BLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF TYPE WITH ARBITRARY POSITIVE INITIAL ENERGY
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1 Electronic Journal of Differential Equations, Vol. 6 6, No. 33, pp. 8. ISSN: URL: or BLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF TYPE WITH ARBITRARY POSITIVE INITIAL ENERGY ZHIFENG YANG, ZHAOGANG GONG Abstract. We consider the viscoelastic equation Z t u ttx, t M u ux, t + gt s ux, sds + u t = u p u with suitable initial data and boundary conditions. Under certain assumptions on the kernel g and the initial data, we establish a new blow-up result for arbitrary positive initial energy, by using simple analysis techniques.. Introduction The wave equation u tt u + hu t = fu. with suitable initial data and boundary conditions has been extensively studied and several results concerning existence and blow-up have been established see [,,, 6]. Here h represents the friction or damping, and f the source. To describe the nonlinear vibrations of an elastic string, the so-called Kirchhoff equation u tt M u u + hu t = fu. was introduced [8], where Ms = m + bs γ is a positive C -function m >, b, γ >, s. In this case the existence and blow-up of solutions have been discussed by many authors see [5, 6, 4, 5, ] and the references cited therein. When we take the viscoelastic materials into consideration, the models. and. become and u tt u + u tt M u u + gt s usds + hu t = fu.3 gt s usds + hu t = fu.4 respectively, where g represents the kernel of the memory. For.3, many existence and blow-up results have been proved. See in this regard [7,,, 7, 8, ]. For example, Messaoudi [] studied.3 with hu t = u t m u t and fu = u p u and proved a blow-up result for solutions with negative initial energy if p > m and a global existence result for p m. Mathematics Subject Classification. 35L5, 35L55, 35L7. Key words and phrases. Viscoelastic equation; blow-up; arbitrary positive initial energy. c 6 Texas State University. Submitted June, 6. Published December 8, 6.
2 Z. YANG, Z. GONG EJDE-6/33 This result has been improved by the same author in [] to the case of positive initial energy. In [7], Song and Zhang consider.3 with hu t = u t and fu = u p u and prove a blow-up result for solutions with positive initial energy by using potential well theory introduced by Payne and Sattinger[6]. Later, Song [8] obtained the blow-up result of.3 in the case of hu t = u t m u t. The model.4 states that the dynamic equilibrium of a body depends not only on the present state of deformation, but also on the previous history of the deformation[3]. This model was first studied by Torrejón and Young [9], who proved the existence of weakly asymptotic stable solution for a large analytical datum. Later, Munoz Rivera [3] showed the global existence for small datum and the total energy decays to zero exponentially under some restrictions. In [] and [], Wu and Tsai studied the model.4 with strong damping and nonlinear damping respectively and proved the existence and blow-up of solutions. In [], a blow-up result of the model.4 with m =, hu t = a u t ν u t +a u t m u t and fu = u p u is obtained under some assumptions on the kernel g, the exponential p and the initial data. But this result holds only in the case E < E, where E is the initial energy of the solution and E is some a positive constant. Recently, by using concavity method, Liu and Liang [9] improved the results of [] to the case of arbitrary positive initial energy. They considered the following initial-boundary value problem u tt M u u + gt s usds + u t = fu, x, t, T, ux, t =, x, t, T, ux, = u x, u t x, = u x, x,.5 where is a bounded domain in R n with a smooth boundary. u and u are given initial data. M and g are two functions which stated as in. and.3. For this model, they obtained a blow-up result under some basic assumptions on f, g, M and the initial data u, u. Readers can see [9, Conditions A A4,.3 and.4]. However, we find that [9, conditions A4 and.4] are inessential. Moreover, it is difficult to construct a concrete model according to all the assumptions in [9], especially for A4 and.4. So, motivated by [8,, 9], we try to consider the blow-up properties of the model.5 with m = and fu = u p u. That is, we study the following problem u tt M u u + gt s usds + u t = u p u, x, t, T, ux, t =, x, t, T, ux, = u x, u t x, = u x, x,.6 where Ms = + bs γ b, γ >, s is a positive C -function. We hope to get some more concise sufficient conditions.
