Research Article Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

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1 International Differential Equations Volume 24, Article ID , 5 pages Research Article Global an Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Bounary Conitions Ülkü Dinlemez an Esra AktaG 2 Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, Turkey 2 Incirli Mahallesi, Karaelmas Sokak, Yunusemre Caesi 5/8, İncirli, Ankara, Turkey Corresponence shoul be aresse to Ülkü Dinlemez; ulku@gazi.eu.tr Receive 24 December 23; Revise 7 March 24; Accepte 2 March 24; Publishe 3 April 24 Acaemic Eitor: D. D. Ganji Copyright 24 Ü. Dinlemez an E. Aktaş. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. We consier an initial-bounary value problem to a nonlinear string equations with linear amping term. It is prove that uner suitable conitions the solution is global in time an the solution with a negative initial energy blows up in finite time.. Introuction We stuy the ampe nonlinear string equation with source term u α u: u tt +u t =(σ( u x 2 )u x ) x + u α u, (x, t) (, ) [, T], where <α, σ(s) is a smooth function for swith the initial conitions u (x, ) =u (x), u t (x, ) =u (x), x [, ], (2) an bounary conitions u (, t) =, t (, T), σ( u x (, t) 2 u x (, t)) u t (, t) =2φ(t), t [, T], σ( u x (, t) 2 u x (, t))+u t (, t) =2ψ(t), t [, T]. The problem () (3) can be regare as moelling a nonlinear string with vertical isplacement function u(x, t) in R. An this problem has nonlinear mechanical amping of the form u α u.therightenofthestringmakesitsteay.theinput φ(t) function an the output ψ(t) function are applie on the left. Wu an Li [] stuie the motion for a nonlinear beam moel with nonlinear amping a φ t m φ t an external () (3) forcing b φ p φ terms. They showe that this moel has a unique global solution an blow-up solution uner the same conitions. Levine et al. [2] an Levine an Serrin [3] stuie abstract version. Georgiev an Toorova [4]stuie nonlinear wave equations involving the nonlinear amping term u t m u t an source term of type u t p u t.theyprove global existence theorem with large initial ata for <p m. Hao an Li [5] stuie the global solutions for a nonlinear string with bounary input an output. Dinlemez [6]prove the global existence an uniqueness of weak solutions for the initial-bounary value problem for a nonlinear wave equation with strong structural amping an nonlinear source terms in R. Alotofpapersinconnectionwithblowup, global solutions an existence of weak solutions were stuiein[7 5]. In this paper we first fin energy equation for the problem () (3). Then we prove the solutions of the problem () (3) are global in time uner some conitions on the function σ(s), inputφ(t), anheoutputψ(t). Finallyweestablisha blow-upresultforsolutionswithanegativeinitialenergy.our approach is similar to the one in [5]. 2. Main Results Now we give the following lemma for energy equation for the problem () (3).

2 2 International Differential Equations Lemma. Let <αan u(x, t) be a solution of the problem () (3).Thentheenergyequationoftheproblem() (3) is E (t) = 2 u t 2 2 α+2 u α+2 α σ (ξ) ξ x, E (t) =φ2 (t) ψ 2 (t) u t 2 2. (5) Proof. Multiplying ()withu t an integrating over (, ),then we get { 2 u t 2 2 α+2 u α+2 α+2 } (4) (6) = (σ ( u x 2 )u x ) x u t x u t 2 2. Applying integration by parts in the right han sie of (6), we fin (σ ( u x 2 )u x ) x u t x = σ ( u x (, t) 2 )u x (, t) u t (, t) u 2 x 2 σ (ξ) ξ x. (7) Anusingbounaryconitionsinequality (7), we obtain { 2 u t 2 2 α+2 u α+2 α =φ 2 (t) ψ 2 (t) u t 2 2. Hence the proof is complete. σ (ξ) ξ x} Next we give the following theorem for global solutions in time. Theorem 2. Assume that u(x, t) is a solution of the problem () (3) with <αan (i) σ(s) satisfies the following conition: (8) s α σ(s), for s R + {}, (9) (ii) the input an the output functions satisfy Then the solution u(x, t) is global in time. Proof. Let G (t) := E (t) + 2 α+2 u α+2 α+2 = 2 u t φ 2 (t) ψ 2 (t). () σ (ξ) ξ x + α+2 u α+2 α+2. () Differentiating G(t) with respect to t an using (5), we get G (t) =φ2 (t) ψ 2 (t) u t u α uu t x. (2) Using the Cauchy-Schwarz inequality in the last term of (2), we obtain 2 u α uu t x 2 u α+ u t x (3) u 2(α+) 2(α+) + u t 2 2, an it follows from (2), (3), an ()thatwehave G (t) u 2(α+) 2(α+) + u t 2 2. (4) By assumption (9) an integrating over (, ) an (, ), respectively, we yiel α+ u x 2(α+) Furthermore, we have u (x, t) 2(α+) = an then 2(α+) x 2(α+) u ξ (ξ, t) ξ u 2(α+) ξ (ξ, t) x ξ σ (ξ) ξ x. (5) u x (x, t) 2(α+) x = u x (x, t) 2(α+) 2(α+), (6) u (x, t) 2(α+) 2(α+) u x (x, t) 2(α+) 2(α+). (7) Combining (), (4), (5), an (7), we get G (t) G (t), (8) ξ where ξ = min{/2, /(α + )}. Using Gronwall s inequality, we have G (t) G() e (/ξ )t. (9) Therefore together with the continuation principle an the efinition of G(t) we complete the proof of Theorem 2. Then we give the following theorem for the blow-up solutions of the problem () (3). Theorem 3. Let u(x, t) be a solution of the problem () (3) with <α. Assume that (i) there exists < ε < (α + 2)/2 such that the function σ(s) satisfies σ (s) s ε s 2 σ (ζ) ζ for sεr + {}, (2)

