Global existence of weak solutions for strongly damped wave equations with nonlinear boundary conditions and balanced potentials
aa r X i v : . [ m a t h . A P ] D ec GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR STRONGLY DAMPED WAVEEQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS AND BALANCEDPOTENTIALS
JOSEPH L. SHOMBERG
Department of Mathematics and Computer Science, Providence College, Providence, Rhode Island02918, USA, [email protected]
Abstract.
We demonstrate the global existence of weak solutions to a class of semilinear stronglydamped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. Our work as-sumes ( − ∆ W ) θ ∂ t u with θ ∈ [ ,
1) and where ∆ W is the Wentzell-Laplacian. Hence, the associated linearoperator admits a compact resolvent. A balance condition is assumed to hold between the nonlinearitydefined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary(supercritical) polynomial growth on each potential, as well as mixed dissipative/anti-dissipative be-havior. Moreover, the nonlinear function defined on the interior of the domain is assumed to be only C . Contents
1. Introduction 12. Formulation of the model problem 33. Global existence 6Appendix A. 9Acknowledgments 10References 101.
Introduction
Our aim in this article is to show the global existence of global weak solutions to the fractional stronglydamped wave equation with nonlinear hyperbolic dynamic boundary conditions. We establish the globalexistence of weak solutions under a balance condition imposed on the nonlinear terms. This conditionis motivated by [20, Lemma 3.1]. In the present article, both nonlinearities are allowed supercriticalpolynomial growth. Special attention is given to obtaining the compact resolvent for the associatedlinear operator which contains (fractional) Wentzell-Laplacians.Let Ω be a bounded domain in R with smooth boundary Γ := ∂ Ω. Throughout we assume θ ∈ [ , ω ∈ (0 ,
1] and α ∈ (0 , u = u ( t, x ), ∂ t u − ω ∆ θ ∂ t u + ∂ t u − ∆ u + u + f ( u ) = 0 in (0 , ∞ ) × Ω , (1.1) ∂ t u + ω∂ θ n ∂ t u + ∂ n u − αω ∆ Γ ∂ t u + ∂ t u − ∆ Γ u + u + g ( u ) = 0 on (0 , ∞ ) × Γ . (1.2)Additionally, we impose the initial conditions u (0 , x ) = u ( x ) and ∂ t u (0 , x ) = u ( x ) at { } × Ω , (1.3) Date : December 27, 2018.2010
Mathematics Subject Classification.
Primary: 35L71, 35L20; Secondary: 35Q74, 74H40.
Key words and phrases.
Nonlinear hyperbolic dynamic boundary condition, semilinear strongly damped wave equation,balance condition, global existence, weak solution. and u | Γ (0 , x ) = γ ( x ) and ∂ t u | Γ (0 , x ) = γ ( x ) at { } × Γ . (1.4)Above, ∆ Γ denotes the Laplace-Beltrami operator (cf. e.g. [6]).We assume f ∈ C ( R ) and g ∈ C ( R ) satisfy the sign conditionslim inf | s |→∞ f ( s ) s > − M , g ′ ( s ) ≥ − M , ∀ s ∈ R , (1.5)for some M , M >
0, and the growth assumptions, for all s ∈ R , | f ( s ) | ≤ ℓ (1 + | s | r − ) , | g ( s ) | ≤ ℓ (1 + | s | r − ) , (1.6)for some positive constants ℓ and ℓ , and where r , r ≥
2. In addition, we assume there exists ε ∈ (0 , ω )so that the following balance condition holds,lim inf | s |→∞ f ( s ) s + | Γ || Ω | g ( s ) s − C | Γ | ε | Ω | | g ′ ( s ) s + g ( s ) | | s | r > , (1.7)for r ≥ max { r , r − } , where C Ω > k u − h u i Γ k L (Ω) ≤ C Ω k∇ u k L (Ω) , h u i Γ := 1 | Γ | Z Γ tr D ( u ) dσ, (1.8)for all u ∈ H (Ω).Let us provide further context for the balance condition (1.7) in our setting (also see [20] and [12] forother settings). Suppose that for | y | → ∞ , both the internal and boundary functions satisfy the following:lim | y |→∞ f ( y ) | y | r − = ( r − c f , lim | y |→∞ g ′ ( y ) | y | r − = ( r − c g , for some constants c f , c g ∈ R \ { } . In particular, there holds f ( y ) y ∼ c f | y | r , g ( y ) y ∼ c g | y | r as | y | → ∞ . For the case of bulk dissipation (i.e., c f >
0) and anti-dissipative behavior at the boundary Γ (i.e., c g < r > max { r , r − } . Furthermore,if 2 < r < r −
1) = r and c f > ε (cid:18) C Ω | Γ | c g r | Ω | (cid:19) , for some ε ∈ (0 , ω ), then (1.7) is again satisfied. In the case when f and g are sublinear (i.e., r = r = 2in (1.6)), the condition (1.7) is also automatically satisfied provided that (cid:18) c f + | Γ || Ω | c g (cid:19) > ε (cid:18) C Ω | Γ | c g | Ω | (cid:19) for some ε ∈ (0 , ω ). Notation and conventions.
