# Existence and Uniqueness of a Solution in the Space of  BV Functions to the Equation of a Vibrating Membrane with  a “Viscosity” Term

Authors: Koji Kikuchi
Journal: Journal of Calculus of Variations
 (2013)

---

## Abstract

A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term-Δut is added, and it is well known that this equation is well posed in the space of W1,2 functions. In this paper, the viscosity term is changed to -(div(∇u/1+|∇u|2))t, and it is proved that if initial data is slightly smooth (but belonging to W2,2 is sufficient), then a weak solution exists uniquely in the space of BV functions.


---

## Body

## 1. Introduction

LetΩ be a bounded domain in Rn with the Lipschitz continuous boundary ∂Ω. In [1] and in the author’s previous works [2–4], the following:
(1)utt-div(∇u1+|∇u|2)=0,x∈Ω
is investigated, which is in these works referred to as the equation of motion of vibrating membrane. Up to now, neither existence nor uniqueness of a solution to (1) is obtained. In [1–3], we only have that a sequence of approximate solutions to (1) converges to a functionuin an appropriate function space, and that ifusatisfies the energy conservation law, it is a weak solution to (1). In [1], approximate solutions are constructed by the Ritz-Galerkin method and in [2, 3] by Rothe’s method. In [2], the boundary condition is not essentially discussed, and the observation is added in [3]. In these works, the limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet. Instead, in [4], linear approximation for (1) is established. On the other hand, the equation with the strong viscosity term -Δut is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if u(0)∈W01,2(Ω) and ut(0)∈L2(Ω), there exists a unique solution u∈L2((0,T);W01,2(Ω))∩W1,2((0,T);L2(Ω)) for each T>0. Namely, the equation with strong viscosity term is well posed in W1,2(Ω), and since W1,2 is a smaller class than the space of BV functions, this suggests that the influence of the term -Δut is too strong.In this paper, replacing the strong viscosity term-Δut with -(div(∇u/1+|∇u|2))t, we investigate it in the space of BV functions. Namely, our problem of this paper is as follows:
(2)utt-div(∇u1+|∇u|2)-(div(∇u1+|∇u|2))t=0,x∈Ω,
with initial and boundary conditions
(3)u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω,(4)u(t,x)=0,x∈∂Ω.
We should note that the term “viscosity” probably means implying regularity. However, in this paper, we only investigate existence and uniqueness of (2)–(4), regularity is not investigated. This is the reason that in the title there is a quotation mark.A functionu is said to be a function of bounded variation or a BVfunction in Ω if the distributional derivative Du is an Rn valued finite Radon measure in Ω. The vector space of all functions of bounded variation in Ω is denoted by BV(Ω). It is a Banach space equipped with the norm ∥u∥BV=∥u∥L1(Ω)+|Du|(Ω) (see, e.g., [6–8]). We should note that, for u∈BV(Ω), the operator div(∇u/1+|∇u|2) is multivalued. It is usually defined by the use of the subdifferential of the area functional. Namely, for a function u∈BV(Ω)∩L2(Ω), we regard -div(∇u/1+|∇u|2) as
(5)∂J(u)={f∈(BV(Ω)∩L2(Ω))′;J(u+ϕ)-J(u)≥(f,ϕ)foreachϕ∈BV(Ω)∩L2(Ω)(BV(Ω)∩L2(Ω))′},
where
(6)J(u)=1+|Du|2(Ω¯).
Here, readers should note that J(u) is not 1+|Du|2(Ω). We are imposing (4), and in the analysis in the space of BV functions, the most appropriate weak formulation of  (4) is to replace 1+|Du|2(Ω) with 1+|Du|2(Ω¯) (cf. [3], see, also [4, Appendix C]).Now, we present our definition of a weak solution to (2)–(4).Definition 1.
A functionu is a weak solution to (2)–(4) in [0,T)×Ω if u satisfies that(i)
u∈L∞((0,T);BV(Ω)∩L2(Ω)), ut∈L2((0,T)×Ω),(ii)
s-limt↘0u(t,x)=u0(x) in L2(Ω),(iii)
there existf0∈∂J(u0) and f1∈L∞((0,T);(BV(Ω)∩L2(Ω))′) such that for ℒ1-a.e. t (ℒ1 denotes the one-dimensional Lebesgue measure), f1∈∂J(u) and, for any ϕ∈W1,∞((0,T);BV(Ω)∩L2(Ω))∩C00([0,T);L2(Ω)),
(7)∫0T‍∫Ω‍ut(t)ϕt(t)dxdt-∫0T‍(f1,ϕ(t))dt+∫0T‍(f1,ϕt(t))dt+(f0,ϕ(0))+∫Ω‍v0ϕ(0)dx=0.
If a functionu∈L∞((0,∞);BV(Ω)∩L2(Ω)) is a weak solution to (2)–(4) in [0,T)×Ω for each T>0, then we say that u is a weak solution to (2)–(4) in [0,∞)×Ω.Our main theorem is as follows.Theorem 2.
Suppose thatu0∈BV(Ω)∩L2(Ω) and v0∈L2(Ω). We further suppose that J(u0)∩L2(Ω)≠∅. Then, there exists a unique weak solution to (2)–(4) in [0,∞)×Ω.Remark 3.
IfJ(u0)∩L2(Ω)≠∅, then the element is unique. Indeed, for each ϕ∈C0∞(Ω), J(u0+εϕ) is differentiable at ε=0 and (d/dε)J(u0+εϕ)|ε=0=(f0,ϕ) for each f0∈J(u0)∩L2(Ω). Since f0∈L2(Ω) and ϕ is arbitrary, f0 is uniquely determined.


