GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC EQUATION WITH BOUNDARY MEMORY SOURCE TERM FUQIN SUN AND MINGXIN WANG Received 21 January 2005; Accepted 17 August 2005 We study a nonlinear hyperbolic equation with boundary memory source term. By the use of Galerkin procedure, we prove the global existence and the decay property of solu- tion. Copyright © 2006 F. Sun and M. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction This paper deals with a hyperbolic equation with boundary memory source terms: ρ(x)u −Δu −Δu = g(u), x ∈Ω, t>0, u = 0, x ∈ Γ 0 , t>0, ∂u ∂ν + ∂u ∂ν + f (u ) = t 0 K(t −τ)h τ,u(τ) dτ, x ∈Γ 1 , t>0, u(0,x) = u 0 (x), ρu (0,x) = ρu 1 (x), x ∈ Ω, (1.1) where u =u(t,x), Ω is a bounded domain of R N (N ≥1) with sufficiently smooth bound- ary ∂Ω = Γ 0 ∪ Γ 1 , ¯ Γ 0 ∩ ¯ Γ 1 =∅,whereΓ 0 and Γ 1 have positive measures. u = ∂u/∂t, u = ∂ 2 u/∂t 2 . Equations of type (1.1) are of interest in many applications such as in the theory of electromagnetic materials with memory which obey the Ohm’s law. It can also describe the temperature evolution in a rigid conductor with a memory. We refer to [8, 9] to see the details. In many works concerned with equations of type (1.1), we cite Aassila et al. [1], where the following wave equation was considered: u −Δu+ f 0 (∇u) =0, x ∈Ω, t>0, u = 0, x ∈ Γ 0 , t>0, ∂u ∂ν + g u = t 0 K(t −τ)h u(τ) dτ, x ∈Γ 1 , t>0, u(0,x) = u 0 (x), u (0,x) = u 1 (x), x ∈ Ω. (1.2) Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Ar ticle ID 60734, Pages 1–16 DOI 10.1155/JIA/2006/60734 2 Global solutions for a nonlinear hy perbolic equation Under some conditions on nonlinear terms, they acquired the existence and uniform decay of solutions. Recently, Park and Park [12] generalized problem (1.2)byendowing with some discontinuous and multivalued terms. For more related works, we refer to [3, 4, 7, 11, 13] and the references therein. For problem (1.1) without memory source term, we point out the work [6] of Cavalcanti et al., where they investigated the following equation with boundary damping: ρ(x)u −Δu = 0, x ∈ Ω, t>0, u = 0, x ∈ Γ 0 , t>0, ∂u ∂ν + f (u )+g(u) = 0, x ∈ Γ 1 , t>0, u(0,x) = u 0 (x), ρu (0,x) = ρu 1 (x), x ∈ Ω. (1.3) Through a partition of boundary Γ and Galerkin procedures, they acquired the existence and decay behavior of the solution to problem (1.3). In another work of theirs [5], using similar method, they studied problem (1.3)withρ = 1 and the source term g(u) =|u| p u coupled in the first equation. Motivated by the above works, we are devoted to study problem (1.1). By v irtue of the potential well method, and through Galerkin procedures, we acquire the global existence and decay property of perturbed energy of solutions of problem (1.1). The organization of this paper is as follows. In Section 2,wemakeas- sumptions and introduce a potential well, and then state the main results. In Section 3, making use of Galerkin procedures, we study the existence of solution of problem (1.1). And in the last section, we der ive the uniform decay by the perturbed energy method. 