Let us introduce some notation and conventions that are used through-out the article. Norms in the associated space are clearly denoted k · k B where B is the correspondingBanach space. We use the notation ( · , · ) H to denote the inner-product on the Hilbert space H . The dualproduct on H ∗ × H is denoted h· , ·i H ∗ × H . The notation h· , ·i is also used to denote the product on thephase space and various other vectorial function spaces. Denote by ( u, v ) tr the vector-valued function (cid:0) uv (cid:1) . In many calculations, functional notation indicating dependence on the variable t is dropped; forexample, we will write u in place of u ( t ). Throughout the article, C > generic constantwhich may depend on various structural parameters such as | Ω | , | Γ | , M , M , etc, and these constantsmay even change from line to line. Furthermore, Q : R + → R + will be a generic monotonically increasingfunction whose specific dependance on other parameters will be made explicit on occurrence. All of theseconstants/quantities are independent of the perturbation parameters θ, α and ω. Outline of the article.
In the next section we establish the variational formulation of Problem P and define weak solutions. A proof of the existence of global weak solutions is developed in Section 3.Because of the nature of the balance condition, a continuous dependence type estimate is not available.The article continues with some remarks on this difficulty and plans for possible further research. An TRONGLY DAMPED WAVE EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS 3 appendix contains some explicit characterizations for the fractional Wentzell-Laplacian used throughoutthe article, as well as a certain compact embedding result that we need to draw upon.2.
Formulation of the model problem
In this section we first recall the Wentzell-Laplacian defined on vectorial Hilbert spaces. (For thiswe largely refer to [1, Section 2] and [10, Section 2 and Appendix].) Following this, we give the basicfunctional setup in order to formulate the model problem. We also provide various results pertaining tothe problem.To begin, let λ Ω > λ Ω Z Ω u dx ≤ Z Ω ( |∇ u | + u ) dx. (2.1)We will also rely on the Laplace-Beltrami operator − ∆ Γ on the surface Γ . This operator is positive definiteand self-adjoint on L (Γ) with domain D (∆ Γ ). The Sobolev spaces H s (Γ), for s ∈ R , may be defined as H s (Γ) = D ((∆ Γ ) s/ ) when endowed with the norm whose square is given by, for all u ∈ H s (Γ), k u k H s (Γ) := k u k L (Γ) + (cid:13)(cid:13)(cid:13) ( − ∆ Γ ) s/ u (cid:13)(cid:13)(cid:13) L (Γ) . (2.2)On the boundary, let λ Γ > λ Γ Z Γ u dσ ≤ Z Γ (cid:0) |∇ Γ u | + u (cid:1) dσ. (2.3)Next, recall that Ω is a bounded domain of R with boundary Γ, to which we now assume is of class C .To this end, consider the space X = L (Ω , dµ ) , where dµ = dx | Ω ⊕ dσ is such that dx denotes the Lebesguemeasure on Ω and dσ denotes the natural surface measure on Γ. Then X = L (Ω , dx ) ⊕ L (Γ , dσ ) maybe identified by the natural norm k u k X = Z Ω | u ( x ) | dx + Z Γ | u ( x ) | dσ. Moreover, if we identify every u ∈ C (Ω) with U = ( u | Ω , u | Γ ) tr ∈ C (Ω) × C (Γ), we may also define X tobe the completion of C (Ω) with respect to the norm k · k X . Thus, in general, any function u ∈ X willbe of the form u = (cid:0) u u (cid:1) with u ∈ L (Ω , dx ) and u ∈ L (Γ , dσ ). It is important to note that there neednot be any connection between u and u . From now on, the inner product in the Hilbert space X willbe denoted by h· , ·i X . Now we recall that the Dirichlet trace map tr D : C ∞ (Ω) → C ∞ (Γ) , defined by tr D ( u ) = u | Γ extends to a linear continuous operator tr D : H r (Ω) → H r − / (Γ) , for all r > /