## 2. Reduction of the Problem

In order to solve (2)–(4), we give a formal observation. Let us put
(8)f=-div(∇u1+|∇u|2),g=utt,
then (2) becomes ft=-f-g, which can be regarded as an ordinary differential equation to f. By the variation-of-constants formula, we obtain that f(t)=f0e-t-e-t∫0t‍g(s)esds. Noting that g=utt, we have
(9)f(t)=f0e-t-ut+v0e-t+u-u0e-t-e-t∫0t‍u(s)esds,
where f0=f(0)=-div(∇u0/1+|∇u0|2). Hence, formally, (2) is reduced to
(10)ut-div(∇u1+|∇u|2)=f0e-t+v0e-t+u-u0e-t-e-t∫0t‍u(s)esds.
Definition of a weak solution to this equation is as follows.Definition 4.
Letf0∈∂J(u0),  u0∈BV(Ω)∩L2(Ω), and v0∈L2(Ω). A function u is a weak solution to (10) with (3) and  (4) in [0,T)×Ω if u satisfies that(i)
u∈L∞((0,∞);BV(Ω)∩L2(Ω)), ut∈L2((0,T)×Ω),(ii)
s-limt↘0u(t,x)=u0(x) in L2(Ω),(iii)
for anyϕ∈L2(Ω)∩BV(Ω) and for ℒ1-a.e. t,
(11)J(u+ϕ)-J(u)≥-∫Ω‍utϕ(t,x)dx+∫Ω‍uϕ(t,x)dx+e-t(f0,ϕ(t))+e-t∫Ω‍(v0(x)-u0(x)-∫0t‍u(s,x)esds)ϕ(t,x)dx.
Similar to the case of (2), we say that u is a weak solution to (10) with (3) and (4) in [0,∞)×Ω if a function u∈L∞((0,∞);BV(Ω)∩L2(Ω)) is a weak solution to (10) with  (3) and (4) in [0,T)×Ω for each T>0.
The previous observation is just formal. In the following proposition, we show it rigorously.Proposition 5.
Definitions1 and 4 are equivalent.Proof.
It is sufficient to show that, for eachT>0, a function u is a weak solution to (2)–(4) in [0,T)×Ω if and only if it is a weak solution to (10) with (3) and (4) in [0,T)×Ω.
Suppose thatu is a weak solution to (10) with (3) and (4) in [0,T)×Ω. Conditions (i) and (ii) of Definition 1 are the same as those of Definition 4. Thus, we only have to show (iii) of Definition 1. Let
(12)f1=-ut+u+e-tf0+e-t(v0-u0-∫0t‍uesds).
Then, by (iii) of Definition 4, we have that f1∈∂J(u) for ℒ1-a.e. t. Thus, by a direct calculation, we have that u satisfies (iii) of Definition 1.
Next, we suppose thatu is a weak solution to (2)–(4) in [0,T)×Ω. For each ϕ∈L∞((0,T);L2(Ω)∩BV(Ω)) and each ρ∈C00([0,T)), we put
(13)ψ(t)=et∫tT‍e-sρ(s)ϕ(s)ds.
Then ψ∈W1,∞((0,T);BV(Ω)∩L2(Ω))∩C00([0,T);L2(Ω)), and since ψt(t)=ψ(t)-ρ(t)ϕ(t), we have the following by (iii) of Definition 1:
(14)∫0T‍∫Ω‍ut(t)ψ(t)dxdt-∫0T‍ρ(t)∫Ω‍ut(t)ϕ(t)dxdt-∫0T‍ρ(t)(f1,ϕ(t))dt+(f0,ψ(0))+∫Ω‍v0ψ(0)dx=0.
By integration by parts, we have(15)∫0T‍∫Ω‍ut(t)ψ(t)dxdt=[∫Ω‍u(t)ψ(t)dx]0T-∫0T‍∫Ω‍u(t)ψt(t)dxdt=-∫Ω‍u0ψ(0)dx-∫0T‍∫Ω‍u(t)ψ(t)dxdt+∫0T‍ρ(t)∫Ω‍u(t)ϕ(t)dxdt.
Furthermore, we have the following by Fubini’s theorem:
(16)∫0T‍∫Ω‍u(t)ψ(t)dxdt=∫0T‍ρ(s)∫Ω‍∫0s‍u(t)etdte-sϕ(s)dxds.
Finally, noting that ψ(0)=∫0T‍e-sϕ(s)ds, we have the following by (14), (15), and (16):
(17)∫0T‍ρ(t)(f1,ϕ(t))dt=-∫0T‍ρ(t)∫Ω‍ut(t)ϕ(t)dxdt+∫0T‍ρ(t)∫Ω‍u(t)ϕ(t)dxdt+∫0T‍e-tρ(t)(f0,ϕ(t))dt+∫0T‍e-tρ(t)∫Ω‍v0ϕ(t)dxdt-∫0T‍e-tρ(t)∫Ω‍u0ϕ(t)dxdt+∫0T‍ρ(t)∫Ω‍∫0t‍u(s)esdse-tϕ(t)dxdt.
Since ϕ and ρ are arbitrary, we have, for ℒ1-a.e. t,
(18)f1=-ut+u+e-tf0+e-t(v0-u0-∫0t‍uesds),
which means that u satisfies Definition 4, (iii).Now, Theorem2 is reduced to the following.Theorem 6.
Suppose thatu0∈BV(Ω)∩L2(Ω) and v0∈L2(Ω). We further suppose that J(u0)∩L2(Ω)≠∅ and let f0∈J(u0)∩L2(Ω). Then, there exists a unique weak solution to (10),  (3), and (4) in [0,∞)×Ω.Our strategy of proving Theorem6 is the contracting mapping theorem. For this purpose, given that u~∈L2((0,T)×Ω), we solve
(19)ut-div(∇u1+|∇u|2)=f0e-t+v0e-t+u~-u0e-t-e-t∫0t‍u~(s)esds
and show that the map u~↦u is a contraction. A weak solution to (19) with (3) and (4) is defined as follows.Definition 7.
Letf0∈∂J(u0). A function u is a weak solution to (19) with (3) and (4) in [0,T)×Ω if u satisfies that (i)
u∈L∞((0,T);BV(Ω)∩L2(Ω)), ut∈L2((0,T)×Ω),(ii)
s-limt↘0u(t,x)=u0(x) in L2(Ω),(iii)
for anyϕ∈L2(Ω)∩BV(Ω) and for ℒ1-a.e. t,
(20)J(u+ϕ)-J(u)≥-∫Ω‍utϕ(t,x)dx+∫Ω‍u~ϕ(t,x)dx-e-t(f0,ϕ(t))+e-t∫Ω‍(∫0t‍v0(x)-u0(x)-∫0t‍u~(s,x)esds)ϕ(t,x)dx.
The proof of Theorem6 consists of two parts. The first part is solving (19), and the second part is to show that the map u~↦u is a contraction.