2. Assumptions and main results In this section, we first give the notations used throughout this paper: (u,v) = Ω u(x)v(x)dx,(u,v) Γ 1 = Γ 1 u(x)v(x)dΓ, · p =· L p (Ω) , ·=· L 2 (Ω) , · Γ 1 ,p =· L p (Γ 1 ) , · Γ 1 =· L 2 (Γ 1 ) , (2.1) and r denotes the conjugate exponent of r>1. Define V = u ∈H 1 (Ω):u =0onΓ 0 . (2.2) Since the measure of Γ 0 is positive, Poincar ` e inequality holds and trace embedding theo- rem holds (see [2]), we know that ∇u is equivalent to the norm on V.Letμ 1 and μ 2 be the optimal constants such that u≤μ 1 ∇u, u Γ 1 ≤ μ 2 ∇u∀u ∈V. (2.3) Now we make the following assumptions. F. Sun and M. Wang 3 (A 1 ) f ∈ C(R), f (s)s ≥ 0, and there exist positive constants k 1 and k 2 such that k 1 |s| q−1 ≤|f (s)|≤k 2 |s| q−1 , (2.4) where 2 <q< ∞ if N =1, 2; 2 <q≤ 2(N −1)/(N −2) if N ≥3. (A 2 ) g ∈C(R), g(s) s ≥0, and there exists positive constant k 3 such that g(s) ≤ k 3 |s| p , (2.5) where 1 <p< ∞ if N =1, 2; 1 <p≤ N/(N −2) if N ≥3. (A 3 ) K : R + → R + is a continuously differentiable function verifying K (t) ≤−k 4 K(t) ∀t ≥0, K(0) > 0, 1 −μ 2 2 ∞ 0 K(s)ds L>0, (2.6) where k 4 > 0. (A 4 ) h(τ,s)ismeasurablewithτ and continuous with s, and it satisfies h(τ,s) −s ≤ K(τ) K(0) |s|∀s ∈ R, τ ≥ 0. (2.7) (A 5 ) ρ(x) ≥0, ρ ≡0andρ ∈L ∞ (Ω). (A 6 ) Assume that the initial data u 0 ,u 1 ∈ V ∩H 3/2 (Ω) (2.8) and satisfy the compatibility conditions −Δ u 0 + u 1 = g u 0 , x ∈Ω, u 0 = 0, u 1 = 0, x ∈ Γ 0 , ∂u 0 ∂ν + ∂u 1 ∂ν + f u 1 = 0, x ∈Γ 1 . (2.9) Remark 2.1. (i) The assumptions (A 3 )and(A 4 )implythath(τ,s) ≈ (1 + K(τ))s. (ii) Given u 1 ∈ V ∩H 3/2 (Ω), by the assumption (A 2 ) and the theory of elliptic prob- lems, we see that problem (2.9) admits a weak solution u 0 ∈ V ∩H 3/2 (Ω). Let B ∗ > 0 be the optimal constant such that v p+1 ≤ B ∗ ∇v∀v ∈V, (2.10) where p is the number given in the assumption (A 2 ). 4 Global solutions for a nonlinear hy perbolic equation If we define B ∞ sup v∈V ,v=0 (1/(p +1))v p+1 p+1 ∇v p+1 , (2.11) then B ∞ ≤ B p+1 ∗ p +1 , 1 p +1 v p+1 p+1 ≤ B ∞ ∇v p+1 ∀v ∈V. (2.12) Now for some function u,wedefine J(u) = L 2 ∇u 2 − k 3 p +1 u p+1 p+1 , E(t) = 1 2 ρu 2 + 1 2 ∇u 2 − Ω G(u)dx − 1 2 t 0 K(τ)dτ u(t) 2 Γ 1 + 1 2 (K u)(t), (2.13) where G(s) = s 0 g(η)dη,(Ku)(t) = t 0 K(t −τ) h τ,u(τ) − u(t) 2 Γ 1 dτ. (2.14) Putting d inf u∈V,u=0 sup λ>0 J(λu) , H(λ) L 2 λ 2 −k 3 B ∞ λ p+1 , λ>0. (2.15) We have the following result. Proposition 2.2. Let the assumptions (A 2 )–(A 4 ) be fulfilled. It holds that d = max λ>0 H(λ) = H λ ∞ = (p −1)L 2(p +1) λ 2 ∞ , (2.16) where λ ∞ = (L/(p +1)k 3 B ∞ ) 1/(p−1) . If ∇u <λ ∞ , then J(u) ≥ 0, ∇u 2 ≤ 2(p +1) (p −1)L E(t). (2.17) Proof. From H (λ) =Lλ −(p +1)k 3 B ∞ λ p = L −(p +1)k 3 B ∞ λ p−1 λ, (2.18) F. Sun and M. Wang 5 we see that λ ∞ = [L/((p +1)k 3 B ∞ )] 1/(p−1) is the maximum point of H.Hence, max λ>0 H(λ) = H λ ∞ = (p −1)L 2(p +1) λ 2 ∞ . (2.19) Note the definition of B ∞ , by the direct computation, we have d = inf u∈V,u=0 sup λ>0 J(λu) = L 2 L k 3 2/(p−1) − k 3 p +1 L k 3 (p+1)/(p−1) inf u∈V,u=0 ⎛ ⎝ ∇ u p+1 u p+1 p+1 ⎞ ⎠ 2/(p−1) = (p −1)L 2(p +1) L (p +1)k 3 B ∞ 2/(p−1) = (p −1)L 2(p +1) λ 2 ∞ . (2.20) Thus the first conclusion is valid. If ∇u <λ ∞ ,thenweobtain E(t) ≥ J u(t) ≥ L 2 ∇u 2 −k 3 B ∞ ∇u p+1 > ∇u 2 L 2 −k 3 B ∞ λ p−1 ∞ =∇ u 2 L 2 − L p +1 = (p −1)L 2(p +1) ∇u 2 . (2.21) Thus the second conclusion is valid. Remark 2.3. The number d defined in Proposition 2.2 is the Mountain Pass level related to the elliptic problem −LΔu =k 3 |u| p−1 u, x ∈Ω, u = 0, x ∈ Γ 0 , ∂u ∂ν = 0, x ∈ Γ 1 , (2.22) see [5]or[14]. In fact, d is equal to the number inf α∈Λ sup t∈[0,1] J α(t) , (2.23) where Λ = α ∈C [0,1];V ; α(0) = 0, J(α(1)) < 0 . (2.24) 6 Global solutions for a nonlinear hy perbolic equation Now we are in a position to state the main results of this paper. Theorem 2.4. Let the assumptions (A 1 )–(A 6 ) hold. If in addition, the initial data satisfy ∇ u 0 <λ ∞ , E(0) <d, (2.25) then for any T>0,problem(1.1) admits a solution u ∈ L ∞ (0,T;V) and satisfies √ ρu ∈ L ∞ (0,T;L 2 (Ω)), u ∈ L 2 (0,T;V), ρu ∈ L q (0,T;L 2 (Ω)). Theorem 2.5. Let u bethesolutionobtainedinTheorem 2.4.Ifq = 2,thenthesolutionu verifies the following decay estimate: E(t) ≤ 3d exp − 2 3 Ct ∀ t ≥ 0 (2.26) for some positive constant C. 3. Proof of Theorem 2.4 In this section, we will use Faedo-Galerkin procedure to prove Theorem 2.4. Step 1. Let {ω k } ∞ k=1 be a basis in V, which is orthogonal in L 2 (Ω). For fixed n,let V n = ω 1 , ,ω n (3.1) be the linear span of {ω k } n k =1 ,andlet ρ ε = ρ + ε (ε>0), u εn (t) = n k=1 q εkn (t)ω k ∈ V n . (3.2) Consider the Cauchy problem: ρ ε u εn ,ω + ∇ u εn (t),∇ω + ∇ u εn (t),∇ω + f u εn ,ω Γ 1 = g u εn ,ω + t 0 K(t −τ) h τ,u εn (τ) ,ω Γ 1 dτ, ∀ω ∈V n , (3.3) u εn (0) = n k=1 q εkn (0)ω k −→ u 0 strongly in V, (3.4) u εn (0) = n k=1 q εkn (0)ω k −→ u 1 strongly in L 2 (Ω). (3.5) By the standard method of ordinary differential equations, s ystem (3.3)-(3.4) has a local solution u εn (t)oninterval(0,t εn )withq εkn (t) ∈W 2,1 (0,t εn ). The extension of this solu- tion to the whole interval [0, ∞) will be deduced by a series a priori estimates. F. Sun and M. Wang 7 Using the method exploited in the paper [15], we can construct the energy function and the energy identity associated to problem (3.3)-(3.4)asfollows: E εn (t) = 1 2 ρ ε u εn 2 + 1 2 ∇ u εn 2 − Ω G u εn dx − 1 2 t 0 K(τ)dτ u εn (t) 2 Γ 1 + 1 2 Ku εn (t), (3.6) E εn (t) −E εn (s) =− t s Γ 1 f u εn (η) u εn (η)dΓ dη − t s ∇u εn (η) 2 dη + 1 2 t s K u εn (η)dη + K(0) 2 t s h η,u εn (η) − u εn (η) 2 Γ 1 dη − 1 2 t s K(η) u εn (η) 2 Γ 1 dη (3.7) for 0 ≤ s ≤t<t εn . Using the assumption (A 4 ), it is easily known that K(0) 2 t s h η,u εn (η) − u εn (η) 2 Γ 1 dη − 1 2 t s K(η) u εn (η) 2 Γ 1 dη ≤ 0. (3.8) Then using the assumption (A 3 ), (3.7)and(3.8)implythatE εn (t) is a decreasing function. By