The existence and properties of the solution of the wave equation on graph
aa r X i v : . [ m a t h . A P ] A ug THE EXISTENCE AND PROPERTIES OF THE SOLUTION OF THEWAVE EQUATION ON GRAPH
YONG LIN AND YUANYUAN XIE
Abstract.
Let G = ( V, E ) be a locally finite weighted graph, and Ω ⊆ V be a boundeddomain such that Ω ◦ = ∅ . In this paper, we study the following wave equation ∂ t u ( t, x ) − ∆ Ω u ( t, x ) = f ( t, x ) , ( t, x ) ∈ (0 , ∞ ) × Ω ◦ ,u (0 , x ) = g ( x ) , x ∈ Ω ◦ ,∂ t u (0 , x ) = h ( x ) , x ∈ Ω ◦ ,u ( t, x ) = 0 , ( t, x ) ∈ [0 , ∞ ) × ∂ Ω , where Ω ◦ and ∂ Ω denote the interior and the boundary of Ω respectively, and ∆ Ω denotesthe Dirichlet Laplacian on Ω ◦ . Using the Rothe’s method, we prove that the above waveequation has a unique solution. Also we prove that the above wave equation has infinite wavepropagation speed, if f (0 , x ) is a negative function on Ω ◦ . Contents
1. Introduction 12. Preliminaries 32.1. Laplacian on graph 42.2. Green’s formula 52.3. Sobolev embedding theorem on graph 52.4. Spaces involving time 52.5. Dominated Convergence Theorem 52.6. Infinite propagation speed 63. Proof of Theorem 1.1 64. Proof of Theorem 1.2 and Theorem 1.3 18References 21 Introduction
Wave equations have been extensively studied in different fields, such as Euclidean space,Riemannian manifold, fractal, graph and so on. Using ´Asgeirsson’s mean value theorem (see [9,Theorem II.5.28]), Helgason [9] solved the wave equation ∂ t u ( t, x ) = ∆ x u ( t, x ) ,u (0 , x ) = g ( x ) ,∂ t u (0 , x ) = h ( x ) Mathematics Subject Classification.
Key words and phrases.
Rothe’s method, wave equation, graph. on Euclidean spaces R n (see [9, Exercise II.F.1]). In 1998, Evans [2] also studied the wave equa-tion on R n . Meirose [14] discussed two propagation results for the wave group of a subellipticnon-negative self-adjoint second order differential operator on a compact manifold.Lee [12] proved that waves propagate with infinite speed on some p.c.f. fractals. The proofuses the sub-Gaussian heat kernel estimates and the relation between heat equation and waveequation. Ngai et al. [15] studied the wave propagation speed problem on metric measurespaces, emphasizing on self-similar sets that are not postcritically finite. They prove that asub-Gaussian lower hear kernel estimates leads to infinite propagation speed, and also formulateconditions under which a Gaussian upper heat kernel estimate leads to finite propagation speed.Chan et al. [1] studied one dimensional wave equations defined by some fractal Laplacians. Usingthe finite element and central difference methods, they obtained the numerical approximationsof the weak solutions, and proved that the numerical solutions converge to the weak solution.Kuchment [11] presented a brief survey on graph models for wave propagation in thin struc-tures. Friedman and Tillich [3] developed a wave equation for graphs that has many of theproperties of the classical Laplacian wave equation. The wave equation on graph is basedon the edge-based Laplacian. They also gave some applications of wave equation on graphs.Schrader [17] studied the solution of the wave equation on metric graphs, and established thefinite propagation speed.In recent years, the study of equations on graphs has attracted attention of researchers invarious fields. Grigoryan et al. [5, 7], using the mountain pass theorem due to Ambrosetti-Rabinowitz, studied the existence of solutions to Yamabe type equation and some nonlinearequations on graphs, respectively. They [6] also considered the Kazdan-Warner equation ongraph. The proof use the calculus of variations and a method of upper and lower solutions.In [13], Lin and Wu proved the existence and nonexistence of global solutions of the CauchyProblem for ∂ t u = ∆ u + u α with α > G = ( V, E ) be a locally finite weighted graph, where V and E denote the vertex set andthe edge set of G , respectively. Given a non-empty bounded domain Ω ⊆ V , the boundary ofΩ is defined by ∂ Ω := { x ∈ Ω : there exists y ∈ Ω c such that y ∼ x } , (1.1)and the interior of Ω is defined by Ω ◦ := Ω \ ∂ Ω. In this paper, we assume that Ω ◦ = ∅ , andconsider the following initial boundary value problem ∂ t u ( t, x ) − ∆ Ω u ( t, x ) = f ( t, x ) , ( t, x ) ∈ (0 , ∞ ) × Ω ◦ ,u (0 , x ) = g ( x ) , x ∈ Ω ◦ ,∂ t u (0 , x ) = h ( x ) , x ∈ Ω ◦ ,u ( t, x ) = 0 , ( t, x ) ∈ [0 , ∞ ) × ∂ Ω , (1.2) AVE EQUATION ON GRAPH 3 where ∆ Ω is the Dirichlet Laplacian on Ω ◦ . Our main results are as follows: Theorem 1.1.