## 3. Existence and Uniqueness of a Solution to (19)

Letu0 and v0 be as in Theorem 6. In this section, we show that there exists a unique solution to (19) with (3) and (4) in [0,T)×Ω for each T>0.Uniqueness is easy. Suppose thatu and v are solutions to (19) with (3) and (4) in [0,T)×Ω, and inserting ϕ=v-u to (iii) of Definition 7, integrating it from 0 to T, obtaining another inequality by replacing u and v, and adding these two inequalities, we have
(21)0≥∫0T‍∫Ω‍(vt-ut)(v-u)dxdt=12(∫Ω‍|v(T)-u(T)|2dx-∫Ω|v(0)-u(0)|2‍dx).Sinceu(0)=v(0)=u0, we have the uniqueness of a solution to (19).It is sufficient to show the existence in[0,T)×Ω for ℒ1-a.e. T. Approximate solutions are constructed by Rothe’s time semidiscretization method. In Rothe’s method, we should solve elliptic equations with respect to space variables. Here, we solve them by a direct variational method (namely, this is the method of discrete Morse semiflow, cf. [9] and references cited therein).Suppose thatu0∈BV(Ω)∩L2(Ω) with J(u0)∩L2(Ω)≠∅ and v0∈L2(Ω), and let f0∈J(u0)∩L2(Ω). For a positive number h, we construct a sequence {ul}l=0∞ in the following way. For l=0, we let u0 be as in (3), and for l≥1, it is defined as a minimizer of the following functional:
(22)ℱl(v)=12∫Ω‍|v-ul-1|2hdx+J(v)+1h∫(l-1)hlh‍(G,v)dt,
in the class L2(Ω)∩BV(Ω), where
(23)G(t,x)=-e-tf0-v0e-t-u~+u0e-t+e-t∫0t‍u~(s,x)esds.
Since
(24)1h∫(l-1)hlh‍(G,v)dt≥-(12h∫(l-1)hlh‍∫Ω‍|G(t,x)|2dxdt2222222222+12∫Ω‍|v|2dx{12h∫(l-1)hlh‍}),Fl is bounded from below, and hence, the existence of a minimizer of ℱl follows.Lemma 8 (energy inequality).
consider the following:(25)∑j=1l‍[12∫Ω‍|uj-uj-1|2hdx+∫(j-1)hjh‍(G,uj-uj-1h)dt{12∫Ω‍|uj-uj-1|2hdx}]+J(ul)≤J(u0).Proof.
Sinceul is a minimizer of ℱl, we have
(26)ℱl(ul)≤ℱl(ul-1)=J(ul-1)+1h∫(l-1)hlh‍(G,ul-1)dt.
Hence, for each l,
(27)12∫Ω‍|ul-ul-1|2hdx+J(ul)+∫(l-1)hlh‍(G,ul-ul-1h)dt≤J(ul-1).
Thus, by induction on l, we have the conclusion.Next, we define approximate solutionsuh(t,x) and u¯h(t,x) for (t,x)∈(-h,∞)×Ω as follows: for (l-1)h<t≤lh,
(28)uh(t,x)=t-(l-1)hhul(x)+lh-thul-1(x),u¯h(t,x)=ul(x).
Then Lemma 8 shows for each T>0(29)12∫0T‍∫Ω‍|uth(t,x)|2dxdt+∫0T‍(G,uth)dt+J(u¯h(T,·))≤J(u0).Now, we estimate the second term of the left hand side of (29). Then,
(30)∫0T‍∫Ω‍|e-t∫0t‍u~(s,x)esds|2dxdt≤∫0T‍∫Ω‍e-2t∫0t‍e2sds∫0t‍|u~(s,x)|2dsdxdt≤∫0T‍∫Ω‍e-2t2-1(e2t-1)∫0t‍|u~(s,x)|2dsdxdt=∫0T‍∫sT‍2-1(1-e-2t)dt∫Ω‍|u~(s,x)|2dsdx=∫0T‍2-1[t+2-1e-2t]sT∫Ω‍|u~(s,x)|2dsdx=2-1∫0T‍(T+2-1e-2T-s-2-1e-2s)∫Ω‍|u~(s,x)|2dsdx≤C(T)∫0T‍∫Ω‍|u~(s,x)|2dsdx,
where C(T)=2-1(T+2-1e-2T), and thus, it is easy to see that
(31)|∫0T‍(G,uth)dt|≤[∥f0∥L2(Ω)+∥v0∥L2(Ω)+∥u0∥L2(Ω)+(1+C(T))∥u~∥L2((0,T)×Ω)∥f0∥L2(Ω)]∥uth∥L2((0,T)×Ω).
By (29), we have, for each ε>0 and for each T>0,
(32)1-ε2∫0T‍∫Ω‍|uth(t,x)|2dxdt+J(u¯h(T,·))≤C0(u0,v0,f0,u~,ε,T),
where
(33)C0(u0,v0,f0,u~,ε,T)=12ε[∥f0∥L2(Ω)+∥v0∥L2(Ω)+∥u0∥L2(Ω)222222222.+(1+C(T))∥u~∥L2((0,T)×Ω)∥f0∥L2(Ω)]2+J(u0).Proposition 9.
It holds that(1)