the assumption (A 2 ), we see that G(u) ≤ C 1 |u| p+1 , (3.9) Here and in the sequel C i , i = 1,2, , will denote various constants independent of ε and n. Exploiting the continuity of the Nemyskii operator and (3.4), it follows that Ω G u 0εn dx −→ Ω G u 0 dx as n −→ ∞. (3.10) Therefore, using (3.4)and(3.5), it entails E εn (0) −→ E(0) as n −→ ∞ , ε −→ 0. (3.11) Define B n sup u∈V n ,u=0 (1/(p +1))u p+1 p+1 ∇u p+1 , λ n L (p +1)k 3 B n 1/(p−1) , d n (p −1)L 2(p +1) λ 2 n . (3.12) 8 Global solutions for a nonlinear hy perbolic equation By the assumption (A 2 ), it follows that 0 <B n ≤ B n+1 ≤···≤B ∞ , λ ∞ ≤···≤λ n+1 ≤ λ n ,d ≤···≤d n+1 ≤ d n , n ≥ 1. (3.13) By ( 2.25), (3.4), (3.5), (3.11), and (3.13), we know that, for sufficiently large n 0 and suf- ficiently small ε 0 , ∇ u εn (0) <λ n , E εn (0) <d n , n ≥ n 0 , ε ≤ ε 0 . (3.14) From now on, we may assume t hat n ≥ n 0 and ε ≤ε 0 .By(3.6) and the assumptions (A 2 ) and (A 3 ), we deduce that E εn (t) ≥ L 2 ∇ u εn (t) 2 −k 3 B n ∇ u εn (t) p+1 = H n ∇ u εn (t) , (3.15) where H n (λ) =(L/2)λ 2 −k 3 B n λ p+1 has the similar property of the function H defined in Proposition 2.2. It is easy to verify that H n is increasing for 0 <λ<λ n and decreasing for λ>λ n , H n (λ n ) =d n ,andH n (λ) →−∞as λ → +∞.SinceE εn (0) <d n , there exist λ 1 n <λ n < λ 2 n such that H n (λ 1 n ) =H n (λ 2 n ) =E εn (0). From (3.7)and(3.8), we have E εn (t) ≤ E εn (0) ∀t ∈ 0,t εn . (3.16) Denote λ 0 n =∇u εn (0),soλ 0 n <λ n .By(3.15), we have H n (λ 0 n ) ≤E εn (0), thus λ 0 n <λ 1 n . We claim that ∇u εn (t)≤λ 1 n for all t ∈ [0,t εn ). Suppose, by contradiction, that ∇u εn (t 0 ) >λ 1 n for some t 0 ∈ (0,t εn ). From the continuity of ∇u εn (·),wecansup- pose that ∇u εn (t 0 ) <λ n .Thenby(3.15), E εn (t 0 ) ≥H n (∇u εn (t 0 )) >H n (λ 1 n ) =E εn (0), which contradicts (3.16). From (3.14)and(3.16), it yields E εn (t) <d n for t ∈ [0,t εn ). Then using (3.13), one gets ∇ u εn (t) ≤ λ 1 , E εn (t) <d 1 (3.17) for t ∈ [0,t εn ). By (3.17), the assumption (A 2 ), and the Sobolev embedding theorem, we deduce that Ω G u εn dx ≤ k 3 p +1 u εn p+1 p+1 ≤ C 2 . (3.18) Therefore, from (3.6), (3.17), and (3.18), it follows that ρ ε u εn (t) ≤ C 3 . (3.19) Estimates (3.17)and(3.19)implythatt εn =∞. F. Sun and M. Wang 9 For any T>0andforallt ∈ [0,T], by the assumptions (A 1 ), (A 3 ), and (A 4 ), we get from (3.7), (3.14), and (3.16)that t 0 ∇ u εn (τ) 2 dτ ≤ C 4 , t 0 u εn (τ) q Γ 1 ,q dτ ≤ k −1 1 t 0 f u εn (τ) ,u εn (τ) Γ 1 dτ ≤ C 5 . (3.20) Then using t he assumptions (A 1 )-(A 2 ), the Sobolev embedding theorem, (3.13), and (3.20), we derive that, for all t ∈ [0,T], t 0 f u εn q Γ 1 ,q dτ ≤ C 6 , g u εn (p+1) (p+1) ≤ C 7 . (3.21) Similarly, by the assumptions (A 3 )-(A 4 ) and the Sobolev embedding theorem, it leads to t 0 h τ,u εn (τ) Γ 1 dτ ≤ C 8 , t 0 K(t −τ)h τ,u εn (τ) dτ Γ 1 ≤ C 9 ∀t ∈ [0,T]. (3.22) Replacing ω in (3.3)withv ∈ V, and exploiting the H ¨ older inequality, the Sobolev