Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay

Transcription

1 mathematics Article Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term an Delay Danhua Wang *, Gang Li an Biqing Zhu College of Mathematics an Statistics, Nanjing University of Information Science an Technology, Nanjing 1, China; ligang@nuist.eu.cn G.L.; brucechu@163.com B.Z. * Corresponence: mathwang@yeah.net; Tel.: Acaemic Eitors: Reza Abei an Inranil SenGupta Receive: 7 January 16; Accepte: 6 June 16; Publishe: 9 June 16 Abstract: In our previous work Journal of Nonlinear Science an Applications 9: 1 115, 16, we stuie the well-poseness an general ecay rate for a transmission problem in a boune omain with a viscoelastic term an a elay term. In this paper, we continue to stuy the similar problem but without the frictional amping term. The main ifficulty arises since we have no frictional amping term to control the elay term in the estimate of the energy ecay. By introucing suitable energy an Lyapunov functionals, we establish an exponential ecay result for the energy. Keywors: wave equation; transmission problem; exponential ecay; viscoelastic; elay MSC: AMS Subject Classification : 35B37, 35L55, 93D15, 93D 1. Introuction In our previous work 1], we consiere the following transmission system with a viscoelastic term an a elay term: t u tt x, t au xx x, t + v tt x, t bv xx x, t =, gt su xx x, ss +µ 1 u t x, t + µ u t x, t τ =, x, t, + x, t, L, + where < < L < L 3, =, L, L 3, a, b, µ 1, µ are positive constants, an τ > is the elay. In that work, we first prove the well-poseness by using the Faeo Galerkin approximations together with some energy estimates when µ µ 1. Then, a general ecay rate result was establishe uner the hypothesis that µ < µ 1. As for the previous results an evelopments of transmission problems, an the research of wave equations with viscoelastic amping or time elay effects, we have state an summarize in great etail in our previous work 1], thus we just omit it here. The reaers, for a better unerstaning of present work, are strongly recommene to 1] an the reference therein see 33]. It is worth pointing out that, in our previous work, the assumption µ < µ 1 " plays an important role in the proof of the above-mentione general ecay result. In this paper, we inten to investigate 1 Mathematics 16,, ; oi:1.339/math

2 Mathematics 16,, of 13 system 1 with µ 1 =. That is, we stuy the exponential ecay rate of the solutions for the following transmission system with a viscoelastic term an a elay term but without the frictional amping: t u tt x, t au xx x, t + v tt x, t bv xx x, t =, gt su xx x, ss +µ u t x, t τ =, x, t, + x, t, L, + uner the bounary an transmission conitions an the initial conitions u, t = ul 3, t =, ul i, t = vl i, t, i = 1, t a gss u x L i, t = bv x L i, t, i = 1, ux, = u x, u t x, = u 1 x, x u t x, t τ = f x, t τ, x, t, τ] vx, = v x, v t x, = v 1 x, x, L 3 where µ is a real number, a, b are positive constants an u 1 x = f x,. The main ifficulty in ealing with this problem is that in the first equation of system, we have no frictional amping term to control the elay term in the estimate of the energy ecay. To overcome this ifficulty, our basic iea is to control the elay term by making use of the viscoelastic term. In orer to achieve this goal, a restriction of the size between the parameter µ an the relaxation function g an a suitable energy is neee. This is motivate by Dai an Yang s work 3], in which the viscoelastic wave equation with elay term but without a frictional amping term was stuie an an exponential ecay result was establishe. In the work here, we will establish an exponential ecay rate result for the energy. The remaining part of this paper is organize as follows. In Section, we give some notations an hypotheses neee for our work an state the main results. In Section 3, uner some restrictions of µ see 35 below, we prove the exponential ecay of the solutions for the relaxation function satisfying assumption H 1 an H.. Preliminaries an Main Results In this section, we present some materials that shall be use in orer to prove our main result. Let us first introuce the following notations: g ht := g ht := g ht := t t t gt shss gt sht hss gt s ht hs s