{∥uth∥L2((0,∞)×Ω)}  is uniformly bounded with respect to  h;(2)
for any  T>0,  {∥u¯h∥L∞((0,T);L2(Ω)∩BV(Ω))}  is uniformly bounded with respect to  h;(3)
for any  T>0,  {∥uh∥L∞((0,T);L2(Ω)∩BV(Ω))}  is uniformly bounded with respect to  h;Then there exist a sequence  {hj}  with  hj→0  as  j→∞  and a function  u  such  that(4)
uthj  converges to  ut  as  j→∞  weakly in  L2((0,∞)×Ω);(5)
for any  T>0,  uhj  converges to  u  as  j→∞  weakly star in  L∞((0,T);L2(Ω));(6)
for any  T>0,  uhj  converges to  u  as  j→∞  strongly in  L∞((0,T);Lp(Ω))  for each  1≤p<1*=n/(n-1);(7)
for any  T>0,  u¯hj  converges to  u  as  j→∞  strongly in  L∞((0,T);Lp(Ω))  for each  1≤p<1*;(8)
u∈L∞((0,∞);BV(Ω));(9)
for  ℒ1-a.e.  t∈(0,∞),  Du¯hj(t,·)  converges to  Du(t,·)  as  j→∞  in the sense of distributions;(10)
s-limt↘0u(t)=u0  in  L2(Ω).Proof.
Assertion (1) immediately follows from (32). Since we have
(34)uh(t,x)-uh(t′,x)=∫t′t‍uth(s,x)ds,
for each t, t′≥0, Assertion (1) implies that, for each T>0, {∥uh∥L∞((0,T);L2(Ω))} is uniformly bounded with respect to h. Given that t>0, we let l be an integer such that (l-1)h<t≤lh. Then,
(35)∥u¯h(t)-uh(t)∥L2(Ω)=∥ul-t-(l-1)hhul-lh-thul-1∥L2(Ω)=∥lh-th(ul-ul-1)∥L2(Ω)≤∥ul-ul-1∥L2(Ω).
By (32),
(36)∫0Lh‍∫Ω‍|uth(t,x)|2dxdt=∑l=1L‍∫Ω‍|ul-ul-1|2hdx≤21-εC0(u0,v0,f0,u~,ε,T).
Thus, we have
(37)∥ul-ul-1∥L2(Ω)2≤C(u0,v0,f0,u~,ε,T)h,
for each l, where C(u0,v0,u~,ε,T)=(2/(1-ε))C0(u0,v0,f0,u~,ε,T). Hence,
(38)supt>0∥u¯h(t)-uh(t)∥L2(Ω)≤C(u0,v0,f0,u~,ε,T)h.
Now, we have that {∥u¯h∥L∞((0,T);L2(Ω))} is uniformly bounded with respect to h since
(39)∥u¯h∥L∞((0,T);L2(Ω))≤∥u¯h-uh∥L∞((0,T);L2(Ω))+∥uh∥L∞((0,T);L2(Ω)).
Since C0 is increasing with respect to T, Assertion (2) follows from (32). Since J is convex, we have
(40)J(uh(t,·))≤t-(ℓ-1)hhJ(u¯h(t,·))22222222221.+ℓh-thJ(u¯h(t-h,·)),
and Assertion (3) also holds.
Assertion(4) is a direct consequence of Assertion (1). Assertion (5) follows from Assertion (3). Furthermore, (34) and Assertion (1) imply that the function t↦uh(t,·)∈L2(Ω) is equicontinuous with respect to h. By Sobolev’s theorem BV(Ω)⊂Lp(Ω) compactly for each 1≤p<1*. This means that, for any T>0, {uh(t,·)} is contained in a sequentially compact subset of Lp(Ω) which is independent of h and t∈[0,T]. Thus, by the Ascoli-Arzela theorem, we obtain Assertion (6).
Now, we have, for1≤p<1*,
(41)∥u¯h-u∥L∞((0,T);Lp(Ω))≤∥u¯h-uh∥L∞((0,T);Lp(Ω))+∥uh-u∥L∞((0,T);Lp(Ω)),
the right hand side of which converges to 0 as h→0 by  (38) and Assertion (6). Now, we have Assertion (7). Assertions (2) and (7) imply Assertions (8) and (9).
Lettingt′=0 in (34), we have
(42)∥uh(t)-u0∥L2(Ω)≤∥uth∥L2((0,∞)×Ω)t.
Thus, by Assertion (1) the left hand side is uniformly bounded with respect to h and, hence, passing to a subsequence if necessary, {uh(t)-u0}h>0 converges weakly in L2(Ω), and by Assertion (6), the weak limit is u(t)-u0. Then, by the lower semicontinuity of L2 norm, we have
(43)∥u(t)-u0∥L2(Ω)≤liminfh→0∥uh(t)-u0∥L2(Ω)≤Constantt,
