Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations

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1 Mathematics Statistics 6: 9-9, 04 DOI: 0.389/ms Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations Erhan Pişkin Dicle Uniersity, Department of Mathematics, 80 Diyarbakır, Turkey Corresponding Author: episkin@dicle.edu.tr Copyright c 04 Horizon Research Publishing All rights resered. Abstract In this work, we consider the initial boundary alue problem for the Kirchhoff-type equations with damping source terms u tt M m u dx m u u t p u t = f u,, tt M m dx m t q t = f u, in a bounded domain. We proe the blow up of the solution with positie initial energy by using the technique of [6] with a modification in the energy functional due to the different nature of problems. This improes earlier results in the literature [3, 9, 3, ]. Keywords Blow up, Higher-order Kirchhoff Type Equations, Nonlinear Damping Source Terms Mathematics Subject Classification 00: 35B44, 35G3 Introduction We consider the initial-boundary alue problem for the following coupled nonlinear higher-order Kirchhoff-type equations with damping source terms P u tt M P tt M u P u u t p u t = f u,, x, t 0, T, P t q t = f u,, x, t 0, T, u x, 0 = u 0 x, u t x, 0 = u x, x, x, 0 = 0 x, t x, 0 = x, x, i u ν i = i ν i = 0, i = 0,,..., m, x, where P = m, m is a natural number, p, q are real numbers, is a bounded domain with smooth boundary in R n, ν is the outer normal, M s = β β s γ, s, γ 0, β, β > 0. Without loss of generality, we can assume.

2 0 Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations that β = β = in the problem.. In [6], Kirchhoff firstly proposed a model gien by the equation ρh u t δ u t g { u = ρ t 0 Eh L L 0 u dx} u f u,. x x for f = g = 0, 0 < x < L, t 0, where u x, t is the lateral displacement, E is the Young modulus, ρ is the mass density, h is the cross-section area, L is the length, ρ 0 is the initial axial tension, δ is the resistance modulus, f g are the external forces. Moreoer,. is called a degenerate equation when ρ 0 = 0 nondegenerate one when ρ 0 > 0. The problem. is a generalization of a model introduced by Kirchhoff. The single higher-order Kirchhoff-type equation of the problem. u tt M m u dx m u u t p u t = u r u.3 hae studied local existence blow up of the solution [3]. In case of M s = m =, the equation.3 becomes a nonlinear wae equation u tt u u t p u t = u r u..4 Many authors hae been established the local existence, blow up asymptotic behaior, see [4, 7, 8,, 5, ]. The interaction between damping u t p u t the source term u r u makes the problem more interasting. Leine [7, 8] first studied the interaction between the linear damping p = source term by using Concaity method. But this method can t be applied in the case of a nonlinear damping term. Georgie Todoroa [4] extended Leine s result to the nonlinear case p >. They showed that solutions with negatie initial energy blow up in finite time. Later, Vitillaro in [6] extended these results to situations where the nonlinear damping the solution has positie initial energy. When M s = m =, the equation.3 becomes a Petrosky equation which has been extensiely studied seeral results concerning existence, blow up asymptotic behaior hae been established [, 5]. In case M s =, m, Ye [7] obtained the global existence asymptotic behaior of solutions for the equation.3. Also, Zhou et. al. [9] extended the results of [7]. Ono [6] considered equation.3 with M s = s γ, m = showed that the solution blow up if the initial energy is negatie. Wu Tsai [3] showed that the solution blow up under the condition of positie upper bounded initial energy, for m = in.3. When M s = s γ, m equation.3 becomes the higher-order Kirchhoff-type equation which has been discussed by many authors [9, 3, 0, 8]. Recently, Agre Rammaha [] studied the existence blow up of the solution for the problem. with M s = m =, by using the same techniques as in [4]. After that, Houari [5] showed the global existence decay of the solution for the problem. Li et. al. [0] showed the global existence, blow up decay of the solution for the problem. for M s = m =. In [7, 8], it was shown the global existence, decay blow up of solutions for the problem. with M s =, m p = q =. Later, Pişkin Polat [9] showed the global existence, decay of solutions for the problem. with M s = m. Very recently, Pişkin Polat [] studied the decay of the solution blow up the solution with the negatie initial energy of the problem.. Motiated by the aboe researches, in this work, we analyze the influence of the damping terms source terms on the solutions of the problem.. In fact, when both nonlinear damping source terms are present, then the analysis of their interaction is more difficult []. Blow up of the solution with positie initial energy was proed for r > max {γ, p, q } by using the technique of [6] with a modification in the energy functional. This work is organized as follows: In the next section, we present some lemmas, the local existence theorem. In section 3, we show the blow up properties of solutions.