3 EJDE-6/33 BLOW-UP OF SOLUTIONS 3. Preliminaries and statement of main result In this article, C denotes a generic positive constant. It may be different from line to line. And we use the standard Lebesgue space L p with their usual norms p. Moreover, we denote by, the usual L inner product. We first state the general assumptions on g and p as follows: A g C [, is a non-negative and non-increasing function satisfying < k := gsds <.. A If the space dimension n =,, then γ + < p < ; If n 3, then To simplify the notation, we set φ ψt := γ + < p n n. φt s ψt ψs dxds, where ψ may be a scalar, or a vector valued function. A direct computation shows that, for any g C R and u H, T, L, the following identity holds: gt s us, u t t ds = g ut gt ut d { } g ut gsds ut. dt. Now, we state a local existence theorem that can be established by adopting the arguments of []. Theorem. Local solution. Assume that A and A hold. Let u H and u H be given. Then, there exists a unique weak solution ut of.5 such that u C[, T ]; H C [, T ]; L, u t L [, T ]; H..3 for a small enough T >. The energy functional of the solution u of.5 is defined as Et := u t + + g ut p u p p. gsds u b + γ + u γ+.4 By. and assumption A, direct computations yield E t = g ut gt u u t u t..5 According to [], we can obtain the following blow-up with negative initial energy:
4 4 Z. YANG, Z. GONG EJDE-6/33 Theorem.. Assume that A, A and k < p p 3 hold. if E <, then for all the initial data u H and u H, the corresponding solution ux, t of the problem.5 blows up in finite time. Our main result is a blow-up with positive initial energy that reads as follows. Theorem.3. Assume that A, A and k < pp p hold. Moreover, E > maybe large enough is a given initial energy state. If we choose initial data u H and u H satisfying u u dx > βe,.6 where β = ε, ε, is a positive constant, then the corresponding solution ux, t of the problem.5 blows up in finite time. In [9], the kernel g must be the so-called positive type function. But, we do not need that assumption. Moreover, our kernel function space is bigger than the one in [] since pp p > p p Proof of main result Assume u is a global solution of problem.6. Let Qt = uu t dx. Multiplying the first equation of.6 by u and integrating over, we get uu tt dx + M u u Then, we easily obtain gt s usds udx + uu t dx = u p p. Q t = u t M u u + u p p gt s usds udx uu t dx. 3. For the last term on the right side of 3., using Cauchy inequality, we deduce that gt s usds udx = = gt s gt s us utdxds ut us utdxds + p ε g ut + p ε gsds u gsds u 3.
5 EJDE-6/33 BLOW-UP OF SOLUTIONS 5 for all ε,. By 3. and.4, we have Q t u t gsds u b u γ+ + u p p p ε g ut p ε = p ε + u t + p ε gsds u gsds u p εet + ε u p p p ε bp ε + γ + b u γ+. uu t dx gsds u uu t dx Now, by assumption A, we select ε small enough to ensure that bp ε γ + b >. Moreover, using Hölder inequality and Young inequality, we can get uu t dx u u t ε u + ε u t. Then, by assumption A,.5 and Poincaré s inequality, we have Qt Et Q t + ε ε u t p ε + u t p εet ε u p ε k + k u p ε p ε + u t p εet + fελ ε u where λ is the first eigenvalue of and p ε fε = k k p ε. 3.5 Since k < pp p and p >, we deduce that k > p θ := p k k p >. Moreover, we note that fε θ as ε +. So, we can select ε small enough such that fελ ε >. Then, using Cauchy inequality to 3.4, we have hεqt p εet Qt Et ε = hε Qt and p ε Et, hε 3.6
6 6 Z. YANG, Z. GONG EJDE-6/33 where Denote It is easy to see that p ε hε = + fελ ε. p ε ϕε = + fελ ε. fελ ε θλ, ϕε θλ p +, as ε +, fε, fελ ε, ϕε as ε. Hence, by the continuity of ϕε, there exists ε, such that ϕ ε = and ϕε > for all ε, ε. So, we have h ε = ϕ ε = and hε = ϕε > for all ε, ε. And then, we easily deduce that p ε hε p ε hε p θλ p +, ε +, as ε +, +, ε ε, as ε ε. Thus, using the continuity in ε of p ε hε and ε, there exists ε, ε, such that = p ε. ε hε Now, let β = ε and Ht = Qt βet. 3.7 By using.6,.5 and 3.6, we deduce that Then, we have H = Q βe >, H t Q t hε Ht. Ht e hεt H. Since u is global, by.5 and Theorem., the energy Et remains nonnegative, i.e., Et E for all t [, +. So, we deduce that Qt e hεt H and ut = u + u + = u + H hε Qsds e hεs Hds e hε t. 3.8