2, which isonto for 1 / < r < / . This map also possesses a bounded right inverse tr − D : H r − / (Γ) → H r (Ω) suchthat tr D ( tr − D ψ ) = ψ, for any ψ ∈ H r − / (Γ). We can thus introduce the subspaces of H r (Ω) × H r − / (Γ)and H r (Ω) × H r (Γ), respectively, by V r := { U = ( u, γ ) ∈ H r (Ω) × H r − / (Γ) : tr D ( u ) = γ } , (2.4) V r := { U = ( u, γ ) ∈ V r : tr D ( u ) = γ ∈ H r (Γ) } , for every r > / , and note that V r , V r are not product spaces. However, we do have the following denseand compact embeddings V r ⊂ V r , for any r > r > / V r ⊂ V r ). Naturally, the norm on the spaces V r , V r are defined by k U k V r := k u k H r (Ω) + k γ k H r − / (Γ) , k U k V r := k u k H r (Ω) + k γ k H r (Γ) . (2.5)Here we consider the basic (linear) operator associated with the model problem (1.1)-(1.4), the so-calledWentzell-Laplacian. Let ∆ W (cid:18) u u (cid:19) := (cid:18) ∆ u − u − ∂ n u + ∆ Γ u − u (cid:19) , (2.6)with D (∆ W ) := (cid:26) U = (cid:18) u u (cid:19) ∈ V : − ∆ u ∈ L (Ω) , ∂ n u − ∆ Γ u ∈ L (Γ) (cid:27) . (2.7)By, for example, [10, see Appendix and in particular Theorem 5.3], the operator (∆ W , D (∆ W )) is self-adjoint and strictly positive operator on X , and the resolvent operator ( I + ∆ W ) − ∈ L ( X ) is compact. J. L. SHOMBERG
Since Γ is of class C , then D (∆ W ) = V . Indeed, the map L : U ∆ W U, as a mapping from V into X = L (Ω) × L (Γ) , is an isomorphism, and there exists a positive constant C ∗ , independent of U = ( u, γ ) tr , such that, for all U ∈ V , C − ∗ k U k V ≤ k L ( U ) k X ≤ C ∗ k U k V , (2.8)(cf. Lemma 2.1, see also [7]).The following basic elliptic estimate is taken from [11, Lemma 2.2]. Lemma 2.1.
Consider the linear boundary value problem, (cid:26) − ∆ u = p in Ω , − ∆ Γ u + ∂ n u + u = p on Γ . (2.9) If ( p , p ) ∈ H s (Ω) × H s (Γ) for s ≥ and s + N , then the following estimate holds for some constant C > , k u k H s +2 (Ω) + k u k H s +2 (Γ) ≤ C (cid:0) k p k H s (Ω) + k p k H s (Γ) (cid:1) . (2.10)We also recall the following basic inequality which gives interior control over some boundary terms (cf.[9, Lemma A.2]). Lemma 2.2.
Let s > and u ∈ H (Ω) . Then, for every ε > , there exists a positive constant C ε ∼ ε − such that, k u k sL s (Γ) ≤ ε k∇ u k L (Ω) + C ε ( k u k γL γ (Ω) + 1) , (2.11) where γ = max { s, s − } . We refer the reader to more details to e.g., [5], [7] and [13] and the references therein.Finally, since the operator ∆ W with domain D (∆ W ) is positive and self-adjoint on X , we may definefractional powers of ∆ W (see Appendix A). Indeed, with θ ∈ [ , α ∈ (0 ,
1] and ω ∈ (0 , θW (cid:18) u u (cid:19) := (cid:18) ∆ θ u − u − ∂ θ n u + ∆ Γ u − u (cid:19) and ∆ θ,α,ωW (cid:18) u u (cid:19) := (cid:18) ω ∆ θ u − u − ω∂ θ n u + αω ∆ Γ u − u (cid:19) with domain D (∆ θ,α,ωW ) := (cid:26) U = (cid:18) u u (cid:19) ∈ V : − ω ∆ θ u ∈ L (Ω) , ω∂ θ n u − αω ∆ Γ u ∈ L (Γ) (cid:27) . (2.12)Hence, ∆ θ, , W = ∆ θW . The fractional flux ∂ θ n are defined as follows. Consider ∂ n u = ∇ u · n , and recall ∂ n u ∈ L (Γ) whenever u ∈ H / (Ω). So we can define ∂ θ n u = ∇ θ/ W u · n when u ∈ H + θ (Ω) guaranteeingthe fractional flux ∂ θ n u ∈ L (Γ) . (These fractional flux operators are explicitly