OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

Electronic Journal of Differential Equations, Vol. 27 (27, No. 6, pp. 2. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS YUAN GAO, JIN LIANG, TI-JUN XIAO Communicated by Jerome A Goldstein Abstract. We study a class of wave propagation problems concerning the nonlinearity of dynamic evolution for boundary material. We establish an observability inequality for the related linear system, and make a connection between the linear system and the original nonlinear coupled system. Also, we obtain the desired energy decay rate for the original nonlinear boundary value problem.. Introduction We are concerned with the nonlinear boundary value problem u tt (x, t = u(x, t, x, t > ; (. u(x, t =, x Γ, t > ; (.2 u t (x, t f(z t g(z = x, t > ; (.3 u ν = z t x, t > ; (.4 u(x, = u (x, u t (x, = u (x, x, z(x, = z (x, x ; (.5 where is the Laplacian operator, is a bounded domain in R n with a boundary Γ = Γ (disjoint, closed, and nonempty of class C 2, and f, g are given functions on R. For some similar systems with or without source terms in (., there exist several results about uniform decay rate of the solutions to these systems. For instance, [6, 9,, ] study the porous boundary condition with the interface described by u t f(xz t g(xz =, x, t > ; u ν ρ(u t = z t, x, t >, where ρ is a given function. In this paper, we focus on the investigation of the problem above concerning the nonlinearity of dynamic evolution for boundary material, which is always described by boundary displacement z. We allow for nonlinear damping f(z t and nonlinear potential g(z (f and g may depend on x also, which 2 Mathematics Subject Classification. 35L7, 35B35, 76Q5, 35L2. Key words and phrases. Nonlinear wave; coupled boundary equations; uniform decay rates; observability inequality. c 27 Texas State University. Submitted September 8, 26. Published July 4, 27.

2 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 can be handled similarly in the boundary displacement equation (.3. Such nonlinearity enables our results to possess wide applicability. Since our system is a coupled system and we hope to control the whole coupled system by only using a single boundary damping, which is different from and much more complex than the case of single equations, we will make efforts to establish the observability of the related linear system, to find a useful connection between the linear system and the original nonlinear system, and finally to obtain the decay rate of the energy. We also would like to state that our idea is stimulated by the significant papers [, 2, 4, 6, 7, 8,, 3, 4]. We first present some notation, basic definitions and assumptions (cf., e.g., [, 8]. Throughout this paper, c, c i are as generic constants whose values may change from line to line. We make the following assumptions: (H there exists x R n such that m(x ν(x for x Γ, where m(x = x x and ν(x is the unit normal vector pointing to the exterior of. (H2 The function g C(R is monotone nondecreasing such that g( = ; the function f C (R satisfies f( = and inf s R f (s >, and there exists a continuous strictly increasing odd function ρ C([, ]; R, which is continuously differentiable in a neighbourhood of with ρ( = ρ ( =, such that c ρ( v f(v c 2 ρ ( v, v, a.e. on, c v f(v c 2 v, v, a.e. on. Moreover, g(s is locally Lipschitz continuous such that Also we define (.6 c v g(v c 2 v, v, a.e.. (.7 H(x := xρ( x, x [, r 2 ], (.8 r > being small enough such that H is strictly convex on [, r 2 ]. We define {Ĥ (y/y, if y (,, L(y :=, if y =. (.9 Here Ĥ := sup{xy Ĥ(x} x R stands for the convex conjugate function of Ĥ (the extension of H to R in which Ĥ(x = for x R \ [, r 2 ]. Moreover, we define a function Λ H on (, r 2 ] by and write ψ(x := Λ H (x := H(x xh (x, H (r 2 H (r 2 /x v 2 ( Λ H ((H (v dv, x H (r 2. Then, there exists δ > such that ψ is strictly increasing on [, δ]. Let V ( = {u(x H (, u Γ = },

EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 3 and define the inner products and norms on V (, L 2 (, and L 2 ( respectively as follows ( /2, ((u, v V = u(x v(xdx, u V = u (x dx 2 (u, v = u(xv(xdx, u = ( (u(x 2 dx /2, φ, ψ = φ(xψ(xdγ, φ Γ = ( (φ(x 2 dx /2. Clearly, the V is equivalent to the usual H norm. Define the finite energy space by where the norm on H is given by H := V ( L 2 ( L 2 (, (u, v, z 2 H = u 2 V v 2 z 2. Define the energy of system (.-(.5 by E(t := u 2 u 2 t dx zt 2 dγ G(zdΓ, 2 2 where G(x = x g(sds is the anti-derivative of g. 2. Main results and proofs Rewrite the system (.-(.5 as u u t u u t = u = A u t. (2. t z f ( u t Γ g(z z The action of the operator A is given by the matrix of operators that appears in (2.. The remaining boundary conditions are encoded in the domain of A, given by { u D(A = v H; v V (, u L 2 (, u ν z Γ } = f ( v Γ g(z. From (H2, one knows that f is strictly increasing, and its inverse function f is Lipschitz continuous. Thus, using the standard method of nonlinear monotone operators and the semigroup theory (cf. [3], we can obtain wellposedness of the system. To study the energy decay rates of (.-(.5, we first give an observability inequality of the following linear system, which has the same initial values as the original nonlinear system: P tt (x, t = P (x, t, x, t > ; (2.2 P (x, t =, x Γ, t > ; (2.3 P t (x, t M t (x, t M(x, t =, x, t > ; (2.4 P (x, t = M t, x, t > ; (2.5 ν P (x, = u (x, P t (x, = u (x, x ; (2.6

4 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 M(x, = z (x, x. (2.7 Using the multiplier method (cf., e.g., [2, 4], we can prove the following observability inequality. Theorem 2. (Observability inequality. There is a constant T >, depending only on, such that for T T, there corresponds a positive constant C T satisfying T C T E p ( Mt 2 dx dt, (2.8 where E p (t := 2 is the energy of (2.2-(2.7. P 2 t P 2 dx 2 Proof. The proof is divided into the following 5 steps. M 2 dγ Step : Let ξ(t C (R be the cutoff function defined by, t [, T ] ξ(t = a C function with range in (,, t (, (T, T, t (, (T,, for (, T/2. Let h be a [C 2 ( ] n -vector field, which will be specified later. Then, multiplying (2.2 by h P, integrating in time and space and using the boundary condition, we obtain T ɛ = h P (P tt P dx dt = (h P, P t L 2 ( T ɛ ( h ] 2 P 2 dx dt T ɛ T ɛ J P 2 dx dt = (h P, P t L 2 ( T ɛ Γ T ɛ T ɛ T ɛ T ɛ T ɛ h 2 (P 2 t P 2 dx dt h P M t dγdt, where J := hi(x x j. By (H we can take h such that [ ( h 2 (P 2 t h 2 P 2 t h P M t dγdt h ν = on Γ, h 2 P 2 dx dt h ν 2 (P t 2 P 2 dγdt T ɛ J P 2 dx dt J = h i(x x j ρ I on,

EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 5 for some constant ρ >. Hence, T ɛ ρ P 2 dx dt Since T ɛ T ɛ J P 2 dx dt h P M t dγdt (h P, P t L2 ( we have T ɛ ρ T ɛ T ɛ P 2 = (M 2 t P τ 2, P 2 dx dt T ɛ T ɛ h 2 (P t 2 P 2 dx dt [ T ɛ C h Mt 2 dγdt h ν 2 (P 2 t P 2 dγdt h 2 (P 2 t P 2 dx dt. E p = Mt 2 dγ, P 2 t P τ 2 dγdt ] C h E p (, where := (, T, and C h is a positive constant depending on h. Write l.o.t(p, M := (P, P t, M C([,T ];H ɛ ( H ɛ ( H ɛ (, (2.9 for ɛ >. Multiplying (2.2 by P h, integrating in time and space, and using the boundary condition and Sobolev Trace Theory, we obtain T ɛ h(pt 2 P 2 dx dt = P t, P h H ɛ ( H ɛ ( T ɛ C ɛ M 2 t dγdt ɛ P hm t dγdt T ɛ T T ɛ ɛ P P ( h dx dt P 2 dx dt l.o.t(p, M. Let min{ h} = d >. Combining (2. and (2.9 gives T ɛ P 2 Pt 2 dx dt C ɛ,h { M 2 t dγdt T ɛ C h E p ( l.o.t(p, M. (Pt 2 P } τ 2 dγdt (2. (2.