7 EJDE-6/33 BLOW-UP OF SOLUTIONS 7 By.5, Theorem., and Hölder inequality, we obtain ut u + u + t / u s s ds / u s s ds u + t / E Et / u + t / E / 3.9 which contradicts 3.8. As a simple example, we consider a one-dimension model with Ms = +s, = [, π] and p = 5. Let u = ξ sinηx, u = ξη sinηx, where ξ > and η is a positive integer. Then, we have Q = u, u = ξ η π and E = u + u + 4 u 4 5 u 5 5 = π ξη sinηx dx 5 = ξ η 4 π 3 75 ξ5. π ξ sinηx 5 dx Now, we choose η > /β and ξ = η πη β. Then, we can deduce that Q = βe > βe. According Theorem.3, the corresponding solution blows up in finite time. Acknowledgments. The author would like to thank the anonymous referees for their invaluable comments and suggestions. This research was supported by the Natural Science Foundation of China 678, the Science and Technology Plan Project of Hunan Province 6TP, the Key Construction Disciplines of Hunan Province and the Starting Project of Hengyang Normal University6D. References [] J. Ball; Remarks on blow up and nonexistence theorems for nonlinear evolutions equations. Q. J. Math., 977, 8: [] V. Barbu, M. Iannelli; Controllability of the heat equation with memory. Differ. Integral Equ.,, 3: [3] C. Giorgi, G. Gentili; Thermodynamic properties and stability for the heat flux equation with linear memory. Q. Appl. Math., 993, 5: [4] A. Haraux, E. Zuazua; Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal., 988, 5: 9-6. [5] M. Hosoya, Y. Yamada; On some nonlinear wave equations II: global existence and energy decay of solutions. J. Fac. Sci., Univ. Tokyo, Sect. IA, Math., 99, 38: [6] R. Ikehata; A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms. Differ. Integral Equ., 995, 8: [7] M. Kafini, S. A. Messaoudi; A blow-up result in a Cauchy viscoelastic problem. Appl. Math. Lett., 8, : [8] G. Kirchhoff; Vorlesungen über Mechanik. Teubner, Leipzig, 883.
8 8 Z. YANG, Z. GONG EJDE-6/33 [9] J. Liu, F. Liang; Blow-up of solution for an integro-differential equation with arbitrary positive initial energy. Boundary Value Problems, 5, 596: -. [] S. A. Messaoudi; Blow up in a nonlinearly damped wave equation. Math. Nachr.,, 3: -7. [] S. A. Messaoudi; Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr., 3, 6: [] S. A. Messaoudi; Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl., 6, 3: [3] J. Munoz Rivera; Global solution on a quasilinear wave equation with memory. Boll. Unione Mat. Ital., B, 994, 78: [4] K. Ono; On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Methods Appl. Sci., 997, : [5] K. Ono; On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. J. Math. Anal. Appl., 997, 6: [6] L. Payne, D. Sattinger; Saddle points and instability on nonlinear hyperbolic equations. Isr. J. Math., 975, : [7] H. Song, C. Zhong; Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal.,, : [8] H. Song; Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal.: RWA. 5, 6: [9] R. Torrejón, J. Young; On a quasilinear wave equation with memory. Nonlinear Anal., 99, 6: [] Y. Wang; A global nonexistence theorem for viscoelastic equations with arbitrarily positive initial energy. Appl. Math. Lett., 9, : [] S-T. Wu, L-Y. Tsai; Blow-up of solutions for some nonlinear wave equations of Kirchhoff type with some dissipation. Nonlinear Anal., 6, 65: [] S-T. Wu, L-Y. Tsai; Blow-up of positive-initial-energy solutions for an integro-differential equation with nonlinear damping. Taiwan. J. Math.,, 4: Zhifeng Yang College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan, 4, China address: zhifeng yang@6.com Zhaogang Gong corresponding author College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan, 4, China address: zhaogang gong@6.com
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