3 Mathematics 16,, 3 of 13 We easily see that the above operators satisfy t g ht = gss ht g ht t g ht gs s g ht Lemma 1. For any g, h C 1 R, the following equation hols g h]h = g h gt h t For the relaxation function g, we assume H1 g: R + R + is a C 1 function satisfying { t } g h gss h g >, β := a gss = a ḡ > H There exists a positive constant ξ satisfying ξ > ξ > ξ efine by 39 below an g t ξgt, t Accoring to previous results in the literature see 1], we state the following well-poseness result, which can be prove by using the Faeo Galerkin metho. Theorem. Assume that H1 an H hol. Then, given u, v V, u 1, v 1 L, an f L, 1, H 1, Equations have a unique weak solution in the following class: u, v C, ; V C 1, ; L where an V = { u, v H 1 H 1, L : u, t = ul 3, =, ul i, t = vl i, t t } a gss u x L i, t = bv x L i, t, i = 1, L = L L, L To state our ecay result, we introuce the following energy functional: Et = 1 u t x, tx + 1 t a gss u xx, tx + 1 g u x x + 1 L ] v t x, t + bv xx, t x + ζ t e σs t u s x, sxs t τ where σ an ζ are positive constants to be etermine later. 5 Remark 1. We note that the energy functional efine here is ifferent from that of 1] in the construction of the last term. This is motivate by the iea of ], in which wave equations with time epenent elay was stuie. Our ecay results rea as follows:

4 Mathematics 16,, of 13 Theorem 3. Let u, v be the solution of Equations. Assume that H1, H an a > L + L 3 L β, b > L + L 3 L β 6 hol. Let a be the constants efine by 35 below. If µ < a, then there exists constants γ 1, γ > such that 3. Proof of Theorem 3 For the proof of Theorem 3, we use the following lemmas. Et γ e γ 1ξt t, t t 7 Lemma. Let u, v be the solution of Equations. Then, we have the inequality t Et 1 g µ u x tx + + ζ ζ u t x, tx e στ µ u t x, t τx 1 gt u xx, tx σζ t e σt s u s x, sxs t τ Proof. Differentiating 5 an using, we have t Et = + = 1 t u t u tt + a t ζ ζ gt s gss u x u xt 1 ] L gtu x x + u xt u x t u x sxs + 1 g u x x + ζ t e σt s u s x, sxs e στ u t x, t τx σζ g u x x 1 gt e στ u t x, t τx σζ t τ u xx µ u t tu t t τx + ζ t e σt s u s x, sxs t τ v t v tt + bv x v xt ] x u t x, tx u t x, tx 9 By Cauchy inequalities, we get t Et 1 g µ u x tx + + ζ u µ t x, tx + ζ e στ u t x, t τx 1 gt u xx, tx σζ t e σt s u s x, sxs The proof is complete. t τ Remark. In 1] Lemma.1, we prove that the energy functional efine in 1] is non-increasing. µ However, since + ζ u t x, Et may not be non-increasing here. Now, we efine the functional Dt as follows: Dt = Then, we have the following estimate: L uu t x + vv t x

5 Mathematics 16,, 5 of 13 Lemma 5. The functional Dt satisfies t Dt L u t x t gss δ 1 v t x + t δ 1 µ L + δ 1 a g u x x + µ δ 1 gss u t x, t τx L u xx bv xx 1 Proof. Taking the erivative of Dt with respect to t an using, we have t Dt = u t x = u t x L + au x g u x u x x µ t a gss L v t x bv xx u xx L u t x, t τux + g u x u x x µ L v t x bv xx u t x, t τux 11 By the bounary conition 3, we have which implies x u x, t = L1 u x x, tx u xx, tx, x, ] L3 u x, t L 3 L u xx, tx, x L, L 3 ] L u x, tx L u xx, x 1 where L = max{, L 3 L }. By exploiting Young s inequality an 1, we get for any δ 1 > µ u t x, t τux µ u t x, t τx + δ δ 1 µ L u x x 13 1 Young s inequality implies that g u x u x x δ 1 u xx + 1 t gss g u x x 1 δ 1 Inserting the estimates 13 an 1 into 11, then 1 is fulfille. The proof is complete. Now, as in Lemma.5 of ], we introuce the function x, x, ] qx = + L 3 L L x, x, L x L + L 3, x L, L 3 ] { L1 It is easy to see that qx is boune, that is, qx M, where M = max, L } 3 L is a positive constant. In aition, we efine the functionals: F 1 t = qxu t au x g u x x, Then, we have the following estimates. 15 L F t = qxv x v t x 16