3 International Differential Equations 3 (ii) the initial values satisfy E (), < u (x) u (x) x, (2) (iii) the input an output functions satisfy φ 2 (t) ψ 2 (t), t (ψ (t) +φ(t))( (ψ (s) φ(s))s+u ()), (iv) u(x, t) satisfies u. (22) Then the solution u(x, t) blows up in finite time T max,an T max ( α+4 αη )N α/(α+4) (), (23) where η is some positive constant inepenent of the initial value α an N(t) are given by (25). Proof. We efine M (t) := E (t), γ := α 2 (α+2), (24) N (t) := M γ (t) + u (x, t) u t (x, t) x. (25) By virtue of (5), (2), (22), an (24), we get M (t) = u t 2 2 +ψ2 (t) φ 2 (t), (26) M() M(t), for t. (27) From the efinition of M(t) we yiel =εm(t) + ε 2 u t 2 + ε u 2 x 2 σ (ξ) ξ x (3) ε α+2 u α+2 α+2. Combining (29)an(3)in(28), we get N (t) =( γ)m γ (t) ( u t 2 2 +ψ2 (t) φ 2 (t))+ u t u α+2 α+2 uu t x (ψ (t) +φ(t)) t ( (ψ (s) φ(s))s+u ()) σ( u x 2 )u 2 x x + εm (t) + ε 2 u t ε u 2 x 2 σ (ξ) ξ x ε α+2 u α+2 α+2. (3) Using (22)in(3), we obtain ( + ε 2 ) u t 2 2 +( 2 ε α+2 ) u α+2 α u α+2 α+2 + ( ε u 2 x 2 σ (ξ) ξ σ ( u x 2 )u 2 x )x (32) Taking a erivative of (25)anusing(26), we have N (t) +εm(t) uu t x Thanks to Young s inequality, N (t). =( γ)m γ (t) M (t) + u 2 t x + uu tt x (28) AB δp p Ap + δ q q Bq, A,B <δ, p + q =, (33) =( γ)m γ (t) ( u t 2 2 +ψ2 (t) φ 2 (t))+ u t uu tt x. Multiplying () by u an integrating over the interval [, ] anhenusingbounaryconitions(3), we obtain uu tt x = u α+2 α+2 uu t x (ψ (t) +φ(t)) t ( (ψ (s) φ(s))s+u ()) σ( u x 2 )u 2 x x. (29) for uu tx with p=q=2an γ=2,anhenweget uu t x uu t x u t u 2. (34) From embeing for L p (, ) an using (iv), we have u 2 2 an putting (34)in(32)wehave u α+2 α+2 ( + ε 2 ) u t 2 2 +( 2 ε α+2 ) u α+2 α u ( ε u 2 x 2 σ (ξ) ξ σ ( u x 2 )u 2 x )x+εm(t) u t u 2 2 N (t). (35)