Let G = ( V, E ) be a locally finite weighted graph, Ω be a bounded domain of V such that Ω ◦ = ∅ , and ∆ Ω defined as in (2.9) . Let M = . If k f ( t, · ) − f ( s, · ) k L (Ω ◦ ) ≤ c | t − s | α for all x ∈ Ω ◦ , t, s ∈ [0 , ∞ ) , and for T > , sup t ∈ [0 ,T ] k f ( t, · ) k L (Ω ◦ ) ≤ e c ( T ) , then the wave equation (1.2) has a unique solution, where α, c, e c ( T ) are some positive constants. It is easy to see that f ( t, x ) = tx β and f ( t, x ) = x β · sin t satisfy the hypotheses of Theorem1.1, where β ∈ R \{ } . Theorem 1.2.
Let G = ( V, E ) be a locally finite weighted graph, Ω be a bounded domain of V such that Ω ◦ = ∅ , and ∆ Ω defined as in (2.9) . Let { ϕ k } Nk =1 be an orthonormal basis of W , (Ω) consisting of the eigenfunctions of − ∆ Ω such that − ∆ Ω ϕ k = λ k ϕ k for k = 1 , . . . , N , where N = ◦ . Let f ( t, x ) = N X k =1 b k ( t ) ϕ k ( x ) , g ( x ) = N X k =1 g k ϕ k ( x ) , h ( x ) = N X k =1 h k ϕ k ( x ) . (1.3) Then the solution of (1.2) is given by u ( t, x ) = N X k =1 − √ λ k Z t sin( p λ k s ) b k ( s ) ds · cos( p λ k t ) ϕ k ( x )+ N X k =1 √ λ k Z t cos( p λ k s ) b k ( s ) ds · sin( p λ k t ) ϕ k ( x )+ N X k =1 g k cos( p λ k t ) ϕ k ( x ) + N X k =1 √ λ k ( h k − b k (0)) sin( p λ k t ) ϕ k ( x ) . Theorem 1.3.
Assume the hypotheses of Theorem 1.2. Let Ω ◦ = { x , . . . , x N } and ϕ k ( x ) satisfying ϕ k ( x j ) = ( , if k = j, , if k = j. (1.4) If g = h = 0 and f (0 , x ) is a negative function on Ω ◦ , then the wave equation (1.2) does nothave a finite wave propagation speed. The rest of the paper is organized as follows. In Section 2, we introduce some definitions,notations and Sobolev embedding theorem on graph. Section 3 is devoted to the proof ofTheorem 1.1. In Section 4, we give the proofs of Theorem 1.2 and Theorem 1.3.2.
Preliminaries
Let G = ( V, E ) be a locally finite graph. We write y ∼ x if xy ∈ E . For any edge xy ∈ E ,we assume that its weight ω xy > ω xy = ω yx . A couple ( V, ω ) is called a weighted graph . Y. LIN AND Y. XIE
Furthermore, let µ : V → R + be a positive finite measure. In this paper, we consider weightedgraphs and assume D µ := max x ∈ V m ( x ) µ ( x ) < ∞ , (2.5)where m ( x ) := P y ∼ x ω xy .2.1. Laplacian on graph.