3 Mathematics Statistics 6: 9-9, 04 Preliminaries In this section, we shall gie some assumptions lemmas which will be used throughout this work. Let.. p denote the usual L norm L p norm, respectiely. Concerning the functions f u, f u,, we take f u, = a u r u b u r u r, where a, b > 0 are constants r satisfies f u, = a u r u b u r r, According to the aboe equalities one can easily erify that < r if n m,. < r 3m n n m if n > m. u f u, f u, = r F u,, u, R,. where We hae the following result. F u, = [ a u r b u r]..3 r Lemma [4]. There exist two positie constants c 0 c such that c 0 u r r r F u, c u r r..4 We define the energy function as follows E t = u t t γ P u P P γ P u γ F u, dx..5 The next lemma shows that our energy functional 3.3 is a nonincreasing function along the solution of.. Lemma E t is a nonincreasing function for t 0 E t = u t p p t q q 0..6 Proof. Multiplying the first equation of. by u t the second equation by t, integrating oer, using integrating by parts summing up the product results, we get E t E 0 = t 0 u τ p p τ q q dτ for t 0. Lemma 3 Sobole-Poincare inequality []. If p n [n m] u p C m u p < if n = m, then for u H0 m holds with some constant C, where we put [a] = max {0, a}, [a] = if [a] = 0.

4 Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations Lemma 4 []. Suppose that p n n, n 3 holds. Then there exists a positie constant C > depending on only such that u s p C u u p p for any u H 0, s p. Next, we state the local existence theorem that can be established by combining arguments of [, 3, 6, 4]. Theorem 5 Local existence. Under condition. there are p, q satisfying p, q if n m,.7 p, q nm n m if n > m further u 0, 0 H m 0 H m u, H m 0 such that problem. has a unique local solution u, C [0, T ; H m 0 H m, u t C [0, T ; L L p [0, T t C [0, T ; L L q [0, T. Moreoer, at least one of the following statements holds i T =, ii u t t P u P P u γ P γ as t T. Remark 6 We denote by C arious positie constants which may be different at different occurrences. 3 Blow up of solutions In this section, we are going to consider the blow up of the solution for the problem.. Lemma 7 Suppose that. holds. Then there exists η > 0 such that for any u, H m H m 0 H m H m 0 the inequality holds. P u r r ur r η P u r 3. Proof. The proof is almost the same that of [4], so we omit it here. For the sake of simplicity to proe our result, we take a = b = introduce B = η r, α = B r r, E = α r, 3. where η is the optimal constant in 3.. Next, we will state proe a lemma which is similar to the one introduced firstly by Vitillaro in [6] to study a class of a single wae equation. Lemma 8 Suppose that. holds. Let u, be the solution of system.. Assume further that E 0 < E P u0 P 0 > α. 3.3

