SELF-SIMILAR MEASURES WITH OVERLAPS
2
WEI TANG AND ZHI-YONG WANG
3
Abstract. The weak well-posedness of strong damping wave equations defined by fractal Laplacians is proved by using Galerkin method. These fractal Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. In general, the struc- ture of self-similar measures with overlap are complicated and intractable. However, some important information about the structure of the three measures above can be obtained.
We make use of these information to set up a framework for one-dimensional measures to discretize the equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.
Contents
1. Introduction 1
2. Preliminaries 6
3. Existence and uniqueness of weak solution 8
4. The finite element method 13
5. Convergence of numerical approximations 16
References 21
1. Introduction
4
Many phenomena in the real world are best modeled by some exotic geometric structures
5
with a non-smooth appearance. The theory of fractals seeks to provide the mathematical
6
framework for such powerful development. In the last decades years, analysis on fractals has
7
shown an explosive development, due to numerous applications to problems arising in various
8
fields, including physics, chemistry and biology. In this paper, we study the strong damping
9
linear wave equations defined by fractal Laplacians associated with a class of self-similar mea-
10
sures with overlaps. Such measures have attracted considerable attentions because of their
11
Date: August 16, 2021.
2010Mathematics Subject Classification. Primary: 28A80, 35L05; Secondary: 74S05, 65L60, 65L20.
Key words and phrases. Fractal; Laplacian; wave equation; self-similar measure with overlaps.
The first author is supported by the NNSF of China ( Grants No. 11901187 and 11771136). The second author is supported by the NNSF of China ( Grant No. 12001183), the Hunan Provincial NSF ( Grant No.
2020JJ5097), and the SRF of Hunan Provincial Education Department (Grant No. 19B117).
1
relation to classical analysis and their numerous unusual properties, such as non-integer spec-
1
tral dimension [27, 29], Sub-Gaussian heat kernel [19], infinite wave propagation speed [30],
2
and so on. Our long term goal is to combine ideas of Strichartz, including the celebrated
3
Strichartz estimates, in a comprehensive study of nonlinear undamped and damping wave
4
equations on fractals and fractalfolds. In our paper, we investigate the solution of the strong
5
damping linear wave equation theoretically, and also provide numerical examples. We do not
6
discuss the nonlinear wave equation directly, but rather develop approximating tools that
7
may help in this study.
8
LetU ⊆Rd,d ≥1, be a bounded open subset, and letµbe a positive finite Borel measure
9
onRdwith supp(µ)⊆U andµ(U)>0. It is known (see, e.g., [20]) thatµdefines a Dirichlet
10
Laplace operator ∆µ, if the following Poincar´e inequality for a measure (PI) holds: There
11
exists some constant C > 0 such that
12
Z
U
|u|2dµ≤C Z
U
|∇u|2dx for all u∈Cc∞(U) (1.1) (see, e.g., [20,25,26]). We remark that ifn= 1, then (PI) holds for any suchµ, and thus ∆µis
13
well-defined. More recently, the operator ∆µhas been studied extensively in connection with
14
fractal measures by authors including Fujita, M. Solomyak, Verbitsky, Naimark, Freiberg,
15
Lobus, Z¨ahle, Bird et al., Huet al., Andrewset al., Guet al., Ngai, Xie, and the first author
16
(see [1–4, 8, 12–17, 19, 20, 26–30, 33, 37] and the references therein). Many of these papers
17
study the spectral asymptotics of ∆µ, while others study the associated wave, heat, and
18
Schr¨odinger equations. In this paper, we study the following strong damping linear wave
19
equation
20
∂t2u−α∆µu−σ∆µ∂tu=f on U×[0, T],
u= 0 on ∂U ×[0, T],
u(0) =g, ∂tu(0) =h on U× {t = 0},
(1.2) where α ≥ 1 and σ ≥ 0 are real parameters, and u(t) is a Hilbert space valued function
21
of t. In particular, if α = 1 and σ = 0, then equation (1.2) becomes general undamped
22
wave equation. In classical case, various forms of damping wave equations on a domain with
23
smooth enough boundary have been studied extensively (see, e.g., [23, 32, 36]). For a class of
24
one-dimensional fractal measures with overlaps, approximations of the solution of undamped
25
linear wave equations have been studied in [3]. Recently, Dekkers et al. studied boundary
26
valued problems for linear and nonlinear strong damping wave equations on arbitrary and
27
fractal domain (see, [6, 7]).
28
The first objective of this paper is to obtain the weak well-posedness result of the strong
29
damping wave equation (1.2) (see Definition 3.1 for definition of a weak solution). The
30
main ingredient we used is the usual Galerkin method (see, [10] for detials). To perform
31
the Galerkin method, we assume that −∆µ has compact resolvent; that is, there exists a
32
complete orthonormal basis {ϕn}∞n=1 of L2(U, µ) such that −∆µϕn = λnϕn for all n ≥ 1,
33
where the eigenvalues satisfy 0< λ1 ≤ · · · ≤λn ≤ λn+1 ≤ · · · with limn→∞λn =∞. Some
34
sufficient conditions for the existence of an orthonormal basis{ϕn}∞n=1 of L2(U, µ) consisting
1
of the eigenfunctions of −∆µ can be found in [20]. We remark that if n = 1, then −∆µ has
2
compact resolvent. See Section 2 for the definition of domE and L2([0, T], L2(U, µ)).