written in Appendix A.)Moving toward the linear operator associated with the model problem (1.1)-(1.4) Let U = ( u , u ) ∈ V and V = ( v , v ) ∈ X , and let X = ( U, V ). Motivated by [4], we define the unbounded linear operator A θ,α,ω written as A θ,α,ω X := (cid:18) I × ∆ W ∆ θ,α,ωW (cid:19) (cid:18) UV (cid:19) = (cid:18) V ∆ W U + ∆ θ,α,ωW V (cid:19) = (cid:18) V ∆ θ, , W (∆ − θ, , W U + ∆ ,α,ωW V ) (cid:19) with domain D ( A θ,α,ω ) := (cid:26) X = (cid:18) UV (cid:19) ∈ V × X : ∆ − θ, , W U + ∆ ,α,βW V ∈ D (∆ θ, , W ) (cid:27) . By [16, Theorem 3.1 (a)], the resolvent ( I × + A θ,α,ω ) − ∈ L ( V × X ) is compact. Hence, we cansupport the local existence of weak solutions (defined below) with a Galerkin method.Next we define the nonlinear mapping on V × X given by F ( U ) := (cid:18) − f ( u ) (cid:19) , G ( U ) := (cid:18) − g ( γ ) (cid:19) , (2.13) TRONGLY DAMPED WAVE EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS 5 and F ( X ) := (cid:18) F ( U ) G ( U ) (cid:19) = − f ( u )0 − g ( γ ) for U ∈ V . Due to the two embeddings, H (Ω) ֒ → L s (Ω), s ∈ [1 , H (Γ) ֒ → L s (Ω), s ∈ [1 , ∞ ), one canshow that when r ∈ [1 ,
3] in (1.6), then F : V × X → V × X is locally Lipschitz (indeed, cf. e.g. [14,Lemma 2.6]). With r ≥ e V s,r = (cid:8) U = ( u, γ ) tr ∈ [ H s (Ω) ∩ L r (Ω)] × H s (Γ) : tr D ( u ) = γ (cid:9) with the canonical norm whose square is given by k U k e V s,r := k u k H s (Ω) + k u k r L r (Ω) + k γ k H s (Γ) , and also set H := e V ,r × X . The space H is Hilbert with the norm whose square is given by, for X = ( U, V ) ∈ H , kX k H := k U k e V ,r + k V k X = k u k H (Ω) + k u k r L r (Ω) + k v k L (Ω) + k γ k H (Γ) + k δ k L (Γ) = (cid:16) k∇ u k L (Ω) + k u k L (Ω) (cid:17) + k u k r L r (Ω) + k v k L (Ω) + (cid:16) k∇ Γ γ k L (Γ) + k γ k L (Γ) (cid:17) + k δ k L (Γ) . The space H is our weak energy phase space. Moreover, given X = ( U , U ) ∈ H = e V ,r × X , theabstract formulation of Problem P takes the form ddt X ( t ) = A θ,α,ω X ( t ) + F ( X ( t )) t > , X (0) = X . We can now introduce the variational formulation of Problem P . Definition 2.3.
Let θ ∈ [ , , α ∈ (0 , and ω ∈ (0 , . Let
T > and X = ( U , U ) ∈ H . A function X ( t ) = ( U ( t ) , ∂ t U ( t )) = ( u ( t ) , u | Γ ( t ) , ∂ t u ( t ) , ∂ t u | Γ ( t )) satisfying U ∈ L ∞ (0 , T ; V ) , (2.14) ∂ t U ∈ L ∞ (0 , T ; X ) , (2.15) √ ω∂ t u ∈ L (0 , T ; H θ (Ω)) , (2.16) ∂ t U ∈ L ∞ (0 , T ; ( V ) ∗ ) , (2.17) for almost all t ∈ (0 , T ] is called a weak solution to Problem P with initial data X if the followingidentities hold almost everywhere on [0 , T ] , and for all Ξ = (Ξ , Ξ ) ∈ V × V : ddt hX ( t ) , Ξ i V − ×V = hA θ,α,ω X ( t ) , Ξ i H + hF ( X ( t )) , Ξ i H . (2.18) Also, the initial conditions (1.3) - (1.4) hold in the L -sense; i.e., hX (0) , Ξ i H = hX , Ξ i H , for every Ξ ∈ V × V . (2.19) We say X ( t ) = ( U ( t ) , ∂ t U ( t )) is a global weak solution of Problem P if it is a weak solution on [0 , T ] , for any T > . Remark . Observe that we are solving a more general problem because γ and γ , from U and U respectively, may be taken to be initial data independent of u and ∂ t u . However, if ∂ t u ( t ) ∈ H s (Ω), forall t > s > /
2, then γ t ( t ) = ∂ t u | Γ ( t ). J. L. SHOMBERG Global existence
Theorem 3.1.
Let X = ( U , U ) ∈ H satisfy kX k H ≤ R for some R > . Then there exists a globalweak solution to Problem P satisfying the additional regularity, √ αω∂ t u ∈ L (0 , T ; H (Γ)) , (3.1) Proof.