Let C ( V ) be the set of real functions on V . Define ℓ q ( V, µ ) to bethe space of all ℓ q summable functions u ∈ C ( V ) satisfying(1) k u k ℓ q ( V,µ ) = (cid:16) P x ∈ V | u ( x ) | q µ ( x ) (cid:17) /q < ∞ , if 1 ≤ q < ∞ , and(2) k u k ℓ ∞ ( V,µ ) = sup x ∈ V | u ( x ) | < ∞ , if q = ∞ .We define Z V u dµ = X x ∈ V u ( x ) µ ( x ) for u ∈ ℓ ( V, µ ) , (2.6)and ( u, v ) = X x ∈ V u ( x ) v ( x ) µ ( x ) for u, v ∈ ℓ ( V, µ ) . (2.7)It is obvious that ℓ ( V, µ ) is a Hilbert space under the standard inner product (2.7).The µ -Laplacian ∆ of u is defined as follows:∆ u ( x ) = 1 µ ( x ) X y ∼ x ω xy (cid:0) u ( y ) − u ( x ) (cid:1) for any u ∈ C ( V ) . It is well known that D µ < ∞ is equivalent to the µ -Laplacian ∆ being bounded on ℓ q ( V, µ ) forany q ∈ [1 , ∞ ] (see [8]).The associated gradient form is defined byΓ( u, v )( x ) = 12 µ ( x ) X y ∼ x ω xy (cid:0) u ( y ) − u ( x ) (cid:1)(cid:0) v ( y ) − v ( x ) (cid:1) . Write Γ( u ) = Γ( u, u ). We denote the length of its gradient by |∇ u | ( x ) = p Γ( u )( x ) = (cid:16) µ ( x ) X y ∼ x ω xy (cid:0) u ( y ) − u ( x ) (cid:1) (cid:17) / . (2.8)Given a non-empty bounded domain Ω ⊆ V , let ∂ Ω defined as in (1.1), and Ω ◦ = Ω \ ∂ Ω. Forany u ∈ C (Ω ◦ ), the Dirichlet Laplacian ∆ Ω on Ω ◦ is defined as follows: first we extend u to thewhole V by setting u ≡ ◦ and then set∆ Ω u = (∆ u ) | Ω ◦ . Then ∆ Ω u ( x ) = 1 µ ( x ) X y ∼ x ω xy (cid:0) u ( y ) − u ( x ) (cid:1) for any x ∈ Ω ◦ , (2.9)where u ( y ) = 0 whenever y / ∈ Ω ◦ . It is easy to see that − ∆ Ω is a positive self-adjoint operator(see [4, 18]). AVE EQUATION ON GRAPH 5
Green’s formula.Lemma 2.1. [4, Lemma 4.1] For any u, v ∈ C (Ω ◦ ) , we have Z Ω ◦ ∆ Ω u · v dµ = − Z Ω Γ( u, v ) dµ. Sobolev embedding theorem on graph.
Let Ω be a domain of V , ∂ Ω and Ω ◦ beits boundary and interior, respectively. Let W , (Ω) be defined as a space of all functions u : V → R satisfying k u k W , (Ω) = (cid:16) Z Ω ( |∇ u | + | u | ) dµ (cid:17) / < ∞ . (2.10)Denote C (Ω) be a set of all functions u : Ω → R with u = 0 on ∂ Ω. We denote W , (Ω) bethe completion of C (Ω) under the norm (2.10). If we further assume that Ω is bounded, thenΩ is a finite set. Note that if Ω is bounded, then the dimension of W , (Ω) is finite, and so wehave the following result: Theorem 2.2. ( [5]) Let G = ( V, E ) be a locally finite graph, Ω be a bounded domain of V such that Ω ◦ = ∅ . Then W , (Ω) is embedding in L q (Ω) for all ≤ q ≤ ∞ . In particular, thereexists a positive constant C depending only on Ω such that (cid:16) Z Ω | u | q dµ (cid:17) /q ≤ C (cid:16) Z Ω |∇ u | dµ (cid:17) / (2.11) for all ≤ q ≤ ∞ and for all u ∈ W , (Ω) . Moreover, W , (Ω) is pre-compact, namely, if u k isbounded in W , (Ω) , then up to a subsequence, there exists some u ∈ W , (Ω) such that u k → u in W , (Ω) . Spaces involving time.
Let X be a real Banach space, with norm k k , and T > Definition 2.3.
A function u ( t, x ) : [0 , T ] → X is strongly measurable if there exist simplefunctions s k : [0 , T ] → X such that s k ( t, x ) → u ( t, x ) for a.e. ≤ t ≤ T. Definition 2.4.
The space L q ([0 , T ]; X ) consists of all strongly measurable functions u ( t, x ) :[0 , T ] → X with k u k L q ([0 ,T ]; X ) := (cid:16) Z T k u ( t, · ) k q dt (cid:17) /q < ∞ for ≤ q < ∞ , and k u k L ∞ ([0 ,T ]; X ) := ess sup ≤ t ≤ T k u ( t, · ) k < ∞ . Dominated Convergence Theorem.Theorem 2.5.
Assume the functions { u k } ∞ k =1 are integrable and u k → u a.e. Suppose also | u k | ≤ ¯ u a.e., for some summable function ¯ u . Then R R n u k dx → R R n u dx . Y. LIN AND Y. XIE
Infinite propagation speed.Definition 2.6.