3
Theorem 1.1. Let U ⊆Rd, d≥ 1, be a bounded open subset, and let µ be a positive finite
4
Borel measure on Rd with supp(µ) ⊆ U and µ(U) > 0. Assume that µ satisfies (PI) and
5
−∆µ has compact resolvent. If g ∈domE, h∈ L2(U, µ) and f ∈L2([0, T], L2(U, µ)). Then
6
the strong damping wave equation (1.2) has a unique weak solution.
7
We also obtain some regularity results of the weak solution of equation (1.2)(see Theo-
8
rem 3.3).
9
The second objective of this paper is to study equation (1.2) from a numerical point of
10
view. We first introduce some definitions and notation that will be used. We call a µ-
11
measurable closed and connected subset I of U a cell (in U) if µ(I) > 0. Clearly, if U is
12
connected, then U itself is a cell. Two cells I, J in U are measure disjoint with respect to
13
µ if µ(I ∩J) = 0. Let I ⊆ U be a cell. We call a finite family Pof measure disjoint cells a
14
µ-partition of I if J ⊆I for all J ∈P, and µ(I) = P
J∈Pµ(J). A sequence of µ-partitions
15
(Pm)m≥1 is refining if for any m ≥ 1, each member of Pm+1 is a proper subset of some
16
member of Pm. A sequence of µ-partitions (Pm)m≥1 of U is compatible if (1) (Pm)m≥1 is
17
refining; (2) for any m ≥ 2 and any J ∈ Pm, there exist similitudes (τI)I∈P1 of the form
18
τI(x) = rIx+bI (rI ∈ (0,1), bI ∈ Rd) and positive constants (cI)I∈P1 such that τ(I) ⊆ J
19
and
20
µ|J = X
I∈P1
cI·µ|I◦τI−1. (1.3)
Intuitively, (1.3) means that the µ measure of each cell in Pm for m ≥ 2 can be expressed
21
as a linear combination of {µ(I) : I ∈ P1}. We remark that (1.3) is more general than
22
second-order (self-similar) identities, which were first introduced by Strichartzet al. in [35].
23
In order to discretize (1.2) and obtain numerical approximations of the weak solution, we
24
will assume that there exists a sequence of compatible µ-partitions (Pm)m≥1. Thus the µ
25
measure of each island in the partition can be computed by using (1.3), making it possible
26
to discretizing the strong damping wave equation (1.2). In order to guarantee that the mass
27
matrix that arises in the finite element method is positive definite (see [3, Proposition 3.1]),
28
we assume that µis a measure on R with supp(µ) = [a, b].
29
Let f ≡ 0 in equation (1.2). Multiplying the first equation in (1.2) by v ∈ domE,
30
integrating both sides with respect todµ, and then integrating by parts, we obtain
31
−α Z b
a
(∂xu)v0(x)dx−σ Z b
a
(∂x∂tu)v0(x)dx= Z b
a
(∂t2u)v(x)dµ, u(0) =g, ∂tu(0) =h,
(1.4) where∂xuand ∂tuare the weak partial derivative of u with respect tox and t, respectively.
32
Theorem 1.2. Let µ be a positive finite Borel measure on R with supp(µ)⊆[a, b]. Assume
1
that there exists a sequence of compatible µ-partitions (Pm)m≥1 of [a, b] and g, h ∈ domE.
2
Then the equation (1.4) can be discretized to a system of second-order ordinary differential
3
equations (4.6) by finite element method. Moreover, if supp(µ) = [a, b] and the integrals
4
R
Ixkdµ, I ∈P1, k = 0,1,2, can be evaluated explicitly, then the equation (4.6) has a unique
5
solution that can be solved numerically.
6
The assumption g, h ∈domE in Theorem 1.2 is used to guarantee that the initial condi-
7
tions u(0) =g and ∂tu(0) =h can be approximated by their linear interpolant. We will use
8
the central difference method to solve the equation (4.6) in section 4.
9
We are mainly interested in fractal measures. Throughout this paper, aniterated function system (IFS) refers to a finite family of contractive similitudes {Si}qi=1 defined on Rd. It is well-known that for each IFS {Si}qi=1 and probability weights {wi}qi=1, there is a unique probability measure, called the self-similar measure, satisfying the following identity
µ=
q
X
i=1
wiµ◦Si−1
(see [11, 21]). An IFS {Si}qi=1 is said to satisfy theopen set condition (OSC) if there exists a
10
non-empty bounded open setO such that S
iSi(O)⊆O and Si(O)∩Sj(O) =∅for alli6=j.
11
IFSs that do not satisfy (OSC), as well as all associated self-similar measures, are said to
12
have overlaps. For a class of self-similar measures with overlaps, the finite element method
13
have been used to compute numerical approximations of the eigenvalues and eigenfunctions
14
of the operator ∆µ in [4].