Step 1. (An a priori estimate.) In (2.18) take Ξ = ( ∂ t U, ∂ t U ) to find the differential identity12 ddt (cid:8) k ∂ t U k X + k U k V + 2( F ( u ) , L (Ω) + 2( G ( u ) , L (Γ) (cid:9) + ω k∇ θ ∂ t u k L (Ω) + k ∂ t u k L (Ω) + αω k∇ ∂ t u k L (Γ) + k ∂ t u k L (Γ) = 0 . (3.2)Using (1.6) and setting ˜ F ′ = f and ˜ G ′ = g , a simple integration by parts on (1.5) shows, for all u ∈ H (Ω),and γ ∈ H (Γ) , ( ˜ F ( u ) , L (Ω) ≥ ( f ( u ) , u ) L (Ω) + M k u k L (Ω) (3.3)and ( ˜ G ( γ ) , L (Γ) ≥ ( g ( γ ) , γ ) L (Γ) + M k γ k L (Γ) . (3.4)To bound the products on the right-hand sides of (3.3) and (3.4) from below, we utilize (1.7). Following[9, (2.22)], [12, (3.34)] and [20, (3.11)], we estimate the products as( f ( u ) , u ) L (Ω) + ( g ( u ) , u ) L (Γ) = Z Ω (cid:18) f ( u ) u + | Γ || Ω | g ( u ) u (cid:19) dx − | Γ || Ω | Z Ω (cid:18) g ( u ) u − | Γ | Z Γ g ( u ) u d σ (cid:19) dx, (3.5)whereby we exploit the Poincar´e inequality (1.8) and Young’s inequality to see that, for all ε > | Γ || Ω | Z Ω (cid:18) g ( u ) u − | Γ | Z Γ g ( u ) udσ (cid:19) dx ≤ C Ω | Γ || Ω | Z Ω |∇ ( g ( u ) u ) | dx = C Ω | Γ || Ω | Z Ω |∇ u ( g ′ ( u ) u + g ( u )) | dx ≤ ε k∇ u k L (Ω) + C | Γ | ε | Ω | Z Ω | g ′ ( u ) u + g ( u ) | dx. (3.6)Then combining (3.5) and (3.6), and applying assumption (1.7) yields( f ( u ) , u ) L (Ω) + ( g ( u ) , u ) L (Γ) ≥ k u k r L r (Ω) − ε k∇ u k L (Ω) − C δ , (3.7)for some positive constants δ and C δ that are independent of t and ε . Hence, together (3.3) and (3.4)become( F ( u ) , L (Ω) + ( G ( u ) , L (Γ) ≥ k u k r L r (Ω) + M k u k L (Ω) + M k u k L (Γ) − ε k∇ u k L (Ω) − C δ . (3.8)Moreover, (3.8) provides a lower-bound to the functional E ( t ) := k ∂ t U ( t ) k X + k U ( t ) k V + 2( F ( u ( t )) , L (Ω) + 2( G ( u ( t )) , L (Γ) . Integrating the identity (3.2) over (0 , t ), yields E ( t ) + 2 Z t (cid:16) ω k∇ θ ∂ t u ( τ ) k L (Ω) + αω k∇ ∂ t u ( τ ) k L (Γ) + k ∂ t U ( τ ) k X (cid:17) dτ = E (0) . (3.9)We can find an upper-bound on E (0) with (1.6). Evidently2( F ( u (0)) , L (Ω) + 2( G ( u (0)) , L (Γ) ≤ ℓ ( k u (0) k L (Ω) + k u (0) k r L r (Ω) ) + ℓ ( k u (0) k L (Γ) + k u (0) k r L r (Γ) ) . (3.10) TRONGLY DAMPED WAVE EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS 7
Hence, (3.10) and the embedding V ֒ → X show E (0) ≤ k ∂ t u (0) k L (Ω) + k∇ u (0) k L (Ω) + k u (0) k L (Ω) + k ∂ t u (0) k L (Γ) + k∇ u (0) k L (Γ) + k u (0) k L (Γ) + ℓ ( k u (0) k L (Ω) + k u (0) k r L r (Ω) ) + ℓ ( k u (0) k L (Γ) + k u (0) k r L r (Γ) ) ≤ k ∂ t U (0) k X + k U (0) k V + C (cid:16) k U (0) k V + k u (0) k r L r (Ω) + k u (0) k r L r (Γ) (cid:17) . (3.11)Thus (3.9) and (3.11) yield, for all t ≥ , k ∂ t U ( t ) k X + k U ( t ) k V + 2( F ( u ( t )) , L (Ω) + 2( G ( u ( t )) , L (Γ) + 2 Z t (cid:16) ω k∇ θ ∂ t u ( τ ) k L (Ω) + αω k∇ ∂ t u ( τ ) k L (Γ) + k ∂ t U ( τ ) k X (cid:17) dτ ≤ k ∂ t U (0) k X + k U (0) k V + C (cid:16) k U (0) k V + k u (0) k r L r (Ω) + k u (0) k r L r (Γ) (cid:17) ≤ k ∂ t U (0) k X + k U (0) k V + C (cid:16) k U (0) k V + k u (0) k r L r (Ω) + 1 (cid:17) , (3.12)where the last inequality follows from Lemma 2.2.Now we see that, for any T >