6 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 Using [2, Lemma 4] to estimate T P τ 2 dγdt in (2., we obtain T ɛ P 2 Pt 2 dx dt { C T,ɛ,h Mt 2 ξ 2 Pt 2 dγdt C h E p ( l.o.t(p, M. T ɛ } Pt 2 dγdt (2.2 Step 2: We estimate T Pt 2 dγdt ξ 2 Pt 2 dγdt. The boundary condition on shows that T ɛ Pt 2 dγdt ξ 2 Pt 2 dγdt 2 Mt 2 M 2 dγdt. By (2.2, we have T ɛ E p (tdt C T,ɛ,h,f (Mt 2 M 2 dγdt C h E p ( l.o.t(p, M. (2.3 From E p = Mt 2 dγ, it follows that [ ] (T 2 E p ( Mt 2 dγdt (T 2 E p (T T ɛ C T,ɛ,h,f E p dt (M 2 t M 2 dγdt C h E p ( l.o.t(p, M. (2.4 Step 3: We estimate M 2 dγdt. Multiplying (2.4 by M and integrating in time and space, we obtain = M(P t M t MdΓdt Σ = MP dγ t=t (M t P MM t M 2 dγdt. t= Hence M 2 dγdt = MM t dγdt M t P dγdt MP dγ t=t t= (2.5 ɛ M Σ 2 dγdt C ɛ Mt 2 dγdt l.o.t.(p, M, where ɛ is arbitrarily small. Combining this with (2.4, we obtain (T 2 C h E p ( C T,ɛ,h Mt 2 dγdt l.o.t(p, M. (2.6 Therefore, for T > T := 2 C h, we almost get (2.8 except for the lower-order terms l.o.t(p, M. Step 5: We claim that for T > T = max{t, 2diam(},

EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 7 there exists a constant C T > such that the solution of (2.2-(2.7 satisfies the inequality l.o.t(p, M C T M t 2 L 2 (Σ. (2.7 Suppose this is false. Then there exists a sequence (P ( n, P t ( n, M( n H, and a corresponding solution sequence (P n, P n t, M n of (2.2-(2.7 such that l.o.t(p n, M n = n, M n t 2 L 2 ( n. Thus, by (2.6, we see that (P ( n, P t ( n, M( n H is bounded for T large enough. Hence there is a subsequence, still denoted by such that (P ( n, P t ( n, M( n, (P (, P t (, M(, (P ( n, P t ( n, M( n (P (, P t (, M(, in H weakly. (2.8 Let (P, Pt, M be the solution corresponding to (P (, P t (, M(. Then from E p = Mt 2 dγ <, it follows that (P n, P n t, M n (P, P t, M, weak star in L (, T ; H. (2.9 Clearly, (P n, P n t, M n C(,T ;H is bounded by the wellposedness of the system. Let X := H ( L 2 ( L 2 (, B := H ɛ ( H ɛ ( H ɛ (, Y := H ɛ ( (H ( H ɛ (. We claim that X B compactly. Indeed, for all s, t R with s > t, for an arbitrary bounded set {ψ n } H s (, we can extend the domain of ψ n to ˆ, such that ψ n ˆ =. It is known that H(ˆ s is compactly embedded in H(ˆ. t Hence, there exists a ψ H(ˆ t such that ψ ni ψ H t (ˆ. Hence ψ n i ψ H t (. We also claim that (P n t, P n tt, M n t L 2 (,T ;Y C uniformly. Indeed, it suffices to estimate Ptt n L2 (,T ;(H (. By (2.2 and the boundary condition, we see that for all t (, T and u H (, P tt, u = P udx = M t udγ P udx (2.2 is meaningful. Hence P tt L (, T ; (H (. We deduce then by a classic compactness result (see [2] that Therefore, (P n, P n t, M n (P, P t, M in L (, T ; B strongly. (P, P t, M C([,T ];H ɛ ( H ɛ ( H ɛ ( =. (2.2