15
Our study of the operator ∆µ is mainly motivated by the effort to extend the current
16
theory of analysis on fractals to include IFSs with overlaps. It is worth pointing out that
17
for general self-similar measures with overlaps, it does not seem possible to discretize the
18
strong damping wave equations (1.2) in the way described in the paper. Thus it is not clear
19
how numerical approximations of the weak solution can be obtain. Theorem 1.2 provides a
20
framework under which discretization can be performed.
21
The following theorem shows that the approximate solutions obtained in Theorem 1.2
22
converge to the actual weak solution, and we also obtain a rate of convergence. Throughout
23
this paper,|E| denotes the diameter of a subsetE ⊆Rd, and let dom ∆µ denote the domain
24
of the operator ∆µ. See Section 2 for the definitions of k · kdomE and k · k2,X, where X is a
25
Hilbert space.
26
Theorem 1.3. Assume the hypotheses of Theorem 1.2. Let g ∈dom ∆µ, h∈dom ∆µ, f ≡0
27
in equation (1.2). If there exist constants r ∈ (0,1) and c > 0 such that max{|J| : J ∈
28
Pm} ≤ c rm for all m ≥ 1, then the approximate solutions um obatained in Theorem 1.2,
29
converge inL2([0, T], L2([a, b], µ))to the actual weak solution u of equation (1.2). Moreover,
30
there exists constant C:=C(u, T, c,)>0 such that for all m≥1
1
kum−uk2,L2([a,b],µ)+k∂tum−∂tuk2,L2([a,b],µ) ≤Crm/2. (1.5) We remark that the condition g ∈ dom ∆µ, h ∈ dom ∆µ, f ≡ 0 implies that ∂tu ∈
2
L2([0, T],domE) (see, Theorem 3.3). It follows that ∂tu can be approximated by its lin-
3
ear interpolant for Lebesgue a.e. t ∈[0, T].
4
Based on Theorems 1.2 and 1.3, we solve the homogeneous strong damping wave equation
5
(1.2) numerically for three different one-dimensional self-similar measures with overlaps. The
6
first measure we study is the infinite Bernoulli convolution associated with the golden ratio:
7 8
µ= 1
2µ◦S1−1+1
2µ◦S2−1, (1.6)
where S1(x) = ρx, S2(x) = ρx+ (1−ρ), and ρ = (√
5−1)/2. The second measure is the three-fold convolution of the Cantor measure, which is defined by the following IFS with overlaps (see [27]):
Si(x) = 1 3x+ 2
3(i−1), i= 1,2,3,4, together with probability weights {1/8,3/8,3/8,1/8}. That is,
9
µ= 1
8µ◦S1−1 +3
8µ◦S2−1+3
8µ◦S3−1+ 1
8µ◦S4−1. (1.7) The third family of self-similar measures that we callessentially of finite type (EFT)(see [29])
10
is defined by the following family of IFSs:
11
S1(x) =r1x, S2(x) =r2x+r1(1−r2), S3(x) = r2x+ 1−r2, (1.8) where the contraction ratiosr1, r2 ∈(0,1) satisfy r1+ 2r2−r1r2 ≤1, i.e.,S2(1)≤S3(0). The
12
first and second measures have been studied very extensively (see, e.g., [22, 27, 29]). They
13
define Laplacians that exhibit many behaviors analogous to Laplacians on post-critically
14
finite fractals, such as sub-Gaussian heat kernel estimates [19] and infinite wave propagation
15
speed [30]. The third class is used in [29] to illustrate self-similar measures satisfying EFT.
16
Corollary 1.4. Let µ be a positive finite Borel measure on R. Assume that g, h∈ dom ∆µ
17
and f ≡0.
18
(a) If µ is the infinite Bernoulli convolution associated with the golden ratio in (1.6),
19
then equation (1.4) can be discretized into a system of ordinary differential equations,
20
which has a unique solution that can be solved numerically. Moreover, the approxi-
21
mate solutions um converge in L2([0, T], L2([0,1], µ)) to the actual weak solution u,
22
and the inequality (1.5) holds.
23
(b) Ifµis the three-fold convolution of the Cantor measure in (1.7), then the conclusions
24
of part (a) also hold.
25
(c) If µ is a self-similar measure defined by the IFS (1.8) with r1+ 2r2−r1r2 = 1, then
26
the conclusions of part (a) also hold.
27
The assumptionr1+2r2−r1r2 = 1 in Corollary 1.4 (c) is used to guarantee that supp(µ) =
1
[0,1]. The numerical results of above three different measures are shown in Figure 2.
2
The rest of this paper is organized as follows. Section 2 summarizes some notation, defi-
3
nitions and results that will be needed throughout the paper. In Section 3, we prove Theo-
4
rem 1.1 and give some regularity results. Section 4 is devoted to the proof of Theorem 1.2.
5
The proof of Theorem 1.3 and Corollary 1.4 are given in Section 5.