5 Mathematics Statistics 6: 9-9, 04 3 Then there exists a constant α > α such that for all t [0, T. E t P u P γ P u γ γ Proof. We first note that by.5, 3. the definition of B, we hae P P u P γ u P where α = P γ α, 3.4 u r r ur r r Bα, 3.5 γ P u P = r = γ u r r ur r P u P γ P P u r r η P P u γ P Br P u r γ γ P u γ γ P γ P γ P γ u P γ γ P u γ P γ γ P γ u γ F u, dx P γ r = α Br r αr = G α, 3.6 u P γ P u γ γ P for 0 < α < α, decreasing for α > α, G α as α, γ. It is not hard to erify that G is increasing G α = α Br r αr = E, 3.7 where α is gien in 3.. Since E 0 < E, there exists α > α such that G α = E 0. P Set α 0 = P u 0 0. Then by 3.6 we get G α0 E 0 = G α, which implies that α 0 α. Now, to established 3.4, we suppose by contradiction that P for some t 0 > 0. By the continuity of P u0 P 0 < α, u P, we can choose t0 such that, P u0 P 0 > α. Again, the use of 3.6 leads to P E t 0 G P u0 0 > G α = E 0. This is impossible since E t E 0 for all t [0, T. Hence 3.4 is established. To proe 3.5, we make use of.5 to get E 0 P u P P u γ γ u r r r ur r γ. P γ

6 4 Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations Consequently, 3.4 yields u r r ur r P P u γ r α E 0 α G α P u γ γ P γ E 0 = Br r αr. 3.8 Therefore, yield the desired result. This completes the proof of Lemma 8. Theorem 9 Assume that. holds. Assume further that r > max {γ, p, q }. Then any solution of the system. with initial data satisfying P u0 P 0 > α E 0 < E cannot exist for all time, where constants α E are defined in 3.. Proof. We suppose that the solution exists for all time we reach to a contradiction. For this purpose, we set By using.5 3.9, we get From we hae 0 < H 0 H t = E Combining we hae We then define P u P H t = E E t. 3.9 u t t P γ P u γ γ F u, dx. 3.0 E u t t γ P u P P γ P u γ c E α r r α c r 0 < H 0 H t c r u r r r r where ε small to be chosen later { r p 0 < σ min, r p u r r r r u r r r r F u, dx. 3. c u r r r r r. 3. Ψ t = H σ t ε uu t t dx, 3.3 r q, r q } r. 3.4 r

7 Mathematics Statistics 6: 9-9, 04 5 Our goal is to show that Ψ t satisfies a differential inequality of the form This, of course, will lead to a blow up in finite time. Ψ t ξψ ζ t, ζ >. Taking the time deriatie of 3.3 using Eq.. we obtain Ψ t = σ H σ t H t ε u t t P ε P u P ε γ P u γ ε r F u, dx ε uu t u t p dx t t q dx. 3.5 From definition of H t, it follows that P γ P u γ = γ H t γ E γ u t t P γ P u γ F u, dx. 3.6 Inserting 3.6 into 3.5, we conclude that Ψ t = σ H σ t H t ε u t t P ε P u ε γ H t γ E ε γ u t t P γ P u ε γ u r r r ur r ε uu t u t p dx t t q dx. Then using 3.5, we hae Ψ t σ H σ t H t ε u t t P ε P u ε γ H t ε γ u t t P γ P u εc u r r ur r ε uu t u t p dx t t q dx, 3.7 where c = γ r γ E Bα r > 0, since α > B r r. In order to estimate the last two terms in 3.7, we make use of the following Young s inequality where X, Y 0, δ > 0, k, l R such that k l XY δk X k k δ l Y l, l =. Consequently, applying the aboe inequality we hae uu t u t p dx δp p up δp p up t t q dx δq q q δq q q p p p pδ p u t p p p p p pδ p H t q q q qδ q t q q q q q qδ q H t,

8 6 Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations where δ, δ are constants depending on the time t specified later. Therefore, 3.7 becomes Ψ t σ H σ t H t ε ε γ P u P ε γ H t u t t εγ u t t εc u r r ur r p ε pδ p p qδ q q H δ p t ε q p up p δq q q q. 3.8 Therefore by choosing δ δ so that δ p p = k H σ t, δ q q = k H σ t, where k, k > 0 are specified later, we get since H t E t F u, dx c σp δ p = k p Hσp t k p cσp u r r r r, 3.9 δ q = k q Hσq t k q cσq u r u r. u r r r Inserting into 3.8, we conclude that Ψ t σ εpk p εqk H σ t H t ε γ H t q P εγ P u ε γ u t t εc u r r ur r εk p cσp u r p r r r εk q cσq q Since r > max {p, q }, we obtain σp u p p r σq, 3.0 σq u r r r q r q. 3. p u p p C up r C u r r q q C q r C u r r q. Thus, Ψ t σ εpk p εqk q P εγ P u ε γ H σ t H t ε γ H t u t t εc u r r ur r εk p cσp C σrpp u p r r εk q cσq C q u r r σrqq, 3. where a b λ C a λ b λ, a, b > 0 is used. From 3.4, we hae σ r p p r, σ r q q r. By using Lemma 4 Sobole-Poincare inequality, we hae r C u u r r P C r u u r u σrpp

