ϕ tt ξ u( ) r Dt α ϕ + γ ud β t ϕ + u( ) l ϕ, (3.1) for any test-function ϕ C 2d 2 x t ( ) with d = max{l, k, r} such that ϕ is positive, ϕ 0 outside a compact K R n, ϕ(x, ) = ϕ t (x, ) = 0 and D β t ϕ, Dα t ϕ C( ). As for the result on the non-existence of a global solution, the constants η, ξ and γ will not play a role, and thus will be taken equal to one. heorem 3.2. Assume that (1) r, k N {0}, l N and 0 < β α < 1; (2) u 0, u 1, L 1 ( ) such that u 0 (x)dx > 0, u 1 (x)dx > 0 2l (3) 1 < p 1 + N + 2l( 1 β 1) =: p c. hen problem (1.1) (1.3) does not admit any global in time nontrivial solution. Proof. he proof is by contraction. Let u be a global weak solution of problem (1.1)-(1.3) and ϕ be a non-negative function (satisfying the conditions of Definition 3.1) that will be specified later. Using ɛ-young s inequality ab ɛa p + c(ɛ)b ep, p > 1, a 0, b 0, p + p = p p, ɛ > 0, we can write uϕ tt ɛ u p ϕ + c(ɛ) ϕ tt ep ϕ ep/p, u( ) k ϕ tt ɛ u p ϕ + c(ɛ) ( ) k1 ϕ tt ep ϕ ep/p, u( ) r Dt α ϕ ɛ u p ϕ + c(ɛ) ( ) r Dt α ϕ ep ϕ ep/p, ud β t ϕ ɛ u p ϕ + c(ɛ) D β t ϕ ep ϕ ep/p, u( ) l ϕ ɛ u p ϕ + c(ɛ) ( ) l ϕ ep ϕ ep/p. (3.2)

5 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 5 Using inequalities (3.2) in (3.1), we obtain the inequality u p ϕ + u 1 (x)ϕ(x, 0) + u 1 (x)( ) k ϕ(x, 0) R N u 0 (x)ϕ t (x, 0) u 0 (x)( ) k ϕ(x, 0) R { N ɛ 1 u p ϕ + C 1 ϕ tt ep ϕ ep/p + ( ) k ϕ tt ep ϕ ep/p } + ( ) r Dt α ϕ ep ϕ ep p + D β t ϕ ep ϕ ep/p + ( ) l ϕ ep ϕ ep/p. Setting A 1 = ϕ tt ep ϕ ep/p, A 2 = ( ) k ϕ tt ep ϕ ep/p, A 3 = ( ) r Dt α ϕ ep ϕ ep/p, A 4 = D β t ϕ ep ϕ ep/p, A 5 = ( ) l ϕ ep ϕ ep/p, and taking ɛ = 1/2, inequality (3.3) becomes u p ϕ + u 1 (x)ϕ(x, 0) + u 1 (x)( ) k ϕ(x, 0) R N u 0 (x)ϕ t (x, 0) u 0 (x)( ) k ϕ t (x, 0) C{A 1 + A 2 + A 3 + A 4 + A 5 }. At this stage, we set (3.3) (3.4) ϕ(x, t) = Ψ ν( t 2 + x 4ρ ) R 4, R > 0, ν 1, ρ > 0, (3.5) where Ψ Cc (R + ) is a decreasing function defined as { 1 if r 1 Ψ(r) = 0 if r 2, with 0 Ψ 1 and r Ψ (r) < C. Note that with this choice of ϕ, we have leading to ϕ t (x, t) = 2νtR 4 Ψ ν 1 ((t 2 + x 4ρ )/R 4 )Ψ ((t 2 + x 4ρ )/R 4 ), We also assume that ϕ satisfies ϕ t (x, 0) = 0. (3.6) ϕ p/ep ( ϕ tt ep + ( ) k ϕ tt ep + ( ) r D α t ϕ ep + D β t ϕ ep + ( ) l ϕ ep ) <, for that, we will choose ν 1. herefore, the inequality (3.4) becomes u p ϕ + u 1 (x)ϕ(x, 0) + u 1 (x)( ) k ϕ(x, 0) C{A 1 + A 2 + A 3 + A 4 + A 5 }. (3.7)