0, there hold U ∈ L ∞ (0 , T ; V ) , (3.13) ∂ t U ∈ L ∞ (0 , T ; X ) , (3.14) √ ω∂ t u ∈ L (0 , T ; H θ (Ω)) , (3.15) √ αω∂ t u ∈ L (0 , T ; H (Γ)) , (3.16) F ( u ) ∈ L ∞ (0 , T ; L (Ω)) , (3.17) G ( u ) ∈ L ∞ (0 , T ; L (Γ)) . (3.18)We have found X ∈ L ∞ (0 , T ; H ). Moreover, since U ∈ L ∞ (0 , T ; V ), we have ∆ W U ∈ L ∞ (0 , T ; ( V ) ∗ )and as √ αω∂ t U ∈ L (0 , T ; V ), we also have ∆ θ,α,ωW ∂ t U ∈ L (0 , T ; ( V ) ∗ ). Therefore, after comparingterms in the first equation of (3.2), we see that ∂ t U ∈ L (0 , T ; ( V ) ∗ ) . (3.19)Hence, this justifies our choice of test function in (3.2). With (3.16), we also find (3.1) as claimed. Thisconcludes Step 1.Step 2. (A Galerkin basis.) According to Section 2, for each θ ∈ [ , , the operator A θ,α,ω admitsa system of eigenfunctions Ψ θ,α,ωi = ( ψ θ,α,ω , φ θ,α,ω , ψ θ,α,ω | Γ , φ θ,α,ω | Γ ) satisfying { Ψ θ,α,ωi } ∞ i =1 ⊂ D ( A θ,α,ω ) ∩ ( C (Ω) × C (Γ) × C (Ω) × C (Γ)) and A θ,α,ω Ψ θ,α,ωi = Λ i Ψ θ,α,ωi , i = 1 , , . . . , where the eigenvalues Λ i = Λ θ,α,ωi ∈ (0 , + ∞ ) may be put into increasing order and counted according totheir multiplicity to form a diverging sequence. This means the pair (Λ i , Ψ i ), Ψ i = Ψ θ,α,ωi is a classicalsolution of the elliptic problem (cid:26) − ∆ ψ i + ψ i + ω ( − ∆) θ φ i + φ i = Λ i ψ i in Ω − αω ∆ Γ φ i | Γ + φ i | Γ − ∆ Γ ψ i | Γ + ψ i | Γ = Λ i ψ i | Γ on Γ . Also due to standard spectral theory, these eigenfunctions form an orthogonal basis in H that is or-thonormal in L (Ω) × L (Ω) × L (Γ) × L (Γ).Let T > n ∈ N , set the spaces H n := span n Ψ θ,α,ω , . . . , Ψ θ,α,ωn o ⊂ H and H ∞ := ∞ [ n =1 H n . Obviously, H ∞ is a dense subspace of H . For each n ∈ N , let P n : H → H n denote the orthogonalprojection of H onto H n . Thus, we seek functions of the form X ( n ) ( t ) = n X i =1 A i ( t )Ψ θ,α,ωi (3.20) J. L. SHOMBERG that will satisfy the associated discretized Problem P n described below. The functions A i are assumedto be (at least) C ((0 , T )) for i = 1 , . . . , n . Precisely, u ( n ) ( t ) = n X i =1 A i ( t ) ψ θ,α,ωi , ∂ t u ( n ) ( t ) = n X i =1 A ′ i ( t ) ψ θ,α,ωi , (3.21)and u ( n ) | Γ ( t ) = n X i =1 A i ( t ) φ θ,α,ωi | Γ , ∂ t u ( n ) | Γ ( t ) = n X i =1 A ′ i ( t ) φ θ,α,ωi | Γ . (3.22)Using semigroup properties of A θ,α,ω , the domain D ( A θ,α,ω ) is dense in H . So to approximate the giveninitial data X ∈ H , we may take X ( n )0 ∈ D ( A θ,α,ω ) such that X ( n )0 → X in H .For T > n ≥
1, the weak formulation of the approximate Problem P n is: tofind X ( n ) given by (3.20) such that, for all X = ( U , V ) ∈ H n , the equation D ∂ t X ( n ) , X E H + D A θ,α,ω X ( n ) , X E H + D P n F (cid:16) X ( n ) (cid:17) , X E H = 0 (3.23)holds for almost all t ∈ (0 , T ), subject to the initial conditions D X ( n ) (0) , X E H = D X ( n )0 , X E H . (3.24)To show the existence of at least one solution to (3.23)-(3.24), we now suppose that n is fixed andwe take X = X ( k ) for some 1 ≤ k ≤ n . Then substituting the discretized functions (3.21)-(3.22) into(3.23)-(3.24), we find a system of ordinary differential equations in the unknowns A k = A k ( t ) on X ( n ) .Also, we recall that D P n F (cid:16) X ( n ) (cid:17) , X ( k ) E H = D F (cid:16) X ( n ) (cid:17) , P n X ( k ) E H = D F (cid:16) X ( n ) (cid:17) , X ( k ) E H . Since f ∈ C ( R ) and g ∈ C ( R ), we may apply Cauchy’s theorem for ODEs to find that there is T n ∈ (0 , T )such that A k ∈ C ((0 , T n )), for 1 ≤ k ≤ n , and (3.23) holds in the classical sense for all t ∈ [0 , T n ]. Thisshows the existence of at least one local solution to the approximate Problem P n and ends Step 2.Step 3. (Boundedness and continuation of approximate maximal solutions.) We begin by noticingthat the a priori estimate (3.12) holds for any approximate solution X ( n ) of Problem P n on the interval[0 , T n ), where T n < T . Thanks to the boundedness of the projector P n , we infer k ∂ t U ( n ) ( t ) k X + k U ( n ) ( t ) k V + 2( F ( u ( n ) ( t )) , L (Ω) + 2( G ( u ( n ) ( t )) , L (Γ) + 2 Z t (cid:16) ω k∇ θ ∂ t u ( n ) ( τ ) k L (Ω) + αω k∇ ∂ t u ( n ) ( τ ) k L (Γ) + k ∂ t U ( n ) ( τ ) k X (cid:17) dτ ≤ k ∂ t U (0) k X + k U (0) k V + C (cid:16) k U (0) k V + k u (0) k r L r (Ω) + k u (0) k r L r (Γ) (cid:17) . (3.25)Since the right-hand side of (3.25) is independent of n and t , every approximate solution may be extendedto the whole interval [0 , T ], and because T > X ( n ) . Thus, U ( n ) ∈ L ∞ (0 , T ; V ) , (3.26) ∂ t U ( n ) ∈ L ∞ (0 , T ; X ) , (3.27) √ ω∂ t u ( n ) ∈ L (0 , T ; H θ (Ω)) , (3.28) √ αω∂ t u ( n ) ∈ L (0 , T ; H (Γ)) , (3.29) F ( u ( n ) ) ∈ L ∞ (0 , T ; L (Ω)) , (3.30) G ( u ( n ) ) ∈ L ∞ (0 , T ; L (Γ)) . (3.31)This concludes Step 3.Step 4. (Convergence of approximate solutions.) We begin this step by applying Alaoglu’s theorem(cf. e.g. [19, Theorem 6.64]) to the uniform bounds (3.26)-(3.31) to find that there is a subsequence of TRONGLY DAMPED WAVE EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS 9 X ( n ) , generally not relabelled, and a function X = ( u, ∂ t u, u | Γ , ∂ t u | Γ ), obeying (3.13)-(3.19), such that as n → ∞ , U ( n ) ⇀ U weakly −∗ in L ∞ (0 , T ; V ) , (3.32) ∂ t U ( n ) ⇀ ∂ t U weakly −∗ in L ∞ (0 , T ; X ) , (3.33) √ ω∂ t u ( n ) ⇀ u weakly in L (0 , T ; H θ (Ω)) , (3.34) √ αω∂ t u ( n ) ⇀ u weakly in L (0 , T ; H (Γ)) , (3.35) ∂ t U ( n ) ⇀ ∂ t U weakly in L (0 , T ; ( V ) ∗ ) . (3.36)Using the above convergences (3.32) and (3.33), as well as the fact that the injection V ֒ → X iscompact, we draw upon the conclusion of the Aubin-Lions Lemma (cf. Lemma A.1) to deduce thefollowing embedding is compact W := { U ∈ L (0 , T ; V ) : ∂ t U ∈ L (0 , T ; X ) } ֒ → L (0 , T ; X ) (3.37)(see, e.g., [22]). Thus, U ( n ) → U strongly in L (0 , T ; X ) , (3.38)and deduce that U ( n ) converges to U , almost everywhere in Ω × (0 , T ). The last strong convergenceproperty is enough to pass to the limit in the nonlinear terms since f, g ∈ C ( R ) (see, e.g., [9, 13]).Indeed, on account of standard arguments (cf. also [5]) we have P n F ( X ( n ) ) ⇀ F ( X ) weakly in L (0 , T ; H ) . (3.39)At this point the convergence properties (3.32)-(3.39) are sufficient to pass to the limit as n → ∞ inequation (3.23). Additionally, we recover (2.18) using standard density arguments. The proof of thetheorem is finished. (cid:3) Concerning uniqueness.