which implies Assertion (10).Now, our purpose is to show thatu is a weak solution to (19). Proposition 9 implies that u satisfies (i) and (ii) of Definition 7.Sinceul is a minimizer of ℱl(v), we have
(44)∂ℱl(ul)=ul-ul-1h+∂J(ul)+1h∫(l-1)hlh‍Gdt∋0.
Let us write, for (l-1)h≤t<lh, Gh(t,x)=(1/h)∫(l-1)hlh‍G(s,x)ds. Then, for each j and for ℒ1-a.e. t∈(0,∞),
(45)uthj(t)+∂J(u¯hj(t))+Ghj(t)∋0,
namely, for each v∈L2(Ω),
(46)J(v)-J(u¯hj(t))≥-∫Ω‍uthj(t,x)(v(x)-u¯hj(t,x))dx-∫Ω‍Ghj(s,x)(v(x)-u¯hj(t,x))dx.
By Proposition 9 (7) we have that, for ℒ1-a.e. t∈(0,∞),
(47)u¯hj(t)⟶u(t)
strongly in L1(Ω) as j→∞. Let T be a number t such that  (46) and  (47) hold. We insert an arbitrary function v∈L2((0,T)×Ω) in  (46). Integrating it from 0 to T, we have the following by Proposition 9 (4), (7), Fatou’s lemma, and the lower semicontinuity:
(48)∫0T‍(J(v(t))-J(u(t)))dt≥-∫0T‍∫Ω‍ut(t,x)v(t,x)dt2222222.+liminfj→∞∫0T‍∫Ω‍uth(t,x)u¯hj(t,x)dxdt2222222.-limsupj→∞∫0T‍∫Ω‍Ghj(s,x)(v(t,x)2222222.-u¯hj(t,x))dxdt.
For a while, we write hj=h for simplicity. First, we note the following identity:
(49)∫Ω‍ul(x)-ul-1(x)hul(x)dx=12h∫Ω‍|ul(x)|2dx+∫Ω‍|ul(x)-ul-1(x)|22hdx-12h∫Ω‍|ul-1(x)|2dx.
Let L be the integer such that (L-1)h<T≤Lh. By (49), we have
(50)∫0T‍∫Ω‍uth(t,x)u¯h(t,x)dxdt=-∫TLh‍∫Ω‍uth(t,x)u¯h(t,x)dxdtxxxxxx+∫0Lh‍∫Ω‍uth(t,x)u¯h(t,x)dxdt=-(Lh-T)∫Ω‍uL(x)-uL-1(x)huL(x)dxxxxxxx+h∑l=1L‍∫Ωul(x)-ul-1(x)h‍ul(x)dx=-Lh-Th∫Ω‍(uL(x)-uL-1(x))uL(x)dxxxxxxx+12∫Ω‍|uL(x)|2dxxxxxxx+h∑l=1L‍∫Ω‍|ul(x)-ul-1(x)|22hdx-12∫Ω‍|u0(x)|2dx=:-I+II+III-IV.
By  (42) and Proposition 9 (1), ∥uh(t)∥L2(Ω) is uniformly bounded with respect to t∈[0,T] and h. On the other hand, since (L-1)h<T≤Lh,
(51)uh(T)=T-(L-1)hhuL+Lh-ThuL-1=uL-Lh-Th(uL-uL-1).
Hence, we have by (37)
(52)∥uL∥L2(Ω)≤∥uh(T)∥L2(Ω)+Lh-Th∥uL-uL-1∥L2(Ω)≤C′′(u0,v0,u~,ε),
and thus by (37) again
(53)|I|≤∥uL-uL-1∥L2(Ω)∥uL∥L2(Ω)≤C′(u0,v0,u~,ε)×C′′(u0,v0,u~,ε)h⟶0,
as h→0. By Proposition 9 (1),
(54)|III|=h·h∑l=1L‍∫Ω‍|ul(x)-ul-1(x)|22h2dx=h2∫0Lh‍∫Ω‍|uth|2dxdt≤h2∫0∞‍∫Ω‍|uth|2dx⟶0,
as h→0. Since u¯h(T), u(T)∈L2(Ω),  (47) implies that u¯hj(T)⇀u(T) in L2(Ω). In particular, we have liminfj→∞∥u¯hj(T)∥L2(Ω)2≥∥u(T)∥L2(Ω)2. Then, since uL(x)=u¯h(T,x), we have
(55)liminfj→∞(II-IV)=liminfj→∞(∥u¯hj(T)∥L2(Ω)2-∥u0∥L2(Ω)2)≥∥u(T)∥L2(Ω)2-∥u0∥L2(Ω)2=∫0T‍ddt∥u(t)∥L2(Ω)2dt=∫0T‍∫Ω‍ut(t,x)u(t,x)dxdt.
Summing up, we have
(56)liminfj→∞∫0T‍∫Ω‍uthj(t,x)u¯hj(t,x)dxdt≥∫0T‍∫Ω‍ut(t,x)u(t,x)dxdt.
It is not difficult to show that Gh→G strongly in L2((0,T)×Ω). Hence, by (48), we finally have
(57)∫0T‍(J(v)-J(u))dt≥-∫0T‍∫Ω‍ut(v-u)dxdt-∫0T‍∫Ω‍G(v-u)dxdt,
for each v∈L2((0,T)×Ω). By the convexity of J, for each function ψ∈L2(0,T) with 0≤ψ≤1, we have ψJ(v)-ψJ(u)≥J(u+ψ(v-u))-J(u). Thus, (57) implies that
(58)∫0T‍ψ(t)[J(v)-J(u)]dt≥∫0T‍[J(u+ψ(v-u))-J(u)]dt≥-∫0T‍ψ(t)∫Ω‍ut(v-u)dxdt-∫0T‍ψ(t)∫Ω‍G(v-u)dxdt.
It is easy to extend this inequality to all nonnegative functions ψ∈L2(0,T). Hence, (iii) of Definition 7 holds for ℒ1-a.e. t∈(0,∞).