6 6 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 Let us pass to the scaled variables y = R 1/ρ x, τ = R 2 t and the function ϕ given by ϕ(x, t) = ϕ(y, τ). In doing so, it follows that ϕ t = R 2 ϕ τ, ϕ tt = R 4 ϕ ττ, D β t ϕ = R 2β D β τ ϕ, ( ) m ϕ = R 2m ρ ϕ, where = R 2 and is a positive constant. Now, let us set = {(y, τ), 0 τ 2 + y 4ρ 2}. Using these definitions, A 1,..., A 5 can be rewritten as A 1 R 4ep+ N ρ +2 ϕ ττ ep ϕ ep/p, In a short form, we can write where A 2 R ( 2k ρ +4)ep+ N ρ +2 ( ) k ϕ ττ ep ϕ ep p, A 3 R ( 2r ρ +2α)ep+ N ρ +2 ( ) r Dτ α ϕ ep ϕ ep/p, A 4 R 2βep+ N ρ +2 D β τ ϕ ep ϕ ep p, A 5 R 2l ρ ep+ N ρ +2 ( ) l ϕ ep ϕ ep/p. A i = C i R θi for i = 1, 2,..., 5, θ 1 = 4 p + N ρ + 2, As β α < 1, we observe that θ 2 = ( 2k ρ + 4) p + N ρ + 2, θ 3 = ( 2r ρ + 2α) p + N ρ + 2, θ 4 = 2β p + N ρ + 2, θ 5 = 2l ρ p + N ρ + 2. θ 2 θ 1 θ 4 and θ 3 θ 4. (3.8) (3.9) For R 1, we have R θi R θ4 + R θ5 for i = 1, 2,..., 5 and then inequality (3.7) becomes u p ϕ + u 1 (x)ϕ(x, 0) + u 1 (x)( ) k ϕ(x, 0) K(R θ4 + R θ5 ), (3.10) where K is a positive constant. We have u 1 (x)( ) k ϕ(x, 0) = u 1 (x)( ) k ϕ(x, 0) R = ρ 2k ρ u 1 (R 1/ρ y)( k ) ϕ(y, 0), where R = {(x, 0); R 1/ρ x 2 1/(4ρ) R 1/ρ }, = {(y, 0); 1 y 2}. We assume that ϕ satisfies ( k ) ϕ(, 0) <,

7 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 7 for that, we will choose ν 1. It follows that u 1 (x)( ) k ϕ(x, 0) C ρ 2k ρ u 1 (R 1/ρ y) CR 2k ρ u 1 (x), R then, passing to the limit as R +, we have u 1 (x)( ) k ϕ(x, 0) 0. (3.11) Now, if θ 4 < 0 and θ 5 < 0, (i.e. max(θ 4, θ 5 ) < 0), that means which is equivalent to 2βρ p < p 1 (ρ) = 1 + N + 2ρ(1 β), 2l p < p 2 (ρ) = 1 + N + 2(ρ l) p < p(ρ) = min(p 1 (ρ), p 2 (ρ)). As p 1 is a decreasing function and p 2 is an increasing function, the maximum value of the function min(p 1, p 2 ) will be at the point ρ = l β, when the two functions p 1 and p 2 are equal herefore, for p 1 ( l β ) = p 2( l β ) = 1 + 2l N + 2l( 1 β 1) =: p c, (i.e. θ 4 = θ 5 ). 1 < p < p c, (3.12) we have R θ4 +R θ5 tends to zero when R and the inequalities (3.10) and (3.11) yield u p + u 1 (x) 0. As u 1 (x) > 0, we get a contradiction. his proves the theorem in the case (3.12). For the border case where p = p c which corresponds to θ 4 = θ 5 = 0 and ρ = l β, let,r = {(x, t), R 4 t 2 + x 4ρ 2R 4 }, if we use the Hölder inequality in the estimate of uϕ tt instead of the ɛ-young inequality, we obtain ( ) 1/p ( ) 1/ep uϕ tt = uϕ tt u p ϕ ϕ ep/p ϕ tt ep,r,r,r (A 1 ) 1/ep ( u p ϕ) 1/p,,R and similarly u( ) k ϕ tt = u( ) k ϕ tt (A 2 ) 1/ep u p ϕ,,r,r ( u( ) r Dt α ϕ = u( ) r Dt α ϕ (A 3) 1/ep u p ϕ,r,r ) 1/p,

8 8 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 hus, we obtain u p ϕ + ( ud β t ϕ = ud β t ϕ (A 4) 1/ep u p ϕ,r,r ( u( ) l ϕ = u( ) l ϕ (A 5 ) 1/ep u p ϕ,r,r As u q < +, we have u 1 (x) ( A 1/ep 1 + A 1/ep 2 + A 1/ep 3 + A 1 ep 4 + A1/ep 5 lim R + ( ) 1/p. C u p ϕ,r,r u q ϕ lim u q = 0. R +,R ) 1/p, ) 1/p. ) (,R ) 1/p u p ϕ Passing to the limit as R +, we find that u q + u 1 (x) = 0, which contradicts u 1 > 0. his prove the theorem in the case p = p c. 4. A pseudo-hyperbolic system his section is concerned with the fractional time pseudo-hyperbolic system (1.4))-(1.5). Definition 