A proof of the following conjecture is needed to show that the weak solutionsto Problem P constructed above depend continuously on initial data, and hence, are unique. Conjecture 3.2.
Let
T > , R > and X = ( U , U ) , X = ( U , U ) ∈ H be such that kX k H ≤ R and kX k H ≤ R . Any two weak solutions, X ( t ) and X ( t ) , to Problem P on [0 , T ] corresponding tothe initial data X and X , respectively, satisfy for all t ∈ [0 , T ] , (cid:13)(cid:13) X ( t ) − X ( t ) (cid:13)(cid:13) H ≤ e Q ( R ) t kX − X k H . (3.40)In order to prove the conjecture, typically one needs to control products of the form( f ( u ) − f ( u ) , ∂ t ¯ u ) L (Ω) and ( g ( u ) − g ( u ) , ∂ t ¯ u ) L (Γ) where u and u are two weak solutions corresponding to (possibly the same) data X = ( U , U ) =( u , γ , u , γ ) and X = ( U , U ) = ( u , γ , u , γ ). A suitable control on k f ( u ) − f ( u ) k L q (Ω) ,for example, is readily available when we assume (1.6) with r ∈ [1 ,
3] (cf. [14, Lemma 2.6])), but this isno longer valid when we assume r ≥ generalized semiflow in the sense of [2, 3] exists. Under certain conditions, such generalizedsemiflows admit global attractors which have similar properties to their well-posed counterparts (cf. [15]). Appendix
A.As introduced in Section 2, the Wentzell-Laplacian ∆ W on X with domain D (∆ W ) := { U = ( u, γ ) tr ∈ V : − ∆ u ∈ L (Ω) , ∂ n u = − γ + ∆ Γ γ ∈ L (Γ) , γ = tr D ( u ) } . is positive, self-adjoint and has compact resolvent [1]. From [18, Theorem A.37 (Spectral Theorem) and(A.28)], we know that for each θ ∈ [ , D (∆ θW ) = U = ( u, γ ) tr ∈ D (∆ W ) : ∞ X j =1 Λ θj | ( U, W j ) | < ∞ where ∆ W W j = Λ j W j , and the sequence (Λ j ) ∞ j =1 contains real, strictly positive eigenvalues, each having finite multiplicity, whichcan be ordered into a nondecreasing sequence in whichlim j →∞ Λ j = + ∞ . We mention some results [10, Theorem 5.2 (c)] concerning the regularity of the eigenfunctions W j . If Γis Lipschitz, then every eigenfunction W j ∈ V , and in fact W j ∈ C (Ω) ∩ C ∞ (Ω), for every j . If Γ is ofclass C , then every eigenfunction W j ∈ V ∩ C (Ω) for every j .Here we remind the reader how we define the fractional powers of the Wentzell-Laplacian with a Fourierseries. Thus, ∆ θW U = ∞ X j =1 Λ θj ( U, W j ) W j , and we can rely on (cf. [8, (2.6)]) to define the fractional flux, where,∆ θ/ W U = ∇ θW U = N X i =1 ∂ θ U∂x θi e i , and d θ Udx θ = 1Γ(1 − θ ) ddx Z x −∞ ( x − y ) − θ U ( y ) dy. The following result is the classical Aubin-Lions Lemma, reported here for the reader’s convince (cf.[17], and, e.g. [21, Lemma 5.51] or [23, Theorem 3.1.1]).
Lemma A.1.
Let
X, Y, Z be Banach spaces where Z ← ֓ Y ← ֓ X with continuous injections, the secondbeing compact. Then the following embeddings are compact: W := { χ ∈ L (0 , T ; X ) , ∂ t χ ∈ L (0 , T ; Z ) } ֒ → L (0 , T ; Y ) . Acknowledgments
The author is grateful to the anonymous referees for their careful reading of the manuscript and fortheir helpful comments and suggestions.
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