## 4. Proof Thatu~↦u Is a Contraction

Letu~, v~ be functions in L2((0,T)×Ω). In this section, we write
(59)G(u~)=e-tf0-v0e-t-u~HHHHL+u0e-t+e-t∫0t‍u~(s,x)esds,G(v~)=e-tf0-v0e-t-v~HHHHL+u0e-t+e-t∫0t‍v~(s,x)esds.
Then,
(60)G(u~)-G(v~)=-(u~-v~)+e-t∫0t‍(u~(s,x)-v~(s,x))esds.
Let u, v be a solution to (19) with (3) and (4) for u~, v~, respectively. By (iii) of Definition 7,
(61)J(v)-J(u)≥-∫Ω‍ut(v-u)dx+∫Ω‍G(u~)(v-u)dx,J(u)-J(v)≥-∫Ω‍vt(u-v)dx+∫Ω‍G(v~)(u-v)dx.
Summing these, we have
(62)0≥∫Ω‍(ut-vt)(u-v)dx-∫Ω‍(G(u~)-G(v~))(u-v)dx.
Integrating from 0 to t, we have the following by  (60) and by the fact that u(0)=v(0)=u0:
(63)0≥12∫Ω‍|u(t)-v(t)|2dx+∫0t‍∫Ω‍(u~-v~)(u-v)dxdτ-∫0t‍e-τ∫Ω‍∫0τ‍(u~(s,x)-v~(s,x))2222222222222222×(u(τ,x)-v(τ,x))esdsdxdτ.
We further integrate this from 0 to T and write it I+II+III. Then,
(64)II=∫0T‍∫0t‍∫Ω‍(u~(τ,x)-v~(τ,x))111111111×(u(τ,x)-v(τ,x))dxdτdt=∫Ω‍dx∫0T‍dτ∫τT‍(u~(τ,x)-v~(τ,x))1111111111111.×(u(τ,x)-v(τ,x))dt=∫Ω‍dx∫0T‍(T-τ)(u~(τ,x)-v~(τ,x))111111111×(u(τ,x)-v(τ,x))dτ≤T2∫Ω‍∫0T‍|u~(τ,x)-v~(τ,x)|2dτdx+T2∫Ω‍∫0T‍|u(τ,x)-v(τ,x)|2dτdx,(65)III=∫0T‍∫0t‍e-τ∫Ω‍∫0τ‍(u~(s,x)-v~(s,x))111111111111111×(u(τ,x)-v(τ,x))esdsdxdτdt=∫Ω‍dx∫0T‍dt∫0t‍e-τ[∫0τ‍(u~(s,x)-v~(s,x))esds]111111111111111..×(u(τ,x)-v(τ,x))dτ.
Here,
(66)∫0τ‍(u~(s,x)-v~(s,x))esds(u(τ,x)-v(τ,x))≤12(∫0τ‍(u~(s,x)-v~(s,x))esds)2111111+12|u(τ,x)-v(τ,x)|2≤12∫0τ‍|u~(s,x)-v~(s,x)|2ds∫0τ‍e2sds111111+12|u(τ,x)-v(τ,x)|2≤14(e2τ-1)∫0τ‍|u~(s,x)-v~(s,x)|2ds111111+12|u(τ,x)-v(τ,x)|2.
Thus,
(67)III≤∫Ω‍dx∫0T‍dt∫0t‍e-τ14(e2τ-1)111111111111..×∫0τ‍|u~(s,x)-v~(s,x)|2dsdτ+∫Ω‍dx∫0T‍dt∫0t‍e-τ12|u(τ,x)-v(τ,x)|2dτ=:III1+III2.
These two terms are estimated as follows:
(68)III1=∫Ω‍dx∫0T‍dt∫0t‍e-τ14(e2τ-1)×∫0τ‍|u~(s,x)-v~(s,x)|2dsdτ=∫Ω‍dx∫0T‍dt∫0t‍[∫st‍14(eτ-e-τ)dτ]hhhhhhhhhhhhh×|u~(s,x)-v~(s,x)|2ds=12∫Ω‍dx∫0T‍dt∫0t‍[∫st‍sinhτdτ]11111111111111..×|u~(s,x)-v~(s,x)|2ds=12∫Ω‍dx∫0T‍dt∫0t‍(cosht-coshs)111111111111111×|u~(s,x)-v~(s,x)|2ds≤12∫Ω‍dx∫0T‍coshtdthhhhhhhhhhh×∫0T‍|u~(s,x)-v~(s,x)|2ds=sinhT2∫Ω‍∫0T‍|u~(s,x)-v~(s,x)|2dsdx,(69)III2=∫Ω‍dx∫0T‍dt∫0t‍e-τ12|u(τ,x)-v(τ,x)|2dτ=∫Ω‍dx∫0T‍dτ∫τT‍e-τ12|u(τ,x)-v(τ,x)|2dt=∫Ω‍dx∫0T‍dτ(T-τ)e-τ12|u(τ,x)-v(τ,x)|2≤T2∫Ω‍∫0T‍|u(τ,x)-v(τ,x)|2dτdx.
Summing up, we have
(70)0≥12∥u-v∥L2((0,T)×Ω)-T2(∥u-v∥L2((0,T)×Ω)+∥u~-v~∥L2((0,T)×Ω))-(sinhT2∥u~-v~∥L2((0,T)×Ω)+T2∥u-v∥L2((0,T)×Ω)).
Hence, when 0<T<1/2, putting
(71)K(T):=T+sinhT1-2T(>0),
we have
(72)∥u-v∥L2((0,T)×Ω)≤K(T)∥u~-v~∥L2((0,T)×Ω).
As T→0, K(T) converges to 0. Thus, if T is sufficiently small, 0<K(T)<1. This means that the map from u~ to u is a contraction in L2((0,T)×Ω). Hence, there is a fixed point u- and it is a solution to (10) with (3) and (4) in [0,T)×Ω.End of the proof of Theorem6.Uniqueness of a Local Solution. Let u-, v- be solutions to  (10) with  (3) and (4). Then, in the same calculus as before, we obtain
(73)(1-K(T))∥u--v-∥L2((0,T)×Ω)≤0.This implies the uniqueness.Existence of a Time Global Solution. Suppose that u¯ is a solution to (10) with (3) and (4) in [0,s]. First, we remark that
(74)∂J(u¯(s,·))∋-u¯t(s)+u¯(s)hhhhhhhhhh+e-tf0+e-s(v0-u0-∫0s‍u¯(σ)eσdσ),
and the right hand side belongs to L2(Ω). Hence, we are able to solve (10) with (3) and (4) from t=s. By the change of variable t=t~+s, (10) becomes
(75)ut~-div(∇u1+|∇u|2)=Use-t~+ut(s)e-t~+u(t~+s)-u(s)e-t~-e-t~∫0t~‍u(σ+s)eσdσ,
where
(76)Us=-ut(s)+u(s)+e-sf0+e-s(v0(x)-u0(x)-∫0s‍u(σ,x)eσdσ).
Hence, solving (10) with (4) and initial condition u(0)=u-(s), ut(0)=u-t(s), we obtain a function u in [0,T]×Ω which solves (10). For t∈[s,s+T], put u-(t,x)=u(t-s,x). Then, u- is a solution to (10) with (3) and (4) in [0,s+T]×Ω. Repeating this process, we obtain a time global solution. Uniqueness of the local solution implies uniqueness of the global solution.