4.1. he couple of functions (u, v), u L q loc ( ) and v L p loc ( ) is a weak solution of (1.4)-(1.5) on with initial data u 0 (x), u 1 (x), v 0 (x) and v 1 (x) L 1 loc (RN ), if it satisfies v p ϕ + u 1 (x)ϕ(x, 0) + η 1 u 1 (x)( ) k1 ϕ(x, 0) R N = u 0 (x)ϕ t (x, 0) + η 1 u 0 (x)( ) k1 ϕ t (x, 0) + uϕ tt (4.1) + u( ) l1 ϕ + η 1 u( ) k1 ϕ tt ξ 1 u( ) r1 D α1 t ϕ + γ 1 ud β1 t ϕ, and u q ϕ + v 1 (x)ϕ(x, 0) + η 2 v 1 (x)( ) k2 ϕ(x, 0) R N = v 0 (x)ϕ t (x, 0) + η 2 v 0 (x)( ) k2 ϕ t (x, 0) + vϕ tt + v( ) l2 ϕ + η 2 v( ) k2 ϕ tt ξ 2 v( ) r1 D α2 t ϕ + γ 2 vd β2 t ϕ, (4.2) for any test-function ϕ C 2l2 x t( ), l = max{l 1, l 2 } being positive, ϕ 0 outside ϕ, Dβ1 t ϕ, Dα2 t ϕ, Dβ2 t ϕ a compact K R n, ϕ(x, ) = ϕ t (x, ) = 0 and D α1 t C( ).

9 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 9 heorem 4.2. Assume that (1) r i, k i N {0}, l i N and 0 < β i α i < 1, i = 1, 2; (2) u 0, u 1, v 0, v 1 L 1 ( ) such that u 0 (x) > 0, u 1 (x) > 0, v 0 (x) > 0 and v 1 (x) > 0; (3) p > 1, q > 1, ( pq min 1 + 2(pβ 2 + β 1 )ρ N + 2(1 β 1 )ρ, 1 + 2(qβ 1 + β 2 )ρ ) N + 2(1 β 2 )ρ where ρ = min( l1 β 1, l2 β 2 ). hen problem (1.4)-(1.5) does not admit any global non trivial solution. Proof. he proof is by contraction. Let (u, v) be a global weak solution of (1.4)- (1.5) and ϕ be a non-negative function (satisfying the conditions of Definition 4.1). Applying Hölder inequality to uϕ tt, we obtain ( ) 1/q ( uϕ tt u q ϕ ϕ eq q ϕtt eq) 1/eq (A1 ) 1/eq( ) 1/q u q ϕ and similarly where ( u( ) k1 ϕ tt (A 2 ) 1/eq u q ϕ ( u( ) r1 D α1 t ϕ (A 3) 1 eq u q ϕ ud β1 t ϕ (A 4) 1/eq( 1/q, u ϕ) q ( u( ) l1 ϕ (A 5 ) 1/eq u q ϕ ) 1/q, ) 1/q, ) 1/q, A 1 = ϕ tt eq ϕ eq q dx dt, A 2 = ( ) k1 ϕ tt eq ϕ eq/q dx dt, A 3 = ( ) r1 D α1 t ϕ eq ϕ eq/q dx dt, A 4 = D β1 t ϕ eq ϕ eq/q dx dt, A 5 = ( ) l1 ϕ eq ϕ eq/q dx dt. (4.3) Now, let ϕ be the test function defined by the expression (3.5). Using the previous estimates (4.3) and the properties (3.5) and (3.6) of the function ϕ in equation (4.1), we obtain the inequality v p ϕ + u 1 ϕ(x, 0) ( ) 1/q [ u q ϕ A 1/eq 1 + A 1/eq 2 + A 1/eq 3 + A 1/eq 4 + A 1/eq 5 ]. (4.4)

10 10 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 Similarly, for the equation (4.2), we have u p ϕ + v 1 ϕ(x, 0) ( ) (4.5) 1/q[B v q 1/eq ϕ 1 + B 1/eq 2 + B 1 eq 3 + B1/eq 4 + B 1 ] eq 5, where B 1 = ϕ tt eq ϕ eq q dx dt, B 2 = u( ) k2 ϕ tt eq ϕ eq q dx dt, B 3 = ( ) r2 D α2 t ϕ eq ϕ eq q dx dt, B 4 = D β2 t ϕ eq ϕ eq q dx dt, B 5 = ( ) l2 ϕ eq ϕ eq q dx dt. Now, we estimate A 1,..., A 5 and B 1,..., B 5 in the same way as in Section 3, we obtain inequalities similar to those given in (3.7) and (3.8) where A i = C i R θi, B i = D i R δi for i = 1, 2,..., 5, (4.6) θ 1 = 4 q + N ρ + 2, δ 1 = 4 p + N ρ + 2, θ 2 = ( 2k 1 ρ + 4) q + N ρ + 2, δ 2 = ( 2k 2 ρ + 4) p + N ρ + 2, N N ) q + ) p + θ 3 = ( 2r 1 ρ + 2α 1 ρ + 2, δ 3 = ( 2r 2 ρ + 2α 2 ρ + 2, (4.7) θ 4 = 2β 1 q + N ρ + 2, δ 4 = 2β 2 p + N ρ + 2, θ 5 = 2l 1 ρ q + N ρ + 2, δ 5 = 2l 2 ρ p + N ρ + 2. If we set ( ) 1/p I = v p ϕ and J = ( u q ϕ) 1/q, inequalities (4.4) and (4.5) become I P + u 1 ϕ(x, 0) J(C 1 R θ1/eq + C 2 R θ2/eq + C 3 R θ3/eq + C 4 R θ4/eq + C 5 R θ5/eq ), J q + v 1 ϕ(x, 0) I(D 1 R δ1 eq + D2 R δ2/eq + D 3 R δ3/eq + D 4 R δ4/eq + D 5 R δ5/eq ). We can observe that under the conditions on α i and β i, we have θ 2 θ 1 θ 4, θ 3 θ 4, δ 2 δ 1 δ 4, δ 3 δ 4. (4.8) (4.9)

11 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 11 Hence, for R 1, we have R θi R θ4 +R θ5 and R δi R δ4 +R δ5, and consequently, inequalities (4.8) and (4.9) can be rewritten as I p + u 1 ϕ(x, 0) CJ ( R θ4 eq + R θ ) 5/eq, (4.10) R N J P + v 1 ϕ(x, 0) DI(R δ4 δ 5eq eq + R ), (4.11) where C = 5 C i, and D = i=1 5 D i. Since u 1 ϕ(x, 0) 0 and v 1 ϕ(x, 0) 0, inequalities (4.10) and (4.11) yield i=1 I p CJ ( R θ4/eq + R θ5//eq), (4.12) J P DI ( R δ4/eq + R δ5/eq). (4.13) he constants C and D will be updated at each step of the calculation and will not play a role. his implies that I pq CI(R δ4/ep + R δ5/ep )(R θ4/eq + R θ 5 eq ) q, J pq CJ(R θ 4 eq + R θ 5eq )(R δ 4/ep + R δ5/ep ) p, leading to I pq 1 C(R δ4/ep + R δ5/ep )(R θ4/eq + R θ 5 eq ) q, J pq 1 C(R θ 4 eq + R θ 5eq )(R δ 4/ep + R δ5/ep ) p. (4.14) Now, let S 1 = 1 p max(δ 4, δ 5 ) + q q max(θ 4, θ 5 ), S 2 = 1 q max(θ 4, θ 5 ) + p p max(δ 4, δ 5 ). If S 1 < 0, and S 2 < 0, (4.15) we have (R δ4/ep +R δ5/ep )(R θ4/eq +R θ5/eq ) q 0 and (R θ 4 eq +R θ 5/eq )(R δ4/ep +R δ5/ep ) p 0 as R. Hence, by (4.14), both I and J vanish as R. his implies that J q = u q ϕ converges to u q ϕ = 0 and I p = v p ϕ converges to v p ϕ = 0. Consequently, u 0 and v 0. As S 1 < 0, then we have max(θ 4, θ 5 ) < 0 or max(δ 4, δ 5 ) < 0. Suppose that max(θ 4, θ 5 ) < 0 and let again R. By (4.10), we obtain u 1 = 0, which contradicts u 1 > 0. Now, we return to the condition (4.15) that lead to the contradiction. Inequalities S 1 < 0 and S 2 < 0 are equivalent to ( S 1 = 2 q min(β 1, l 1 ρ ) + min(β 2, l 2 ρ ) ) + ( N ρ + 2)pq 1 p < 0 ( S 2 = 2 p min(β 2, l 2 ρ ) + min(β 1, l ) 1 ρ ) + ( N ρ + 2)pq 1 < 0. q Let us take ρ = ρ = min( l1 β 1, l2 β 2 ). We have min ( β 1, l 1 ρ ) = β1 and min ( β 2, l 2 ) = β2. ρ (4.16)

12 12 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 he inequalities in (4.16) can now be written as which are equivalent to 1 < pq < min S 1 = 2(qβ 1 + β 2 )ρ + (N + 2ρ) pq 1 p S 2 = 2(pβ 2 + β 1 )ρ + (N + 2ρ) pq 1 p < 0 < 0, ( 1 + 2(pβ 2 + β 1 )ρ N + 2(1 β 1 )ρ, 1 + 2(qβ 1 + β 2 )ρ ). N + 2(1 β 2 )ρ Let us now consider the border case where ( pq = min 1 + 2(pβ 2 + β 1 )ρ N + 2(1 β 1 )ρ, 1 + 2(qβ 1 + β 2 )ρ ), N + 2(1 β 2 )ρ which