Electronic Journal of Differential Equations, Vol. 17 (17), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS FOR SYSTEMS OF SEMILINEAR HYPERBOLIC EQUATIONS WITH POSITIVE INITIAL ENERGY AKBAR B. ALIEV, GUNAY I. YUSIFOVA Communicated by Mokhtar Kirane Abstract. In this paper we study the Cauchy problem for a system of semilinear hyperbolic equations. We prove a theorem on the nonexistence of global solutions with positive initial energy. 1. Introduction We study the solution of some Cauchy problems for systems containing nonlinear wave equations, from mathematical physics problems in [4, 8, 5, 31]. We consider the system of nonlinear Klein-Gordon equations with initial conditions u ktt u k + u k + γu kt = f k (u 1,..., u m ) k = 1,,..., m, (1.1) u k (, x) = u k (x), u kt (, x) = u k1 (x), x R n, k = 1,..., m, (1.) where f k (u 1,..., u k ) = u 1 ρ 1k u ρ k... u m ρ mk u k, ρ jk = p j + 1, ρ kk = p k 1, k, j = 1,,..., m, (u 1, u,..., u m ) are real functions depending on t R + and x R n, p 1, p,..., p m are real numbers. System (1.1) describes the model of interaction of various fields with single masses [8]. The goal of this paper is to investigate nonexistence of global solutions of problem (1.1), (1.). Before going further, we briefly introduce some results for the wave equation u tt u = f(u), (1.3) with f(u) ( + ε)f (u), (1.4) where F (u) = u f(s)ds. The general nonlinearity f(u) satisfying (1.4) was firstly considered for some abstract wave equations in [1], where Levine proved the blowup result when the initial energy is negative. But most results concerning the Cauchy problem of the wave equation were established for the typical form of nonlinearity as f(u) = u p 1 u where 1 < p < n+ n+1 as n 3 and 1 < p < as n = 1,. Here we note that the above power satisfies the condition (1.4). For the 1 Mathematics Subject Classification. 35G5, 35J5, 35Q51. Key words and phrases. Semilinear hyperbolic equations; nonexistence of global solutions; Cauchy problem; blow up. c 17 Texas State University. Submitted August 17, 17. Published September 11, 17. 1
A. B. ALIEV, G. I. YUSIFOVA EJDE-17/11 nonlinearity satisfying (1.4), the wave equations with damping term were studied by many authors [5, 6, 7, 9, 1, 11, 1, 13, 14, 15,,, 3, 3]. For existence and non-existence of global solutions for the Cauchy problem of equation (1.3) with a damping term, we refer the reader to [7, 14, 6, 7]. In particular, recently the wave equation with damping term was considered in [14], where Levine and Todorova showed that for arbitrarily positive initial energy there are choices of initial data such that the local solution blows up in finite time. Subsequently, Todorova and Vitillaro [6] established more precise result regarding the existence of initial values such that the corresponding solution blows up in finite time for arbitrarily high initial energy. More recently, Gazzola and Squassina [7] established sufficient conditions of initial data with arbitrarily positive initial energy such that the corresponding solution blows up in finite time for the wave equation with linear damping and in the mass free case on a bounded Lipschitz subset of R n. A fairly comprehensive picture of the studies in this direction can be gained from the monograph []. In [15], [18] the authors obtained sufficient conditions on initial functions for which the initial boundary value problem for second-order quasilinear strongly damped wave equations blow up in a finite time. The nonexistence of global solutions of a generalized fourth-order Klein-Gordon equation with positive initial energy was analyzed in [11]. A mixed problem for systems of two semilinear wave equations with viscosity and with memory was studied in [1, 1, 4, 9], where the nonexistence of global solutions with positive initial energy was proved. The nonexistence of global solutions of the problem u 1tt u 1 + u 1 + γu 1t = g 1 (u 1, u ), u tt u + u + γu t = g (u 1, u ), (1.5) with u i (, x) = u i (x), u it (, x) = u i1 (x), x R n, i = 1,, (1.6) where g 1 (u 1, u ) = u 1 p 1 u p+1 u 1, g (u 1, u ) = u 1 p+1 u p 1 u, with negative initial energy was studied in [1], [7]. In the case when g 1 (u 1, u ) = u 1 p 1 u q+1 u 1, g (u 1, u ) = u 1 p+1 u q 1 u. The absence of global solutions for problem (1.5), (1.6) was investigated in [1, ]. Recently, more investigations were carried out in this field [1, 7, 4, 9]. The absence of global solutions with positive arbitrary initial energy for systems of semilinear hyperbolic equations u itt u i + u i + γu it = i, i j u j pj u i pi u i i = 1,,..., m was investigated in [], where n, p j, j = 1,,..., m, and k=1 p k n 3. n if
EJDE-17/11 NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS 3. Formulation of the problem and main results To state our main results, we briefly mention some facts, notation, and well known results. We denote the norm on the space L (R n ) by, the inner product on L (R n ) by,, and the norm on the Sobolev space H 1 = W 1 (R n ) by u = [ u + u ] 1. The constants C and c used throughout this paper are positive generic constants that may be different in various occurrences. Assume that p j >, j = 1,,..., m, m =, 3,... ; (.1) p k + m if n 3. (.) n k=1 Let E(t) be the energy functional p j + 1 [ E(t) = u jt(t, ) + u j (t, ) + γ m u j (t, x) pj+1 dx. We also set I(u 1,..., u m ) = R n p j + 1 r=1 p r + m u j(t, ) R n ] u jt(s, ) ds The main result of this article is stated in the following theorem. m u j (t, x) pj+1 dx. (.3) Theorem.1. Let conditions (.1) and (.) be satisfied. Assume u k H 1 and u k1 L (R n ), k = 1,,..., m, and E() >, (.4) I(u 1, u,..., u m ) <, (.5) u k, u k1, (.6) k=1 p j + 1 u j > p j + m p E(). (.7) j Then the solution of the Cauchy problem (1.1), (1.) blows up in finite time. Note that, in the case of m =, this result was obtained in [1], and in the case m =, p 1 = p 1, it was obtained in [9]. 3. Auxiliary assertions In the Hilbert space H = L (R n ) L (R n ) L (R n ) we write problem (1.1), (1.) as the Cauchy problem w + Bw + Aw = F (w), (3.1) w() = w, w () = w 1, (3.)
4 A. B. ALIEV, G. I. YUSIFOVA EJDE-17/11 where u 1 u 1 (x) u 11 (x) w = u..., w = u (x)..., w 1 = u 1 (x)... u m u m (x) u m1 (x) Here A and B are linear operators in H defined by + 1... A = + 1.................. + 1 D(A) = H = H H H, γ... γ... B =............... γ,, D(B) = L (R n ) L (R n ) L (R n ), f 1 (u 1, u,..., u m ) F (w) = f (u 1, u,..., u m ).... f m (u 1, u,..., u m ) Note that A is a self-adjoint positive definite operator, B is a linear bounded operator acting in H and conditions (.1), (.) imply that F (w) is a nonlinear operator acting from H 1 = H 1 H 1 H 1 to H. Lemma 3.1. Let n = 1,, p j 1, j = 1,,..., n, m =, 3,... or n = 3, m =, p 1 = p = 1. Then the nonlinear operator w F (w) : H1 H satisfies the local Lipchitz condition, that is for any w 1, w H 1 we have F (w 1 ) F (w ) H c(r) w 1 w H1, (3.3) where c( ) C(R + ), c(r), r = i=1 wi H1. Proof. Let us take w j = (u j 1, uj,..., uj m) H 1, j = 1,. Then, by the mean value theorem we have F (w 1 ) F (w ) H c ( u 1 j (ρjk 1) + u j (ρjk 1) ) k=1 R n (3.4) m ( u 1 j ρ jk + u j ρ jk ) u 1 j u j dx. i=1,i j Let n. By Holder inequality with exponents, αk i r=1 = p r + m if i j, i = 1,..., m, ρ ki α j k = r=1 p r + m, αk = ρ kj 1 p r + m r=1
EJDE-17/11 NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS 5 and using interpolation inequalities of Gagliardo and Nirenberg in the case n = or Sobolev inequality in case n = 3 we have ( F (w 1 ) F (w ) H e c w 1 P m r=1 pr+m 1 + w P m r=1 ) w pr+m 1 1 w eh eh1. (3.5) 1 eh 1 In case n = 1, from (3.4) using embedding theorem we again obtain (3.1). By the theorem of solvability of the Cauchy problem for the evolution equation [3], we have the following local solvability theorem for problem (.3), (.4). Theorem 3.. Let n = 1,, p j 1, j = 1,,..., m, m =, 3,... or n = 3, m =, p 1 = p = 1. Then for arbitrary w H 1, w 1 H, there exists T > such that problem (3.1), (3.) has a unique solution w( ) C([, T ]; H 1 ) C 1 ([, T ]; H). If T max = sup