8 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 On the other hand, by (2.8, we have Mt =. Differentiating (2.4 in time, we obtain Ptt Γ =. Let a(t, x = Ptt(t, x such that a tt = a, in (, T, a ν = ( P ν tt =, on Γ (, T, a =, on. Using Holmgren s Uniqueness Theorem [], with T > 2diam(, Then from a(t, x = P tt(t, x =, in (, T. P =, in, P P Γ =, ν =, we know that P =. So we obtain M = due to (2.4. Thus (P, M = (, contradicts (2.2. A combination of Steps -5 completes the proof. Next we show a connection between linear and nonlinear systems. Theorem 2.2. Assume that (u, u t, z and (P, P t, M are solutions of system (.- (.5 and (2.2-(2.7 respectively. Then Mt 2 dγdt C zt 2 f(z t 2 dγdt. (2.22 Proof. Set ξ = u P, η = z M. Then (ξ, ξ t, η is the solution of ξ tt (x, t = ξ(x, t, x, t > ; ξ(x, t = x Γ, t > ; ν ξ t (x, t f(z t M t g(z M = x, t > ; ξ ν (x, t = η t(x, t x, t > ; ξ(x, =, ξ t (x, =, x ; η(x, =, x. Multiplying (2.23 by ξ t, integrating in time and space, we obtain t 2 ξ 2 t dx dt 2 t ξ = ν ξ tdγdt = η(m t f(z t M g(zdγdt Γ = (z t M t [M t f(z t M g(z]dγdt. Take ɛ > small enough. (.7 implies that there exist c >, c 2 > such that c v g(v c 2 v, v ɛ, a.e.. (2.23 (2.24

EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 9 Assuming z > ɛ, we have, by (2.24 and z t f(z t dγ, t t 2 ξ 2 t dx dt Mt 2 M t f(z t z t M t dγdt 2 t max{ ηη t, c ηη t }dγdt Therefore t t t t t 2 ξ 2 t dx dt 2 {η t } {η t<} t M t f(z t z t M t dγdt {η t } {η t<} max{ ηη t, c 2 ηη t }dγdt. M 2 t dγdt max{ ( η2 2 t, c ( η2 2 t}dγdt max{ ( η2 2 t, c 2 ( η2 2 t}dγdt. Noting the initial values and using Young s inequality, we obtain t t 2 ξ 2 t dx dt Mt 2 dγdt 2 t, (2.25 c f(z t 2 zt 2 dγdt giving (2.22. Similarly, we obtain (2.22 for z < ɛ. Finally, choose ɛ small enough such that g(z cɛ and z ɛ. By (2.24 we have t t 2 ξ 2 t dx dt = (z t M t [M t f(z t M z z g(z]dγdt, 2 and t t 2 ξ 2 2 η2 2 t dx dt [ M 2 t z t M t M t f(z t zz t M t z z t g(z M t g(z]dγdt. By Young s inequality and Hölder s inequality, we obtain t t 2 ξ 2 η2 2 2 t dx dt Mt 2 dγdt t [ Mt 2 C( (zt 2 f(z t 2 ɛ 2 ]dγdt. (2.26 Since the constant in (2.25 dose not depend on ɛ, we can let ɛ in (2.26. Noticing the initial values, we then obtain (2.22.

YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 Theorem 2.3 (Decay rate. Suppose that lim x H (x Λ H (x =, and T is a time such that (2.8 holds. Then the energy of system (.-(.5 satisfies ( E(t C(T, E(L ψ ( t T, T for t large enough; moreover, if then we have E(t C(T, E((H ( lim sup x Λ H (x <, c t T, for t large enough. Here, C(T, E( is a positive constant depending on T and E(, and T > depends on T. Proof. Clearly, we see that G(zdΓ = z {z } g(sds z { z } c z 2 dγ. 2 Setting c = max(c,, we have Let w satisfy c 2 sdsdγ z g(sdsdγ {z } z c sdsdγ E( c E p (. (2.27 H (w(s = sw(s β, s [, βr2, where { E( E( } β > max c L(H (r 2, (2.28, c δ r is as in (.8, and δ > is a constant such that ψ is strictly increasing on [, δ]. Then the definition of L implies ( s w(s = L, s [, βr 2 β. (2.29 From the property of L, it follows that w is a strictly increasing function from [, βr 2 onto [,. Thus, by using the optimal-weight convexity method (cf. [, Lemma 2.], we deduce that w(e p ( zt 2 f(z t 2 dγdt c 3 T H (w(e p ( c 4 (w(e p ( z t f(z t dγdt.

EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS This and Theorems 2. and 2.2 yield C T E p (w(e p ( T T w(e p ( Mt 2 dγdt Cw(E p ( zt 2 f(z t 2 dγdt Γ T c 3 H (w(e p ( c 6 (w(e p ( z t f(z t dγdt Σ E p (w(e p ( T c 5 c 6 (H (r 2 β z t f(z t dγdt, E( c L(H (r 2 where we used (2.29 and β > (2.27, we have ( CT c 5T E( β Thanks to β > T c5 C T, we set and deduce that [ E(T E( ρ T w( E( ] c Denoting E k := E(kT c β, we obtain c in the last inequality. From this and ( E( w E( E(T. c ρ T := ( CT T c 5 > (2.3 c β E E [ ρ T L (E ]. [ = E( ρ T L ( E( ] c β. From the invariance by time translation t kt for system (.-(.5 and (2.2-(2.7, we have E k E k [ ρ T L (E k ]. Because β > E( c δ, we can apply [, Theorem.5] to complete the proof. Remark 2.4. Under the assumptions of Theorem 2.3, we have ( L ψ ( t T, as t. T Moreover, by taking special f and g, we can see clearly the meaning of the decay rate (please see the examples in [, Section 4]. Acknowledgments. This work was supported partly by the National Natural Science Foundation of China (3795, 57229, and the Shanghai Key Laboratory for Contemporary Applied Mathematics. References [] F. Alabau-Boussouira, K. Ammari; Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 26 (8 (2, 2424-245. [2] G. Avalos, I. Lasiecka; Exact controllability of structural acoustic interactions, J. Math. Pures Appl., 82 (8 (23, 47-73. [3] V. Barbu; Nonlinear differential equations of monotone types in Banach spaces, Springer, New York, 2.

2 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 [4] J. T. Beale, S. I. Rosencrans; Acoustic boundary conditions, Bull. Amer. Math. Soc., 8 (6 (974, 276-278. [5] C. L. Frota, J. A. Goldstein; Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations, 64 ( (2, 92-9. [6] C. L. Frota, N. A. Larkin; Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, Progr. Nonlinear Differential Equations Appl. 66(26, 297-32. [7] C. Gal, G. R. Goldstein, J. A. Goldstein; Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ., 3 (4 (23, 623-635. [8] Y. Gao, J. Liang, T. J. Xiao; A new method to obtain uniform decay rates for damped wave equations with nonlinear acoustic boundary conditions, submitted. [9] P. J. Graber; Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system, Nonlinear Anal., 73(9 (2, 358-368. [] P. J. Graber; Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions, Nonlinear Anal., 74 (2, 337-348. [] P. J. Graber, B. Said-Houari; On the wave equation with semilinear porous acoustic boundary conditions, J. Differential Equations, 252 (9 (22, 4898-494. [2] J. Simon; Compact sets in the space L p (, T ; B, Ann. Mat. Pura Appl., (4 48 (987, 65-96. [3] T. J. Xiao, J. Liang; Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions, J. Differential Equations, 252 (22, 3953-397. [4] T. J. Xiao, J. Liang; Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254(5 (23, 228-257. Yuan Gao Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 2433, China E-mail address: gaoyuan2@fudan.edu.cn Jin Liang School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 224, China E-mail address: jinliang@sjtu.edu.cn Ti-Jun Xiao (corresponding author Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 2433, China E-mail address: tjxiao@fudan.edu.cn