corresponds to (4.17) S 1 = 0, S 2 0 or S 1 0, S 2 = 0. (4.18) Let us take the case S 1 = 0, S 2 0 (the second case: S 1 0, S 2 = 0 is similar). We have From (4.19) and (4.20), we have p qs 1 = q max(δ 4, δ 5 ) + q p max(θ 4, θ 5 ) = 0, (4.19) p qs 2 = p max(θ 4, θ 5 ) + p q max(δ 4, δ 5 ) 0. (4.20) p max(θ 4, θ 5 ) = q max(δ 4, δ 5 ) q and q q (pq 1) max(δ 4, δ 5 ) 0, which implies max(δ 4, δ 5 ) = max(δ 1, δ 2,..., δ 5 ) 0. Moreover, using Young s inequality (a+b) r 2 r 1 (a r +b r ) for r 1, the inequalities in (4.14) lead to I pq 1 2R 1/ep max(δ4,δ5) 2 q 1 R q eq max(θ4,θ5) = 2 q R S1 = 2 q, and similarly J pq 1 2 p R S2 = 2 p, for every (0, + ). Hence, I < + and J < + for every (0, + ), and thus u q < + and v q < +. Now, let,r = { (x, t), R 4 t 2 + x 4ρ 2R 4}. For the first inequality of (4.3), we obtain ( uϕ tt = uϕ tt,r u q ϕ,r (A 1 ) 1/eq(,R ) 1/q (,R ϕ eq q u q ϕ) 1/q C1 R θ1/eq. ϕtt eq) 1/eq Doing the same for the remaining inequalities of (4.3), we obtain a new estimate of (4.1), 0 < u 1 (x) R N v p ϕ + {u 1 (x) + u 0 (x)}ϕ(x, 0),R

13 EJDE-2016/20 SEMI-LINEAR PSEUDO-HYPERBOLIC SYSEMS 13 ( ) 1/q[C1 u q ϕ R θ1/eq + C 2 R θ2/eq + C 3 R θ3/eq + C 4 R θ4/eq + C 5 R θ5/eq],r ( ) 1/q C u q ϕ (R θ4/eq + R θ5/eq),r ( ) 1/q. C u q ϕ,r Let R. Since u q < +, the right-hand side of the above inequality approaches zero when R, while the left-hand side u 1 (x) is assumed to be positive, this is a contradiction. Similarly, the second case S 1 0, S 2 = 0 leads to a contradiction. Acknowledgements. he authors would like to thank aibah University for grant No 4113 which made this work possible. Special thanks also go to Prof. M. Kirane for his continuous support and guidance, which have helped improve the quality of this paper. References [1] A. B. Aliev, A. A. Kazymov; Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations, Inter. J. of Modern Nonlinear heory and Appl. 2013, 2, pp [2] A. B. Aliev, A. A. Kazymov; Existence non existence and asymptotic behavior of global solutions of the Cauchy problem for of semi-linear hyperbolic equations with damping terms, Nonlinear Analysis. 2012, 75, pp [3] A. B. Aliev, A. A. Kazymov; Global Solvability and Behavior of Solutions of the Cauchy Problem for a System of hree Semilinear Hyperbolic Equations with Dissipation, in: ransactions of National Academy of Sciences of Azerbaijan, Series of Physical-echnical and Mathematical Sciences, Vol. XXX, Mathematics, No 4, 2011, pp [4] A. B. Aliev, A. A. Kazymov; Global Solvability and Behavior of Solutions of the Cauchy Problem for a System of wo Semilinear Hyperbolic Equations with Dissipation, Differ. Equations, 2013, Vol. 49, No 4, pp [5] A. B. Aliev, A. A. Kazymov; Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations, Differential Equations 2009, Vol 45, pp [6] A. B. Aliev, B. H. Lichaei; Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear pseudo-hyperbolic equations, Nonlinear Analysis: heory, Methods & Applications, Vol 72, Issues 7 8, 1 April 2010, Pages [7] A. B. Aliev, F. V. Mamedov; Existence and nonexistence of global solutions of the Cauchy problem for semi-linear hyperbolic equations with dissipation and anisotropic elliptic part, Differ. Equations, 2010, Vol. 46, No 3, pp [8] A. B. Aliev, K. S. Mammadzada, B. H. Lichaei; Existence of a global weak solution of the Cauchy problem for systems of semilinear hyperbolic equations, in: ransactions of National Academy of Sciences of Azerbaijan, Series of Physical-echnical and Mathematical Sciences, Vol. XXX, Mathematics and Mechanics, No1, 2010, pp [9] F. John; Blow-up of solutions of nonlinear wave equations in three space dimensions, manuscripta mathematica, Vol. 28, Issue 1-3,(1979), pp [10]. Kato; Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl.Math.,1980, Vol. 33, pp [11] A. A. Kilbas, H. M. Srivastava, J. J. rujillo; heory and Applications of Fractional Differential Equations. Elsevier (2006). [12] M. Kirane, M. afsaoui; Fujita s exponent for a semilinear wave equation with linear damping, Advanced Nonlinear Studies 2 (2002), pp [13] J. L. Lions; uelques méthodes de résolution des problèmes aux limites non-linéaires, Paris, Dunod 1969.

14 14 S. ABDELMALEK, M. BAJNEED, K. SIOUD EJDE-2016/20 [14] E. R. Love, L. C. Young; In fractional integration by parts. Proc. London Math. Soc., Ser. 2, 44, (1938) pp [15] E. Mitidieri, S.I. Pokhozhaev; A Priori Estimates and the Absence of Solutions of Nonlinear Partial Differential Equations and Inequalities, r. Mat. Inst. Steklova, 2001, Vol. 234, pp [16] E. Mitidieri, S. I. Pohozaev; Liouville heorems for Some Classes of Nonlinear Nonlocal Problems, Proc. Steklov Math. Inst., Vol. 248 (2005), pp [17] E. Mitidieri, S. I. Pohozaev; owards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., Vol. 72 (2004), [18] S. I. Pokhozhaev; essentially nonlinear capacities induced by differential operators, Dokl Akad Nauk 1997, Vol. 357, No5, pp [19] S. G. Samko, A. A. Kilbas, O.I. Marichev; Fractional Integrals and Derivatives: heory and Applications. Gordon and Beach, Yverdon and New York (1993). [20]. S. Zhang; A blow-up result for a nonlinear wave equation with damping. he critical case, C.R. Acad. Sci. Paris. Sér. I Math., 2001, Vol 333, pp Salem Abdelmalek Department of Mathematics, College of Sciences, Yanbu, aibah University, Saudi Arabia address: sallllm@gmail.com Maha Bajneed Department of Mathematics, Faculty of Sciences at Yanbu, aibah University, Saudi Arabia address: m bajneed@hotmail.com Khaled Sioud Department of Mathematics, Faculty of Sciences at Yanbu, aibah University, Saudi Arabia address: sioudkha@aol.com