T, i.e., T max is the length of the maximal existence interval of the solution w( ) C([, T max ); H 1 ) C 1 ([, T max ); H), then either (i) T max = +, or (ii) lim sup t Tmax [ w( ) H1 + ẇ( ) H] = +. Theorem 3.3. Let conditions (.1) and (.) be satisfied. Then for arbitrary w H 1 and w 1 H there exists T > such that problem (3.1), (3.) has a solution w( ) C([, T ]; H 1 ) C 1 ([, T ]; H) and w(t) is either global or blow-up in a finite time. Proof. We carry out the proof by Galerkin s method, using some considerations from the work [18]. Let {w 1, w,..., w r... } be the basis of the space H 1 and w r (t, ) = r g rj(t)w j, r = 1,,... be defined as a solution of the system (w r (t), ω j ) eh + (Bw r(t), w j ) eh + (w r (t), ω j ) eh1 = (F (w r ), ω j ) eh (3.6) with initial data w r (, ) = w r, w r(, ) = w 1r, (3.7) where w r and w 1r belongs to the subspace [ω 1, ω,..., ω r ] generated by the r first vectors of the basis {ω j }, and w r w in H 1 and w 1r w 1 in H if r. (3.8) By multiplying the equation (3.6) by g rj (t) and summing by k, where k takes the values from 1 to r, we get that 1 d dt [ w rt(t, ) H e + w r (t, ) H1 e ] + (Bw rt (t, ), w rt (t, )) eh (3.9) = (F (w(t, )), w (t, )) eh. Then using Holder s inequalities and (3.5), for y r (t) = w rt (t, ) e H + w r (t, ) e H1 (3.1) from (3.9) we get y r(t) c(y r (t)) P m r=1 pr+m. Integrating this inequality and taking into account the inequality (3.4), we find that there exists T > and r such that y r (t) c, t [, T ], r r. (3.11) From (3.1), (3.11) it follows that there exists a subsequence still denoted by the same symbols, such that w r w weak star in L (, T ; H 1 ),
6 A. B. ALIEV, G. I. YUSIFOVA EJDE-17/11 w r w weak star in L (, T ; H), F (w r ) χ weak star in L (, T ; H). Further, using the method given in [16], we obtain that χ = F (w). Passing to the limit is carried out by the standard method (for example, see [[16, 18]). Thus, problem (3.1), (3.) has the solution w L (, T ; H 1 ), such that w L (, T ; H) and F (w) L (, T ; H). Further applying the linear theory of the hyperbolic equations, considering equation (3.1) as a linear equation with a given right-hand side of χ(t) = F (w) L (, T ; H), we find that w C([, T ] H 1 ) C 1 ([, T ] H) (see [17]). Remark 3.4. labelrmk3.1 If w H and w 1 H 1, then w( ) C([, T max ); H ) C 1 ([, T max ); H 1 ). Lemma 3.5. Let conditions (.1), (.) and (.4)-(.7) be satisfied. Then I(u 1 (t,.), u (t,.),..., u m (t,.)) <, t [, T max ). Proof. By (.5) there exists T 1 >, such that I(u 1 (t, ), u (t, ),..., u m (t, )) <, t [, T 1 ). (3.1) We shall prove that T 1 = T max. Assume that T 1 < T max. Then by the continuity of I(u 1 (t, ), u (t, ),..., u m (t, )) we have I(u 1 (T 1, ), u (T 1, ),..., u m (T 1, )) =. (3.13) We introduce the functional F (t) = (p j + 1) u j (t, ). Taking into account Remark.1 and using (1.1), (1.) we obtain: F (t) = (p j + 1) u j (t, ), u j (t, ), and F (t) = (p j + 1) u j(t, ) (p j + 1) [ u j (t, ) + γ u j (t, ), u j (t, ) ] Therefore, where ϕ(t) = ( m ) + p k + m R n k=1 m u j (t, x) pj+1 dx. F (t) + γ F (t) = ϕ(t), t [, T 1 ), (3.14) ( m ) (p j + 1) u j(t, ) p k + m I(u 1 (t, ), u (t, ),..., u m (t, )). k=1 Taking into account inequality (3.1), we obtain ϕ(t) >, t [, T 1 ). (3.15) It follows from condition (.6) and relations (3.14) and (3.15) that F (t) >, t [, T 1 ). Therefore, the function F (t) is monotone increasing on [, T 1 ). Consequently, F (t) > F () = (p j + 1) u j. (3.16)