6 Mathematics 16,, 6 of 13 Lemma 6. The functionals F 1 t an F t satisfy t F 1t qx ] a ] ] a au x g u x qxu t + + M δ t ] + a + δ M a µ + g δ + + δ µ gss + δ µ M + µ + µ M ] δ δ t + gss + µ δ u t x, t τx u t x g u x x gδ g u x x u xx 17 an t F t + L 3 L L + b L L v t x + L 3 L v xl, t + v x, t bv xx + v t + L 3 L v t L 1 Proof. Taking the erivative of F 1 t with respect to t an using, we get t F 1t = qxu tt au x g u x x qxu t auxt gtu x t + g u x t x = qx ] au x g u x + 1 q a ] xau x g u x x qxu t + a q xu t x qx µ u t x, t τg u x x + qxau x µ u t x, t τx qxu t g u x t gtu x ]x 19 We note that 1 a q xau x g u x x a u xx + g u x x a u xx + t gt su x s u x t + u x ts x t t u xx + gss u xx + gss g u x x Young s inequality gives us for any δ >, = µ qxau x µ u t x, t τx δ M a µ u xx + µ u t x, t τx 1 δ qx µ u t x, t τg u x x δ M µ + µ M δ t g u x qxu t x, t τx + µ t u t x, t τx + µ δ u t x, t τx gss gss qxu t x, t τu x x t g u x x + δ µ gss u xx

7 Mathematics 16,, 7 of 13 an M δ qxu t g u x t gtu x ]x u t x + g δ u xx gδ g u x x 3 Inserting 3 into 19, we get 17. By the same metho, taking the erivative of F 1 t with respect to t, we obtain L t F t = L qxv xt v t x qxv x v tt x = 1 ] L qxv t + 1 L q xv t x + 1 L bq xv xx + b ] L qxv x L 1 + L 3 L L L v t x + bv L xx + v t + L 3 L v t L + b L 3 L v xl, t + v x, t Thus, the proof of Lemma 6 is finishe. In 1], the authors pointe out that if µ < µ 1, then the energy is non-increasing. Thus, the negative term u t x appeare in the erivative energy can be use to stabilize the system. However, in this paper, the energy is not non-increasing. In this case, we nee some aitional negative term u t x. For this purpose, let us introuce the functional Then, we have the following estimate. F 3 t = u t g ux Lemma 7. The functionals F 3 t satisfies t t F 3t gss α + δ µ gl α u t x, t τx + g u x x u t x, tx + δ + δ t ] gss δ a + µ L t δ δ δ Proof. Taking the erivative of F 3 t with respect to t an using, we get t F 3t = = = t + µ t u tt g ux au xx x, t gss t t gss u t x, tx u xx, tx ] gss g u x x u t g ux gt su xx x, ss µ u t x, t τ g ux u t x, tx u t g ux gt su x x, ss g u x x + a u x g u x x t u t x, t τg ux gss u t x, tx u t g ux 5

8 Mathematics 16,, of 13 an Young s inequality implies that for any δ > : δ t t t δ gss t δ gss gt su x x, ss g u x x gt su x t u x s u x ts x + 1 δ u xx, tx + δ + 1 δ u xx, tx + δ + 1 δ t g u x x g u x x gss g u x x a u x g u x x δ u xx, tx + a t gss g u x x 7 δ By Young s inequality an 1, we get for any δ >, α 1 > 6 µ u t x, t τg ux δ µ u t x, t τx + µ L t gss g u x x δ an u t g ux α α u xx, tx + 1 t g ss g ux α u xx, tx gl g u x x 9 α Inserting 6 9 into 5, we get. Now, we are reay to prove Theorem 3. Proof. We efine the Lyapunov functional: Lt = N 1 Et + N Dt + F 1 t + N F t + N 5 F 3 t 3 where N 1, N, N an N 5 are positive constants that will be fixe later. Since g is continuous an g >, then for any t t >, we obtain t gss t gss = g 31