em- bedding theorem, (3.17), (3.21), and (3.22), it entails ρ ε u εn ,v ≤ C 10 ∇ u εn + ∇ u εn + f u εn Γ 1 ,q + g u εn (p+1) + t 0 K(t −τ)h τ,u εn (τ) dτ Γ 1 ∇ v, ρ ε u εn ≤ C 11 1+ ∇ u εn + f u εn Γ 1 ,q (3.23) for all t ∈ [0,T]. Integrating the above inequality over [0,t], using (3.20)-(3.21) and the H ¨ older in- equality, we get t 0 ρ ε u εn q dτ ≤ C 12 t 0 1+ ∇ u εn q + f u εn q Γ 1 ,q dτ ≤ C 13 . (3.24) 10 Global solutions for a nonlinear hyperbolic equation Step 2. The limiting process. From the estimates (3.17), (3.19)–(3.24), using the standard arguments, it yields that, up to a subsequence, as n →∞, u εn −→ u ε weakly ∗ in L ∞ (0,T;V), ρ ε u εn −→ ρ ε u ε weakly ∗ in L ∞ 0,T;L 2 (Ω) , (3.25) u εn −→ u ε weakly in L 2 (0,T;V), (3.26) f u εn −→ γ weakly ∗ in L q (0,T) ×Γ 1 , (3.27) g u εn −→ χ weakly ∗ in L ∞ 0,T;L (p+1) (Ω) , (3.28) h t,u εn −→ ξ weakly in L 2 0,T;L 2 (Ω) , (3.29) ρ ε u εn −→ ρ ε u ε weakly ∗ in L q 0,T;L 2 (Ω) . (3.30) Since V L 2 (Ω)andV L 2 (Γ 1 ), then by the Aubin-Lions compactness lemma [10,The- orem 5.1], we get from (3.25)and(3.26)that,asn →∞, u εn −→ u ε strongly in L 2 0,T;L 2 (Ω) and a.e. on (0,T) ×Ω, (3.31) u εn −→ u ε strongly in L 2 0,T;L 2 Γ 1 and a.e. on (0,T) ×Γ 1 , u εn −→ u ε strongly in L 2 0,T;L 2 (Ω) and a.e. on (0,T) ×Ω, (3.32) u εn −→ u ε strongly in L 2 0,T;L 2 Γ 1 and a.e. on (0,T) ×Γ 1 , (3.33) From ( 3.25), (3.26), and (3.30), we acquire, as n →∞, u εn ,ω −→ u ε ,ω weakly in L 2 [0,T], (3.34) u εn ,ω −→ u ε ,ω weakly in L 2 [0,T], ρ ε u εn ,ω −→ ρ ε u ε ,ω weakly in L q [0,T]. (3.35) Note that W 1,2 [0,T] C[0,T]andW 1,q C[0,T], from (3.35), we get that u εn (0),ω −→ u ε (0),ω , u εn (0),ω −→ u ε (0),ω , (3.36) and hence u ε (0) =u 0 in V, u ε (0) =u 1 in L 2 (Ω). (3.37) Now letting n →∞in (3.3) and using (3.25)–(3.37), we acquire T 0 ρ ε u ε ,v + ∇ u ε ,∇v + ∇ u ε ,∇v +(γ,v) Γ 1 −(χ,v) dt = T 0 t 0 K(t −τ)(ξ,v) Γ 1 dτ dt (3.38) for any v ∈ V. [...]... 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(4.13) Applying the assumption (A1 ) with q = 2, the H¨ lder inequality, the trace embedding o theorem, and the Young inequality, we have − f u (t) ,u(t) Γ1 ≤ k2 u (t) u(t) Γ1 Γ1 2 ≤ k2 μ4 ∇u (t) 2 2 + 1 2 ∇u(t) 4 (4.14) Combining (4.5), (4.10)–(4.14), and the assumptions (A2 ) and (A4 ), we get m (t) ≤ −E(t) 3 ρ 2 2 ∞ μ1 2 + k2 μ4 + 1 2 ∇u (t) 2 + (K u)(t) (4.15) 14 Global solutions for a nonlinear hyperbolic . M.Aassila,M.M.Cavalcanti,andV.N.DomingosCavalcanti,Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,Calculusof Variations and Partial Differential Equations 15 (2002), no Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, Nonlinear Differential Equations and Applications 8 (2001), no. 2, 157–171. 16 Global solutions for a nonlinear hyperbolic. GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC EQUATION WITH BOUNDARY MEMORY SOURCE TERM FUQIN SUN AND MINGXIN WANG Received 21 January 2005; Accepted 17 August 2005 We study a nonlinear hyperbolic