6
2. Preliminaries
7
In this section, we summarize some notation, definitions and facts that will be used
8
throughout the rest of the paper. For a Banach space X, we denote its topological dual
9
by X0. For v ∈ X0 and u ∈ X, we let hv, uiX0,X :=v(u) denote the dual pairing of X0 and
10
X.
11
Definition 2.1. Let X be a separable Banach space with normk·kX. Denote byLp([0, T], X)
12
the space of all measurable functions u: [0, T]→X satisfying
13
(1) kukLp([0,T],X):=
RT
0 ku(t)kpXdt1/p
<∞, if 1≤p < ∞, and
14
(2) kukL∞([0,T],X) := esssup0≤t≤Tku(t)kX <∞, if p=∞.
15
If the interval [0, T] is understood, we will abbreviate these norms as kukp,X and kuk∞,X,
16
respectively.
17
Remark 2.1. For each1≤p≤ ∞,Lp([0, T], X)is a Banach space; moreover,Lp2([0, T], X)⊆
18
Lp1([0, T], X) if 1 ≤ p1 ≤ p2 ≤ ∞. Let X be a separable Banach space with inner product
19
(·,·)X. If (X,(·,·)X) is a separable Hilbert space, then L2([0, T], X) is a Hilbert space with
20
the inner product
21
(u, v)L2([0,T],X):=
Z T 0
(u(t), v(t))Xdt.
Definition 2.2. let X be a Banach space. We define C([0, T], X) (resp. C1([0, T], X)) to
22
be the vector space of all continuous (resp. C1) functions u: [0, T]→X such that
23
kukC([0,T],X) := max
0≤t≤TkukX <∞ (resp.kukC1([0,T],X) :=k∂tukC([0,T],X)<∞).
Similarly we define Ck([0, T], X) for all k≥1. k · kCk([0,T],X) is a norm.
24
Definition 2.3. Let X be a Banach space and u∈L1([0, T], X).
25
(1) We sayv ∈L1([0, T], X) is the weak derivative of u, written ∂tu=v, if Z T
0
φ0(t)u(t)dt=− Z T
0
φ(t)v(t)dt for all scalar test functions φ∈Cc∞(0, T).
26
(2) We sayv ∈L1([0, T], X) is the second weak derivative of u, written ∂t2u=v, if Z T
0
φ00(t)u(t)dt= Z T
0
φ(t)v(t)dt for all scalar test functions φ∈Cc∞(0, T).
1
Definition 2.4. let X be a Banach space andX0 its dual. We say a sequence{um}∞m=1 ⊆X
2
converges weakly to u∈X, written um * u, if
3
hv, umi→hv, ui for each bounded linear functional v ∈X0.
4
For convenience, we summarize the definition of the Dirichlet Laplacian on a bounded
5
domain defined by a measure; details can be found in [20]. LetU ⊆Rd,d≥1, be a bounded
6
open subset and µ be a positive finite Borel measure with supp(µ)⊆U and µ(U)>0. We
7
assume that µ satisfies (PI) (see (1.1)). Then each equivalence class u∈ H01(U) contains a
8
unique (in theL2(U, µ) sense) member ˆuthat belongs toL2(U, µ) and satisfies both conditions
9
below:
10
(1) there exists a sequence {un} in Cc∞(U) such that un → uˆ in H01(U) and un → uˆ in
11
L2(U, µ);
12
(2) ˆusatisfies inequality (1.1).
13
We call ˆu the L2(U, µ)-representative of u. Define a mapping ι : H01(U) → L2(U, µ) by
14
ι(u) = ˆu. ιis a bounded linear operator, but not necessarily injective. Consider the subspace
15
N of H01(U) defined as N :=
u ∈ H01(U) : kι(u)kµ = 0 . Now let N⊥ be the orthogonal
16
complement of N in H01(U). Then ι : N⊥ →L2(U, µ) is injective. Unless explicitly stated
17
otherwise, we will denote the L2(U, µ)-representative ˆu simply by u.
18
Consider the non-negative bilinear form E(·,·) in L2(U, µ) defined by
19
E(u, v) :=
Z
U
∇u· ∇v dx (2.1)
with domain domE =N⊥, or more precisely,ι(N⊥). (PI) implies that (E,domE) is a closed quadratic form in L2(U, µ). Hence there exists a non-negative self-adjoint operator A in L2(U, µ) such that
E(u, v) = A1/2u, A1/2v
and domE = dom (A1/2)
(see, e.g., [18, Theorem 1.3.1]). We write ∆Dµ = −A, and call it the (Dirichlet) Laplacian with respect to µ. If no confusion is possible, we denote ∆Dµ simply by ∆µ. Let u∈domE. Thenu∈dom ∆µ if and only if there exists a uniquef ∈L2(U, µ) such thatE(u, v) = (f, v)µ for all v ∈domE. In this case, −∆µu=f. Throughout this paper, we let domE :=N⊥,
k · kdomE :=k · kH1
0(U) and k · kdom ∆µ :=k∆µ(·)kL2(U,µ).
Note that the spaces domE, L2(U, µ), (domE)0 form a Gelfand triple:
domE ,→L2(U, µ)∼= (L2(U, µ))0 ,→(domE)0,
where we identifyL2(U, µ) with (L2(U, µ))0, and the embeddingL2(U, µ),→(domE)0 is given by
w∈L2(U, µ)7→(w,·)µ ∈(L2(U, µ))0 ⊂(domE)0.
We now remark on the case when −∆µ has compact resolvent. Let {ϕn}∞n=1 be an or- thonormal basis of L2(U, µ) so that −∆µ such that −∆µϕn = λnϕn for all n ≥ 1, where 0 < λ1 ≤ · · · ≤ λn ≤ λn+1 ≤ · · · and limn→∞λn = ∞. Then the domains domE and dom ∆µ can be expressed by using eigenfunctions and eigenvalues as
domE = ∞
X
n=1
anϕn:
∞
X
n=1
a2nλn<∞
and dom ∆µ = ∞
X
n=1
anϕn:
∞
X
n=1
a2nλ2n <∞
.
Note that f =P∞
n=1anϕn ∈ (domE)0 if and only if there exists a unique u =P∞
n=1bnϕn ∈ domE such that E(u, v) = hf, vi for all v ∈ domE, where, and throughout this paper, h·,·i denotes the pairing between domE and (domE)0. Letting v =ϕk,k ≥1, it follows that
ak =hf, ϕki=E(u, ϕk) =bkλk, and sof =P∞
n=1anϕn∈(domE)0 if and only ifP∞
n=1(an/λn)ϕn∈domE, i.e.,P∞
n=1a2n/λn<
∞. Therefore,
(domE)0 =n u=
∞
X
n=1
anϕn :
∞
X
n=1
a2n/λn <∞o . Moreover, hu, vi= (u, v)µ for all u∈domE and v ∈(domE)0.
1
3. Existence and uniqueness of weak solution
2
In this section, we consider the existence and uniqueness of weak solution of equation (1.2).
3
Let U ⊆Rd, d ≥1, be a bounded open subset, and let µ be a positive finite Borel measure
4
with supp(µ) ⊆ U and µ(U) > 0. Assume that (PI) (see (1.1)) holds, and let −∆µ be the
5
Dirichlet Laplace with respect to µ. Let (E,domE) be given as in (2.1).
6
Definition 3.1. Let f ∈ L2([0, T], L2(U, µ)), g ∈ domE, and h ∈ L2(U, µ). A function
7
u ∈ L2([0, T],domE) with ∂tu ∈ L2([0, T],domE) and ∂t2u ∈ L2([0, T],(domE)0) is a weak
8
solution of strong damping wave equation (1.2) if it satisfies the following conditions:
9
(i) h∂t2u, vi+αE(u, v) +σE(∂tu, v) = (f, v)µ for all v ∈ domE and Lebesgue a.e. t ∈
10
[0, T];
11
(ii) u(0) =g and ∂tu(0) =h.
12
We use the Galerkin method to prove the existence and uniqueness of weak solution
1
of (1.2). To perform the Galerkin method, we start by solving a finite dimensional ap-
2
proximation. We thus assume that −∆µ has compact resolvent; that is, there exists a
3
complete orthonormal basis {ϕn}∞n=1 of L2(U, µ) such that −∆µϕn = λnϕn for all n ≥ 1,
4
0< λ1 ≤ · · · ≤λn≤λn+1 ≤ · · · and limn→∞λn =∞. For each positive integer m, define
5
um(t) :=
m
X
k=1
βm,k(t)ϕk, (3.1)
where we will show that the coefficients{βm,k(t)}mk=1 can be chosen to satisfy
6
βm,k(0) = (g, ϕk)µ, βm,k0 (0) = (h, ϕk)µ, (3.2) and for t∈[0, T],
7
∂t2um, ϕk
µ+αE(um, ϕk) +σE(∂tum, ϕk) = (f, ϕk)µ. (3.3) Note thatβm,k(0) andβm,k0 (0) are independent of m.
8
Proposition 3.1. Assume the hypotheses of Theorem 1.1. Then for each m ≥ 1, there
9
exists a unique function um(t) of the form (3.1) with the coefficients βm,k(t) ∈ H2([0, T])
10
(k = 1, . . . , m) satisfying (3.2) and (3.3).
11
Proof. Letum(t) be defined as in (3.1). By the orthogonality of {ϕk}∞k=1,
12
∂t2um, ϕk
µ=βm,k00 (t), k= 1, . . . , m. (3.4) Moreover, for k= 1, . . . , m,
13
E(um, ϕk) = λkβm,k(t), and E(∂tum, ϕk) =λkβm,k0 (t). (3.5) Letting fk := (f, ϕk)µ for k ≥ 1 and using (3.4), (3.5), we can convert equation (3.3) into
14
the following linear system of ODEs
15
βm,k00 (t) +αλkβm,k(t) +σλkβm,k0 (t) =fk, t∈[0, T], k = 1, . . . , m. (3.6) with the initial condition (3.2). Thus there exists a unique vector-valued function βm(t) =
16
(βm,1(t), . . . , βm,m(t)) such that eachβm,k(t)∈H2([0, T]) satisfies (3.2) and (3.6).