9 Mathematics 16,, 9 of 13 Taking the erivative of 3 with respect to t an making the use of the above lemmas, we have {N t Lt 5 g α µ N 1 + ζ } a N + M u t x δ {N β N δ 1 + L δ 1 µ a + ḡ + δ µ ḡ + δ M a µ + g δ N 5 δ + δ ḡ } u xx { µ + N 1 ζ e στ + N µ + δ µ M + µ 1 + M } + N 5 δ δ 1 δ µ { } bl1 + L 3 L L { } N L + N b v L1 + L xx 3 L L L N N v t x b N b L 3 L v xl, t + v L1 x, t a N v t, t + L 3 L N ḡ + + ḡ + µ ḡ + N 5 δ δ 1 δ a + µ L ] ḡ g u x x δ δ N1 + gδ N 5gL ] g u x x α u t x, t τx ] v t L, t 3 At this moment, we wish all coefficients except the last two in 3 will be negative. We want to choose N an N to ensure that a N b N 33 + L 3 L L N N > such that such that For this purpose, since ll + L 3 L < min{a, b}, we first choose N satisfying Then, we pick ll + L 3 L < N min{a, b} α = g, δ 1 < β an δ < 1 g N 5 g α = N 5g, N β N δ 1 > 7N β, an g δ < 1 Once δ is fixe, we take N satisfying Furthermore, we choose N 5 satisfying N > a + ḡ + g δ β a + ḡ + g δ < N β N 5 g > N + a + M δ such that N < N 5β, a < N 5β, M < N 5β δ an N 5g a N + M > δ

10 Mathematics 16,, 1 of 13 such that Then, we pick δ satisfying δ < β N N ḡ N 5 δ + δ ḡ < N β Once the above constants are fixe, we choose N 1 satisfying N 1 > gδ + N 5gL α Now, we nee to choose suitable µ an ζ such that µ K 1 N 1 + ζ > 5β N K µ > 3 µ K 3 N ζ 1 e στ < where K 1 = N 5g a N + M, K = N δ δ 1 L + δ ḡ + δ M a K 3 = N 1 + N δ 1 + δ M + 1 δ + M δ + N 5 δ We first choose ζ satisfying Then, we pick µ satisfying K 1 N 1 ζ > { 5β N µ < N min, 1 ζ K K 3 e στ, K } 1 ζ := a 35 N 1 From the above, we euce that there exist two positive constants α 5 an α 6 such that 3 becomes Multiplying 36 by ξ, we have t Lt α 5Et + α 6 g u x x 36 ξ t Lt α 5ξEt + α 6 ξ g u x x On the other han, by the efinition of the functionals Dt, F 1 t, F t, F 3 t an Et, for N 1 large enough, there exists a positive constant α 3 satisfying N Dt + N 3 F 1 t + N F t + F 3 t η 1 Et which implies that N 1 η 1 Et Lt N 1 + η 1 Et

11 Mathematics 16,, 11 of 13 Exploiting H an, we have Thus, 36 becomes ξ g u x x g u x x t Et + K 1 u t x 37 N 1 ξ t Lt α 5ξEt α 6 t Et + K 1α 6 N 1 Et 3 We a a restriction conition on ξ, that is, we suppose that Then, 3 becomes, for some positive constants Now, we efine functionals L t as ξ > K 1α 6 N 1 := ξ 39 ξ t Lt α 7ξEt α t Et L t = ξlt + α Et It is clear that L t Et Then, we have A simple integration of 1 over t, t leas to t L t α 7ξEt 1 L t L t e cξt t, t t Recalling, Equation yiels the esire result 7. This completes the proof of Theorem 3.. Conclusions The main purpose of present work is to investigate ecay rate for a transmission problem with a viscoelastic term an a elay term but without the frictional amping term. It is base upon our previous work 1], in which we stuie the well-poseness an general ecay rate for a transmission problem in a boune omain with a viscoelastic term an a elay term. The main ifficulty in ealing with the problem here is that in the first equation of system, we have no frictional amping term to control the elay term in the estimate of the energy ecay. To overcome this ifficulty, our basic iea is to control the elay term by making use of the viscoelastic term. In orer to achieve this target, a restriction of the size between the parameter µ an the relaxation function g an a suitable energy is neee. This is motivate by Dai an Yang s work 3], in which the viscoelastic wave equation with elay term but without a frictional amping term was consiere an an exponential ecay result was establishe. In Section, we give some notations an hypotheses neee for our work an state the main results. In Section 3, because the energy is not non-increasing, we introuce the aitional functional to prouce negative term u t x. Then by introucing suitable Lyaponov functionals, we prove the exponential ecay of the solutions for the relaxation function satisfying assumption H 1 an H.