A function u = u ( t, x ) is said to be a solution of wave equation (1.2) in [0 , ∞ ) × Ω , if ∂ t u ( t, x ) exists in (0 , ∞ ) × Ω ◦ and u ( t, x ) satisfies (1.2) . Definition 2.7.
We say that the wave equation (1.2) has infinite propagation speed on Ω , ifthe solution of (1.2) , u ( t, x ) , satisfies the following condition : For a fixed vertex x ∈ Ω ◦ , u (0 , x ) = 0 for any x ∈ Ω ◦ within a positive distance to x . Butfor any small ǫ > , there is t ∈ (0 , ǫ ) such that u ( t, x ) > . Proof of Theorem 1.1
Proof of Theorem 1.1. Uniqueness. If u and u both satisfy (1.2), then v := u − u solves ∂ t u − ∆ Ω u = 0 , ( t, x ) ∈ (0 , ∞ ) × Ω ◦ ,u (0 , x ) = 0 , x ∈ Ω ◦ ,∂ t u (0 , x ) = 0 , x ∈ Ω ◦ ,u ( t, x ) = 0 , ( t, x ) ∈ [0 , ∞ ) × ∂ Ω . Let e ( t ) = Z Ω |∇ u ( t, x ) | dµ + Z Ω ◦ | ∂ t u ( t, x ) | dµ for t ∈ [0 , ∞ ) . It is easy to see that e (0) = R Ω |∇ u (0 , x ) | dµ + R Ω ◦ | ∂ t u (0 , x ) | dµ = 0. Hence e ′ + (0) = lim t → + e ( t ) − e (0) t = lim t → + R Ω |∇ u ( t, x ) | dµ + R Ω ◦ | ∂ t u ( t, x ) | dµt = lim t → + P x ∈ Ω |∇ u ( t, x ) | µ ( x ) + P x ∈ Ω ◦ | ∂ t u ( t, x ) | µ ( x ) t (by (2.6))= X x ∈ Ω lim t → + |∇ u ( t, x ) | t µ ( x ) + X x ∈ Ω ◦ lim t → + | ∂ t u ( t, x ) | t µ ( x )= X x ∈ Ω lim t → + · ∇ u ( t, x ) · ∂ t ( ∇ u ( t, x )) · µ ( x ) + X x ∈ Ω ◦ lim t → + · ∂ t u ( t, x ) · ∂ t u ( t, x ) · µ ( x )= 0 , AVE EQUATION ON GRAPH 7 the last equality we use the fact that ∇ u (0 , x ) = 0 for x ∈ Ω and ∂ t u (0 , x ) = 0 for x ∈ Ω ◦ .Moreover, for t ∈ (0 , ∞ ), e ′ ( t ) = 2 Z Ω ∇ u · ∂ t ( ∇ u ) dµ + 2 Z Ω ◦ ∂ t u · ∂ t u dµ = 2 Z Ω Γ( u, ∂ t u ) dµ + 2 Z Ω ◦ ∂ t u · ∂ t u dµ = − Z Ω ◦ ∆ Ω u · ∂ t u dµ + 2 Z Ω ◦ ∂ t u · ∂ t u dµ = 2 Z Ω ◦ ∂ t u · (cid:0) ∂ t u − ∆ Ω u (cid:1) dµ = 0 , the second equality follows from the fact that ∂ t ( ∇ u ) = ( ∇ u ) − · Γ( u, ∂ t u ). So e ( t ) ≡ constantfor any t ∈ [0 , ∞ ). Combining this with e (0) = 0, we get e ( t ) ≡ t ∈ [0 , ∞ ). That is, e ( t ) = R Ω |∇ u | dµ + R Ω ◦ | ∂ t u | dµ ≡ t ∈ [0 , ∞ ). This leads to ∇ u ( t, x ) ≡ t, x ) ∈ [0 , ∞ ) × Ω and ∂ t u ( t, x ) ≡ t, x ) ∈ [0 , ∞ ) × Ω ◦ , Fixed t ∈ [0 , ∞ ), since ∇ u ( t, x ) ≡ u ( t, x ) ≡ constant for any x ∈ Ω. Fixed x ∈ Ω ◦ , it follows from ∂ t u ( t, x ) ≡ u ( t, x ) ≡ constant for any t ∈ [0 , ∞ ).Combining this with u (0 , x ) = 0 for x ∈ Ω ◦ and u ( t, x ) = 0 for ( t, x ) ∈ [0 , ∞ ) × ∂ Ω, we get u ( t, x ) ≡ t ∈ [0 , ∞ ) and all x ∈ Ω. This completes the proof.
Existence.Step 1.
Choose a fixed number n , for any T >