17
We remark that iffk is continuous on [0, T] for somek ≥1, thenβm,k(t)∈C2([0, T]).
18
In order to obtain a weak solutionu, we need to take the limit asm→ ∞. To this end, we
19
will first obtain a key energy estimate. For convenience, throughout this section, we denote
20
all generic constants, depending only on U and µ, by C.
21
Proposition 3.2. Assume the hypotheses of Theorem 1.1, and let um(t) be of form (3.1)
22
satisfying (3.2) and (3.3). Then there exists a constant C > 0, depending only on U and µ,
23
such that for all positive integer m,
1
t∈[0,T]max
kum(t)k2domE+k∂tum(t)k2L2(U,µ)
+k∂tumk22,domE +k∂t2umk22,(domE)0
≤C kfk22,L2(U,µ)+kgk2domE +khk2µ .
(3.7)
Proof. Fix any m ≥ 0. We multiply equation (3.3) by βm,k0 (t), sum over k = 1, . . . , m and
2
use (3.1) to obtain
3
∂t2um, ∂tum
µ+αE(um, ∂tum) +σE(∂tum, ∂tum) = (f, ∂tum)µ for all t ≥0. (3.8) Note that
4
1 2
d dt
k∂tum(t)k2µ
= ∂t2um, ∂tum
µ and 1
2 d dt
kum(t)k2domE
=E(um, ∂tum). (3.9) Substituting (3.9) into (3.8), we obtain
5
1 2
d dt
k∂tum(t)k2µ+αkum(t)k2domE
+σk∂tumk2domE
= (f, ∂tum)µ≤ kfkµk∂tumkµ≤Ckfkµk∂tumkdomE ≤ C
2σkfk2µ+σ
2k∂tumk2domE,
(3.10) where Cauchy-Schwarz, (PI) (see (1.1)) and Young inequality are used successively. Multi-
6
plying inequality (3.10) by the constant 2, and then integrating both sides with respect to
7
time, we obtain
8
k∂tum(t)k2µ+αkum(t)k2domE +σ Z t
0
k∂tum(τ)k2domEdτ
≤ C 2σ
Z t 0
kf(τ)k2µdτ +kum(0)k2domE+k∂tum(0)k2µ
≤C
kf(t)k22,L2(U,µ)+kgk2domE +khk2µ
fort ≥0,
(3.11)
where the factkum(0)k2domE ≤ kgk2domE andk∂tum(0)k2µ≤ khk2µare used in the last inequality.
9
Since t∈[0, T] was arbitrary, the inequality (3.11) implies that
10
max
t∈[0,T]
kum(t)k2domE +k∂tum(t)k2µ
+k∂tumk22,domE
≤C
kf(t)k22,L2(U,µ)+kgk2domE +khk2µ .
(3.12) Fix any v ∈ domE with kvkdomE ≤ 1, and write v = v1 +v2, where v1 ∈ span{ϕk}mk=1 and (v2, ϕk)µ = 0 for all k = 1, . . . , m. We first note that kv1kdomE ≤ kvkdomE ≤ 1. Then equations (3.1) and (3.3) imply
h∂t2um, vi= (∂t2um, v)µ= (∂t2um, v1)µ
= (f, v1)µ−αE(um, v1)−σE(∂tum, v1).
Thus, by the Cauchy-Schwartz inequality and (PI), we have
|h∂t2um, vi| ≤ kfkµkv1kµ+αkumkdomEkv1kdomE +σk∂tumkdomEkv1kdomE
≤C
kfkµ+kumkdomE +k∂tumkdomE
.
It follows thatk∂t2umk(domE)0 ≤C(kfkµ+kumkdomE+k∂tumkdomE).Consequently, we deduce
1 2
Z T 0
k∂t2um(t)k2(domE)0dt≤C Z T
0
kfk2µ+ max
t∈[0,T]kum(t)k2domE +k∂tumk2domE dτ
≤C
kfk22,L2(U,µ)+kgk2domE+khk2µ ,
(3.13)
where estimate (3.12) has been used in the last inequality. Combining (3.13) and (3.12), we
3
obtain (3.7).
4
Proof of Theorem 1.1. Using the energy estimate (3.7), we obtain a subsequence {uml}∞l=1,
5
together with a function u ∈L2([0, T],domE) satisfying ∂tu ∈ L2([0, T],domE) and ∂t2u ∈
6
L2([0, T],(domE)0), such that
7
uml * u in L2([0, T],domE),
∂tuml * ∂tu in L2([0, T],domE),
∂t2uml * ∂t2u in L2([0, T],(domE)0).