EJDE-17/11 NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS 7 By taking into account the continuity of the function F (t), from condition (.7) and inequalities (3.16), we obtain F (T 1 ) > [ p j + m ] p E(). (3.17) j 3 On the other hand it follows from (1.1) and (1.) that E(t) = E() (3.18) for every t [, T max ). From (3.13) and (3.18) we obtain the inequality It follows that ( 1 ) m p j + m p j + 1 u j (T 1, ) E(). F (T 1 ) [ p j + m ] p E(). (3.19) j + m The resulting contradiction (3.17) with (3.19) shows that our assumption fails. Therefore T 1 = T max. Let T >, T 3 > and k > be some numbers. Consider the functional R(t) = p j + 1 [ u j (t, ) + γ + k(t 3 + t). ] u j (s, ) ds + γ u j (T t) Lemma 3.6. Let (.4) (.7) be satisfied. Then R(t) > for t [, T max ). (3.) Proof. A simple computation gives us R p j + 1[ (t) = uj (t, ), u j(t, ) + γ u j (t, ) γ u j ] + k(t + T 3 ). (3.1) Next, from (3.18), (3.1) by using relations (1.1) and (1.), we obtain R (t) = (p j + 1)[ u j(t, ) u j (t, ) ] m ] + p j + m R n It follows from (.3) and (3.) that m u j (t, x) pj+1 dx + k. m ] R (t) p j + m I(u 1 (t,.),..., u 3 (t,.)) + k, t [, T max ). By Lemma 3.5, for sufficiently small k it holds (3.) R (t) >, t [, T max ). (3.3)
8 A. B. ALIEV, G. I. YUSIFOVA EJDE-17/11 4. Proof of main result We first assume that u i H, u i1 H 1, i = 1,,..., m. We shall prove that under conditions (.1), (.) and (.4)-(.7), T max < +. Suppose the contrary: T max = +. It follows from (1.1) and (1.) that m u j (t, x) pj+1 dx R n = E() + p j + 1 [ u j(t, ) + u j (t, ) + γ Taking into account this relation in (3.), we obtain u j(s, ) ds]. R (t) = p j + m + + p j + m 3 (p j + 1) (p j + 1) u j(t, ) m ] (p j + 1) u j (t, ) + γ p j + m m ] u j(s, ) ds p j + m E() + k. (4.1) By (3.9) we have m R (t) (p j + 1)( u j (t, ) + γ m (p j + 1)( u j(t, ) + γ u j (s, ) ds) + k(t + T 3 ) ] ] u j(s, ) ds) + k. (4.) By choosing a sufficiently large T 3, from Lemma 3.5 and relations (3.19), (4.1), and (4.), we obtain R(t)R (t) p j + m + (R (t)) 4 m R(t) R (t) p j + m + [ R(t) (T 1 t) (p j + 1) u j ] 4 m (p j + 1)( u j(t, ) + γ u j(s, ) ds) + k] { R(t) p j + m (p j + 1) u j(t, ) + p j + m (p j + 1) u j (t, ) + [ p j + m] (p j + 1) m ] u j(s, ) ds p j + m E() + k
EJDE-17/11 NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS 9 where p j + m + ( ) } (p j + 1) u j(t, ) + u j(s, ) ds + k = R(t)y(t) + p j + m u j(s, ) ds, (4.3) y(t) = p j + m (p j + 1) u j (t, ) [ p j + m ] E() p 1 + p + p 3 + 1 k. Having in mind Lemma 3.5, and choosing a sufficiently small k >, we obtain that y(t). Thus, for sufficiently large T >, T 3 >, and for sufficiently small k > we ahve R(t) R (t) p j + m + R (t). (4.4) 4 On the other hand, R () = (p j + 1) u j, u j1 + kt. Therefore, R () >. Using this inequality and (4.4) bya standard procedure, we obtain that there exists < T < + such that lim t T R(t) = +. We obtain a contradiction, which shows that T max < +. If u i H 1 and u i1 L (R n ), i = 1,,..., m, then the justification can be carried out in a standard way, by approximation of the initial data by functions from H and H 1, respectively. Acknowledgements. The authors want to thank the anonymous referees for the careful reading of the paper and his comments for improvements. References [1] A. B. Aliev, A. A. Kazimov; The existence and nonexistence of global solutions of the Cauchy problem for Klein Gordon systems, Doklady Math., 9, no. 3 (14), 1-3. [] A. B. Aliev, A. A. Kazimov; Nonexistence of global solutions of the Cauchy Problem for systems of Klein-Gordon equations with positive initial energy, Differential Equations, 51, no. 1 (15), 1563-1568. [3] A. B. Aliev; Solvability in the large of the Cauchy problem for quasilinear equations of hyperbolic type, Dokl. Akad. Nauk SSSR, 4 no. (1978), 49-5. [4] S. S. Antman; The equations for large vibrations of strings, Am. Math. Mon., 87 (198), 359-37. [5] B. A. Bilgin, V. K. Kalantarov; Blow up of solutions to the initial boundary value problem for quasilinear strongly dampedwave equations, J. Math. Anal. Appl., 43 (13), 89-94. [6] T. Cazenave; Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 6 (1985), 36-55. [7] F. Gazzola, M. Squassina; Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincare, Anal. Non Lincaire, 3 (6), 185-7. [8] S.Cocco, M. Barbu, M. Peyard; Vector nonlinear Klein-Gordon pattices: General derivation os small amplitude envelope solution solutions, Physics Letters A., 53, (1999), 161-167. [9] V. K. Kalantarov, O. A. Ladyzhenskaya; The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1, no. 1 (1978), 53-7.