4 4 International Differential Equations From (2), we get εm (t) + ε 2 u t ( 2 ε α+2 ) u α+2 α u 2 2 N (t). (36) Choosing ε an κ=min{ε/2, (/2 ε/(α+2)), /4},weobtain κ {M (t) + u t u α+2 α+2 + N (t) u 2 2 }. (37) Thanks to (2)an(27), we yiel <N() N(t), < t. (38) Now we estimate [N(t)] /( γ). From Holer s inequality, uu t x u 2 u t2 u α+2 u t2 ; (39) then using Young s inequality again we get uu t x δ2( γ) 2( γ) u t 2( γ) 2 + 2γ 2( γ) δ 2( γ)/( 2γ) u 2( γ)/( 2γ) α+2, (4) where <δan /p + /q = with p=2( γ).ansowe have /( γ) uu t x δ 2 2 /( γ) ( (2 ( γ)) /( γ) u t 2 2 +( 2γ /( γ) 2( γ) ) δ 2/( 2γ) u 2/( 2γ) α+2 ). (4) Choosing β=max{δ 2 /( γ) /( γ), (( 2γ)/( γ)) /( γ) δ 2/( 2γ) },weobtain Therefore we yiel (N(t)) /( γ) /( γ) uu t x β( u t u α+2 α+2 ). (42) =(M γ /( γ) (t) + u (x, t) u t (x, t) x) 2 /( γ) (M (t) + /( γ) u(x, t)u t (x, t)x ) C(M(t) + u t u α+2 α+2 + u 2 2 ), (43) where C epens on δ an α.from(37)an(43), we have η(n (t)) /( γ) N (t), (44) where η=κ/c.integrating(44)over(, t),thenweget (N ()) α/(α+4) (α/ (α+4)) ηt (N(t))α/(α+4). (45) Hence N(t) blows up in finite time T max. T max is given by the inequality as below: T max α+4 αη (N ()) α/(α+4). (46) Consequently the solution blows up in finite time. An the proof of Theorem 3 is now finishe. Conflict of Interests The authors eclare that there is no conflict of interests regaring the publication of this paper. Acknowlegment The authors woul like to thank the referees for the careful reaingofthispaperanforthevaluablesuggestionsto improve the presentation an style of the paper. References [] J.-Q. Wu an S.-J. Li, Global solution an blow-up solution for a nonlinear ampe beam with source term, Applie Mathematics,vol.25,no.4,pp ,2. [2] H. A. Levine, P. Pucci, an J. Serrin, Some remarks on global nonexistence for nonautonomous abstract evolution equations, Contemporary Mathematics, vol. 28, pp , 997. [3] H. A. Levine an J. Serrin, Global nonexistence theorems for quasilinear evolution equations with issipation, Archive for Rational Mechanics an Analysis, vol.37,no.4,pp.34 36, 997. [4] V. Georgiev an G. Toorova, Existence of a solution of the wave equation with nonlinear amping an source terms, Differential Equations, vol.9,no.2,pp , 994. [5] J. Hao an S. Li, Global solutions an blow-up solutions for a nonlinear string with bounary input an output, Nonlinear Analysis:Theory,MethosanApplications,vol.66,no.,pp. 3 37, 27. [6] Ü. Dinlemez, Global existence, uniqueness of weak solutions an etermining functionals for nonlinear wave equations, Avances in Pure Mathematics,vol.3,pp ,23. [7]Y.GuoanM.A.Rammaha, Globalexistenceanecay of energy to systems of wave equations with amping an supercritical sources, Zeitschrift für Angewane Mathematik un Physik,vol.64,no.3,pp ,23. [8] L.Bociu,M.Rammaha,anD.Tounykov, Onawaveequation with supercritical interior an bounary sources an amping terms, Mathematische Nachrichten, vol. 284, no. 6, pp , 2.

5 International Differential Equations 5 [9]C.O.Alves,M.M.Cavalcanti,V.N.DomingosCavalcanti, M. A. Rammaha, an D. Tounykov, On existence, uniform ecay rates an blow up for solutions of systems of nonlinear wave equations with amping an source terms, Discrete an Continuous Dynamical Systems,vol.2,no.3,pp ,29. [] M. M. Cavalcanti, V. N. Domingos Cavalcanti, an I. Lasiecka, Well-poseness an optimal ecay rates for the wave equation with nonlinear bounary amping-source interaction, Journal of Differential Equations,vol.236,no.2,pp ,27. [] C. O. Alves an M. M. Cavalcanti, On existence, uniform ecay rates an blow up for solutions of the 2-D wave equation with exponential source, Calculus of Variations an Partial Differential Equations,vol.34,no.3,pp.377 4,29. [2] M. A. Rammaha, The influence of amping an source terms on solutions of nonlinear wave equations, Boletim a Socieae Paranaense e Matemática,vol.25,no.-2,pp.77 9,27. [3]V.Barbu,I.Lasiecka,anM.A.Rammaha, Existencean uniqueness of solutions to wave equations with nonlinear egenerate amping an source terms, Control an Cybernetics,vol.34,no.3,pp ,25. [4] M. M. Cavalcanti an V. N. Domingos Cavalcanti, Existence an asymptotic stability for evolution problems on manifols with amping an source terms, Mathematical Analysis an Applications,vol.29,no.,pp.9 27,24. [5]M.M.Cavalcanti,V.N.D.Cavalcanti,J.S.PratesFilho,an J. A. Soriano, Existence an uniform ecay of solutions of a parabolic-hyperbolic equation with nonlinear bounary amping an bounary source term, Communications in Analysis an Geometry,vol.,no.3,pp ,22.

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