(3.14)
Now, we fix an integer N and choose a function v ∈C1([0, T],domE) of the form
8
v(t) =
N
X
k=1
dk(t)ϕk, where {dk(t)}Nk=1 ⊆C1([0, T]). (3.15) Lettingm≥N, multiplying (3.3) by dk(t),k = 1,· · · , N, adding the equations up, and then
9
integrating with respect to t, we get
10
Z T 0
(∂t2um, v(t))µ+αE(um, v(t)) +σE(∂tum, v(t))
dt = Z T
0
(f, v(t))µdt. (3.16) Settingm =ml, letting l tend to ∞, and using (3.14), we have
11
Z T 0
(∂t2u, v(t))µ+αE(u, v(t)) +σE(∂tu, v(t)) dt=
Z T 0
(f, v(t))µdt. (3.17) Since the set of functions of the form (3.15) is dense in L2([0, T],domE), (3.17) holds for all
12
v ∈L2([0, T],domE), and thus for all such v and Lebesgue a.e. t∈[0, T],
13
(∂t2u, v)µ+αE u, v
+σE ∂tu, v
= (f, v)µ.
Next we need to verify u(0) =g and ∂tu(0) =h. For this, in (3.17), we choose any function
14
v ∈ C2([0, T],domE), withv(T) =∂tv(T) = 0. Applying integration by parts twice respect
15
tot to the first term of (3.17), we can find
16
Z T 0
(u, ∂t2v(t))µ+αE(u, v(t)) +σE(∂tu, v(t)) dt
= Z T
0
f, v
µdt− u(0), ∂tv(0)
µ+ ∂tu(0), v(0)
µ.
(3.18)
Similarly, using (3.16), we have
1
Z T 0
(um, ∂t2v(t))µ+αE um, v(t)
+σE(∂tum, v(t)) dt
= Z T
0
f, v
µdt− um(0), ∂tv(0)
µ+ ∂tum(0), v(0)
µ. Settingm =ml and recall (3.2) and (3.14), we deduce
2
Z T 0
(u, ∂t2v)µ+αE u, v
+σE(∂tu, v)dt= Z T
0
f, v
µdt− g, ∂tv(0)
µ+ h, v(0)
µ. (3.19) Comparing (3.18) and (3.19), and noting that v(0) and v0(0) were arbitrary, we conclude
3
that u(0) = g and ∂tu(0) = h. Therefore, u is a weak solution of strong damping wave
4
equation (1.2).
5
It suffices to show that the only weak solution of (1.2) with g = h = f ≡ 0 is u ≡ 0
6
in L2([0, T],domE). To prove it , we take ∂tu ∈ L2([0, T],domE). Then for Lebesgue a.e.
7
t∈[0, T],
8
∂t2u, ∂tu
µ+αE u, ∂tu
+σE(∂tu, ∂tu) = f, ∂tu
µ= 0.
By assumption, we have u(0)≡0 and ∂tu(0) ≡0. Moreover, since d
dt 1
2k∂tuk2µ+ α
2kuk2domE
= ∂t2u, ∂tu
µ+αE u, ∂tu , we have for Lebesgue a.e. s∈[0, T],
1
2k∂tu(s)k2µ+ α
2ku(s)k2domE+σ Z s
0
k∂tuk2domEdt = 0,
which implies ∂tu= 0 in L2([0, T],domE) and u= 0 in L2([0, T],domE).
9
Now, we give some regularity results of the weak solution of equation (1.2).
10
Theorem 3.3. Assume the hypotheses of Theorem 1.1. Let u be the weak solution of the
11
strong damping wave equation (1.2). Then
12
(a) u∈L∞([0, T],domE), ∂tu∈L∞([0, T], L2(U, µ))∩L2([0, T],domE), and ess supt∈[0,T] ku(t)k2domE+k∂tu(t)k2µ
+k∂tuk22,domE+k∂t2uk22,(domE)0
≤C kfk22,L2(U,µ)+kgk2domE +khk2µ .
(b) If, in addition, g ∈ dom ∆µ and h ∈ domE, then u ∈ L∞([0, T],dom ∆µ), ∂tu ∈
13
L∞([0, T],domE)∩L2([0, T],dom ∆µ), ∂t2u∈L2([0, T], L2(U, µ)) and
14
ess supt≥0 ku(t)k2dom ∆
µ+k∂tu(t)k2domE
+k∂tuk22,dom ∆
µ
≤C kfk22,L2(U,µ)+kgk2dom ∆µ+khk2domE
. (3.20)
(c) If g ∈dom ∆µ, h∈dom ∆µ, f ≡0, then ∂t2u∈L2([0, T],domE).
15
Proof. (a) Passing to limits in (3.7) as m=ml → ∞, we deduce part (a).
1
(b) Letum(t) be given as in Proposition 3.1. We multiply equation (3.3) byλkβm,k0 (t) and
2
sum over k= 1,· · · , m. Then we have
3
(∂t2um,−∆µ∂tum)µ+αE(um,−∆µ∂tum) +σE(∂tum,−∆µ∂tum) = (f,−∆µ∂tum)µ. (3.21) We remark that the left-hand side of (3.21) equals
1 2
d dt
k∂tumk2domE+αkumk2dom ∆µ
+σk∂tumk2dom ∆µ. It follows that
4
1 2
d dt
k∂tumk2domE +αkumk2dom ∆µ
+σk∂tumk2dom ∆µ
≤ |(f,−∆µ∂tum)µ| ≤ kfkµ· k∆µ∂tumkµ=kfkµ· k∂tumkdom ∆µ
≤ 2
σkfk2µ+σ
2k∂tumk2dom ∆µ.