12 Mathematics 16,, 1 of 13 Acknowlegments: This work was supporte by the project of the Chinese Ministry of Finance Grant No. GYHY966, the National Natural Science Founation of China Grant No , an the Natural Science Founation of Jiangsu Province Grant No. BK Author Contributions: There was equal contribution by the authors. Danhua Wang, Gang Li, Biqing Zhu rea an approve the final manuscript. Conflicts of Interest: The authors eclare no conflict of interest. References 1. Wang, D.H.; Li, G.; Zhu, B.Q. Well-poseness an general ecay of solution for a transmission problem with viscoelastic term an elay. J. Nonlinear Sci. Appl. 16, 9, Alabau-Boussouira, F.; Nicaise, S.; Pignotti, C. Exponential stability of the wave equation with memory an time elay. In New Prospects in Direct, Inverse an Control Problems for Evolution Equations; Springer: Cham, Switzerlan, 1; Volume 1, pp Ammari, K.; Nicaise, S.; Pignotti, C. Feeback bounary stabilization of wave equations with interior elay. Syst. Control Lett. 1, 59, Apalara, T.A.; Messaoui, S.A.; Mustafa, M.I. Energy ecay in thermoelasticity type III with viscoelastic amping an elay term. Electron. J. Differ. Equ. 1, 1, Bae, J.J. Nonlinear transmission problem for wave equation with bounary conition of memory type. Acta Appl. Math. 1, 11, Bastos, W.D.; Raposo, C.A. Transmission problem for waves with frictional amping. Electron. J. Differ. Equ. 7, 7, Berrimi, S.; Messaoui, S.A. Existence an ecay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 6, 6, Cavalcanti, M.M.; Cavalcanti, V.N.D.; Filho, J.S.P.; Soriano, J.A. Existence an uniform ecay rates for viscoelastic problems with nonlinear bounary amping. Differ. Integral Equ. 1, 1, Cavalcanti, M.M.; Cavalcanti, V.N.D.; Soriano, J.A. Exponential ecay for the solution of semilinear viscoelastic wave equations with localize amping. Electron. J. Differ. Equ.,, Datko, R.; Lagnese, J.; Polis, M.P. An example on the effect of time elays in bounary feeback stabilization of wave equations. SIAM J. Control Optim. 196,, Guesmia, A. Well-poseness an exponential stability of an abstract evolution equation with infinite memory an time elay. IMA J. Math. Control Inf. 13, 3, Guesmia, A. Some well-poseness an general stability results in Timoshenko systems with infinite memory an istribute time elay. J. Math. Phys. 1, 55, Guesmia, A.; Tatar, N. Some well-poseness an stability results for abstract hyperbolic equations with infinite memory an istribute time elay. Commun. Pure Appl. Anal. 15, 1, Kirane, M.; Sai-Houari, B. Existence an asymptotic stability of a viscoelastic wave equation with a elay. Z. Angew. Math. Phys. 11, 6, Li, G.; Wang, D.H.; Zhu, B.Q. Well-poseness an general ecay of solution for a transmission problem with past history an elay. Electron. J. Differ. Equ. 16, 16, Li, G.; Wang, D.H.; Zhu, B.Q. Blow-up of solutions for a viscoelastic equation with nonlinear bounary amping an interior source. Azerbaijan J. Math. 16, 7, in press. 17. Lions, J.L. Quelques Méthoes e Résolution es Problèmes aux Limites Non Linéaires; Duno: Paris, France, Liu, W.J. Arbitrary rate of ecay for a viscoelastic equation with acoustic bounary conitions. Appl. Math. Lett. 1, 3, Liu, W.J.; Chen, K.W. Existence an general ecay for nonissipative istribute systems with bounary frictional an memory ampings an acoustic bounary conitions. Z. Angew. Math. Phys. 15, 66, Liu, W.J.; Chen, K.W. Existence an general ecay for nonissipative hyperbolic ifferential inclusions with acoustic/memory bounary conitions. Math. Nachr. 16, 9, Liu, W.J.; Chen, K.W.; Yu, J. Extinction an asymptotic behavior of solutions for the ω-heat equation on graphs with source an interior absorption. J. Math. Anal. Appl. 16, 35,

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