1 A. B. ALIEV, G. I. YUSIFOVA EJDE-17/11 [1] M. Kirane, N. Tatar; Non-existence results for a semilinear hyperbolic problem with boundary condition of memory type, Journal for Analysis and its Applications, 19, no. () 453-468. [11] M. O. Korpusov; Non-existence of global solutions to generalized dissipative Klein Gordon equations with positive energy, Electron. J. Diferential Equations, 1, no. 119 (1), 1-1. [1] H. A. Levine; Instability and nonexistence of global solutions to nonlinear wave equations of the form P u tt = Au + F (u), Trans. Amer. Math. Soc., 19 (1974), 1-1. [13] H. A. Levine; Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J.Math. Anal., 5 (1974), 138-146. [14] H. A. Levine, G. Todorova; Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Am. Math. Soc., 19 (1), 793-85. [15] H. A. Levine, J. Serrin; Global nonexistence theorems for quasilinear evolution equation with dissipation, Arch. Ration Mech. Anal., 137 (1997), 341-361. [16] J. L. Lions; Quelques methodes de resolution de problemes aux limites non lineares, Dunod, Paris, (1969). [17] J. L. Lions, E. Majenes; Problemes aux limites non homogeneous applications, Vol. 1, Dunod, Paris, (1969). [18] M. M. Miranda, L. A. Medioros; On the Existence of global solutions of coupleed nonlinear Kleyn-Gordon equations, Funkcialaj Ekvacioi, 3 (1987), 147-161. [19] Ma J., Mu C., R. Zeng; A Blow-up result for viscoelastic equations with arbitrary positive initial energy, Boundary Value Problems, 6 (11), 1-1. [] S. A. Messaoudi; Blow up in a nonlinearly damped wave equation, Math. Nachr., 31, (1), 15-111. [1] S. A. Messaoudi, B. Said-Houari; Global nonexistence of positive initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365, no. 1 (1), 77-87. [] E. Mitidieri, S. I. Pokhozhaev; A priori estimates and the absence of solutions to nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 34 (1), 1-384. [3] P. Pucci, J. Serrin; Global nonexistence for abstract evolution equation with positive initial energy, J. Diferential Equations, 15 (1998), 3-14. [4] B. Said-Houari; Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Diferential and Integral Equations, 3, no. 1- (1), 79-9. [5] A. N. Tichonov, A. A. Samarskii; Partial differential equations of mathematical physics, Vol. 1, Published by Holden Day, (1964). [6] G. Todorova, E. Vitillaro; Blow-up for nonlinear dissipative wave equations in R n, J. Math. Anal. Appl., 33 (5), 4-57. [7] E. Vitillaro; Global existence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-18. [8] Y. Wang; Non-existence of global solutions of a class of coupled non-linear Klein-Gordon with non-negative potential and arbitrary initial energy, IMA J. Appl. Math., 74 (9), 39-415. [9] W. Liu; Global Existence, asymptotic behavior and blow-up of solutions for Coupled Klein Gordon equations with damping terms, Nonlinear Anal., 73, (1), 44-55. [3] Y. Ye; Global existence and nonexistence of solutions for coupled nonlinear wave equations with damping and source terms, Bull. Korean Math. Soc., 51, no. 6 (14), 1697-171. [31] E. Zauderer; Partial differential equations of applied mathematics, nd edn. Pure and Applied Mathematics. A Wiley-Intersci. Publ. Wiley, New York, 1989. [3] Y. Zhou; Global existence and nonexistence for a nonlinear wave equation with damping and source terms. Math. Nachr., 78 (5), 1341-1358. (Akbar B. Aliev) Azerbaijan Technical University, Baku, Azerbaijan. Institute of Mathematics and Mechanics, NAS of Azerbaijan, Baku, Azerbaijan E-mail address: alievakbar@gmail.com (Gunay I. Yusifova) Ganja State University, Ganja, Azerbaijan E-mail address: yusifova81@inbox.ru