(3.22)
Integrating over [0, t], we obtain
5
1
2k∂tum(t)k2domE +α
2kum(t)k2dom ∆
µ+ σ 2
Z t 0
k∂tum(τ)k2dom ∆
µdτ
≤ 2 σ
Z t 0
kf(τ)k2µdτ +α
2kum(0)k2dom ∆µ+ 1
2k∂tum(0)k2domE
≤ 2
σkfk22,L2(U,µ)+α
2kgk2dom ∆µ+ 1
2khk2domE for all 0< t≤T.
(3.23)
Taking the weak limit of a subsequence of{um}, we find
6
ess sup
t∈[0,T]
k∂tu(t)k2domE +ku(t)k2dom ∆µ
+k∂tuk22,dom ∆µ
≤C
kfk22,L2(U,µ)+kgk2dom ∆
µ +khk2domE .
(3.24) Hence, (3.20) holds,u∈L∞([0, T],dom ∆µ) and∂tu∈L∞([0, T],domE)∩L2([0, T],dom ∆µ).
7
It follows that ∂t2u∈L2([0, T], L2(U, µ)).
8
(c) Let w(t) := ∂tu. It is easy to check that w(t) is the unique weak solution of the following boundary value problem
∂t2w−α∆µw−σ∆µ∂tw= 0, w(0) = h, ∂tw(0) =α∆µg+σ∆µh, w|∂U = 0.
Using part (a), we have ∂tw∈L2([0, T],domE). That is,∂t2u∈L2([0, T],domE).
9
4. The finite element method
10
In this section, we let f ≡ 0 in equation (1.2), and use the finite element method to
11
solve the homogeneous strong damping wave equaion (1.2). Let µbe a positive finite Borel
12
measure on R with supp(µ) = [a, b]. Assume that there exists a sequence of compatible
13
µ-partitions (Pm)m≥1 = ({Im,i}N(m)i=1 )m≥1 of [a, b]. We thus can write Im,i = [xm,i−1, xm,i] for
1
m≥1 and 1≤i≤N(m). It is easy to see thatxm,0 =a and xm,N(m) =b for all m ≥1.
2
We apply the finite element method to approximate the weak solutionu(t) satisfying (1.4)
3
by
4
um(t) =
N(m)
X
i=0
wm,i(t)φm,i, (4.1)
where, for i = 0,1, . . . , N(m), wm,i(t) are functions to be determined and φm,i are the
5
standard piecewise linear finite element basis functions (also called tent functions) defined
6
by
7
φm,i(x) :=
x−xm,i−1
xm,i−xm,i−1
if x∈Im,i, i= 1,2, . . . , N(m), x−xm,i+1
xm,i−xm,i+1 if x∈Im,i+1, i= 0,1, . . . , N(m)−1,
0 otherwise.
(4.2)
Fix any m ≥ 1. We require um(t) to satisfy the integral form of the homogeneous strong
8
damping wave equation
9
Z b a
∂t2um(t)φm,jdµ=−α Z b
a
∂xum(t)φ0m,jdx−σ Z b
a
∂x∂t um(t)
φ0m,jdx (4.3) forj = 1, . . . , N(m)−1, and the Dirichlet boundary conditionum(t) = 0 on{a, b}×[0, T]. We
10
note that φm,i(a) = φm,i(xm,0) = 0 and φm,j(b) =φm,j(xm,N(m)) = 0 for all i = 1, . . . , N(m)
11
and j = 0,1, . . . , N(m)−1. Thus wm,0(t) = wm,N(m)(t) = 0 for all t ∈[0, T]. Using this and
12
substituting (4.1) into (4.3) gives
13
N(m)−1
X
i=1
w00m,i Z b
a
φm,i(x)φm,j(x)dµ
=−α
N(m)−1
X
i=1
wm,i
Z b a
φ0m,i(x)φ0m,j(x)dx−σ
N(m)−1
X
i=1
wm,i0 Z b
a
φ0m,i(x)φ0m,j(x)dx
(4.4)
for 1≤j ≤N(m)−1. We define the mass matrix M=M(m) = (Mij(m)) and stiffness matrix K=K(m) = (Kij(m)), respectively, by
Mij(m) = Z b
a
φm,i(x)φm,j(x)dµ and Kij(m) = Z b
a
φ0m,i(x)φ0m,j(x)dx,
where 1≤i, j ≤N(m)−1. It follows from the definition of φm,j(x) that bothM andK are tridiagonal. Let
w(t) =wm(t) :=
wm,1(t) ... wm,N(m)−1(t)
. Then (4.4) can be put into matrix form as
Mw00 =−αKw−σKw0.