9 Mathematics Statistics 6: 9-9, 04 7 r C r r P C r r. σrqq Thus, Ψ t σ εpk p εqk q P εγ P u ε γ ε ε k p cσp C p k p cσp C p q H σ t H t ε γ H t u t t εc u r r ur r k q cσq C u r r r r k q cσq C q By using the c 0 u r r r r u r r ur r in 3.3 we obtain Ψ t σ εpk p εqk H σ t H t q u t t ε γ H t ε γ ε c 0 c k p cσp C p ε We choose k, k large enough so that γ k p cσp C p c 0 c k p cσp C p γ k p cσp C p k q cσq P u P. 3.3 C q k q cσq C q k q cσq C q k q cσq C q u r r r r P u P. 3.4 > c 0c > γ. Then, we choose ε small enough so that σ εpk p εqk q 0. Thus, we hae Ψ t ε γ u t t ε γ H t ε γ P P u ε c 0c u r r r r η u t t H t P P u r u } γ c0c where η = min {ε γ, ε γ, ε, ε. Consequently we hae r r r, 3.5 Ψ t Ψ 0 = H σ 0 ε u 0 u dx 0 dx > 0, t On the other h, applying Hölder inequality, we obtain uu t dx t dx σ u σ u t σ σ t σ C u σ r u t σ σ r t σ. 3.7

10 8 Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations Young s inequality gies for µ θ uu t dx t dx =. We take θ = σ, to get µ = σ σ By using Lemma 4, we obtain Thus uu t dx uu t dx t dx σ t dx C σ C u µ σ r u t θ σ µ σ σ r t θ r by 3.4. Therefore 3.8 becomes σ C u t t u σ r σ r u t t u r P r r r P u, ] Ψ σ t = [H σ σ t ε uu t dx t dx σ σ H t ε σ σ uu t dx t dx C u t t H t u r P r r r P u. 3.3 By combining of we arrie Ψ t ξψ σ t, 3.3 where ξ is a positie constant. T A simple integration of 3.3 oer 0, t yields Ψ σ σ t σ σ. ξσψ σ σ 0 Ψ σ 0 ξσt σ. Therefore Ψ t blows up in a finite time REFERENCES [] Adams R.A., Fournier J.J.F., Sobole Spaces, Academic Press, 003 [] Agre K., Rammaha M.A., Systems of nonlinear wae equations with damping source terms, Diff. Integral Eqns., 006, 9, [3] Gao Q., Li F., Wang Y., Blow up of the solution for higher order Kirchhoff type equaions with nonlinear dissipation, Cent. Eur. J. Math., 0, 93, [4] Georgie V., Todoroa G., Existence of a solution of the wae equation with nonlinear damping source terms, J. Differential Equations, 994, 09, [5] Houari B.S., Global existence decay of solutions of a nonlinear system of wae equations, Appl. Anal., 0, 93, [6] Kirchhoff G., Vorlesungen über Mechanik, 3rd ed., Teubner, Leipzig, 883 [7] Leine H.A., Instability nonexistence of global solutions to nonlinear wae equations of the form P u tt = Au F u, Trans. Amer. Math. Soc., 974, 9, [8] Leine H.A., Some additional remarks on the nonexistence of global solutions to nonlinear wae equations, SIAM J. Math. Anal., 974, 5, [9] Li F.C., Global existence blow-up of solutions for a higher-order Kirchhoff-type equation with nonlinear dissipation, Appl. Math. Lett., 004, 7,

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