## 5. Uniform Estimates

LetT be the small number presented in the previous section, and let u be a solution to (10) with (3) and (4) in [0,T)×Ω. As we see in the previous section, it is obtained as a fixed point of the map τ:u~↦u in L2((0,T)×Ω), where u is a solution to (19) with  (3) and (4). The fixed point is obtained as in the following way. Let u(0) be an arbitrary element of L2((0,T)×Ω) and put τn(u(0))=u(n). There is a constant σ which is determined by T such that 0<σ<1 and ∥τ(u)-τ(v)∥L2((0,T)×Ω)≤σ∥u-v∥L2((0,T)×Ω). Thus, we have
(77)∥u(k)-u(k-1)∥L2((0,T)×Ω)≤σk-1∥u(1)-u(0)∥L2((0,T)×Ω)
and hence,
(78)∥u(n)-u(m)∥L2((0,T)×Ω)≤∑k=m+1n‍σk-1∥u(1)-u(0)∥L2((0,T)×Ω),
namely, {u(n)} is a Cauchy sequence in L2((0,T)×Ω) and it converges to a function u. Since u(n)=τ(u(n-1)), we have that u=τ(u) by letting n→∞. Thus, u is the fixed point of τ.Lettingm=0 in (78), we have
(79)∥u(n)∥L2((0,T)×Ω)≤∑k=1n‍σk-1∥u(1)-u(0)∥L2((0,T)×Ω)2222222+∥u(0)∥L2((0,T)×Ω).
By the lower semicontinuity, an energy inequality for a solution to  (19) with (3) and (4) is obtained by letting h→0 in (32):
(80)1-ε2∫0T‍∫Ω‍|ut(t,x)|2dxdt+J(u(T,·))≤C0(u0,v0,f0,u~,ε,T).
Since C0 is increasing with respect to T, we have ess.sup0<t≤TJ(u(t,·))≤C0(T). Hence,
(81)1-ε2∫0T‍∫Ω‍|ut(t,x)|2dxdt222222+ess.sup0<t≤TJ(u(t,·))≤2C0(u0,v0,f0,u~,ε,T).
Recall that C0 is presented as in  (33). By the proof of Proposition 9 (5), we have
(82)∥u(1)∥L2((0,T)×Ω)≤∥u0∥L2((0,T)×Ω)+2T1-εC0(u0,v0,f0,u(0),ε,T)=:C1(u0,v0,f0,u(0),ε,T).
Letting n→∞ in  (79), we finally have
(83)∥u∥L2((0,T)×Ω)≤11-σ(∥u(1)∥L2((0,T)×Ω)111111111111.+∥u(0)∥L2((0,T)×Ω))+∥u(0)∥L2((0,T)×Ω)≤11-σ({∥u(0)∥L2(Ω)}C1(u0,v0,f0,u(0),ε,T)111111111111.+∥u(0)∥L2(Ω))+∥u(0)∥L2((0,T)×Ω)=:C2(u0,v0,f0,u(0),ε,T).
Now, a solution u to  (10) is a solution to  (19) for u~=u, the fixed point of τ. Hence, by (81), we have
(84)1-ε2∫0T‍∫Ω‍|ut(t,x)|2dxdt+ess.sup0<t≤TJ(u(t,·))≤2C0(u0,v0,f0,u,ε,T).By  (83) and (84), there exists a constant C3 such that
(85)∫0T‍∫Ω‍|ut(t,x)|2dxdt+ess.sup0<t≤TJ(u(t,·))≤C3(u0,v0,f0,u(0),ε,T).
Now, by  (83) and  (85), we obtain uniform estimates for ∥u∥L2((0,T)×Ω), ∥u∥L∞((0,T);BV(Ω)∩L2(Ω)), ∥ut∥L2((0,T)×Ω) for the small T.Solving  (10) from t=s (0<s<T), we have the following by (83):
(86)∥u∥L2((s,s+T)×Ω)≤C2(u(s),ut(s),Us,u(0),ε,T),
where Us is as in  (76). By  (81) and Chebyshev’s inequality, for sufficiently large R, ℒ1({∥ut(t)∥L2(Ω)≥R})≤(1/R2)(2C0/(1-ε)). Let δ be an arbitrary small positive number and put R=(4C0/δ(1-ε)). Then, there exists an s∈[T-δ,T)=[T-(1/R2)(4C0/(1-ε),T)) such that ∥ut(s)∥L2(Ω)<R. Hereby, we have by  (83) and  (85) that, for such an s, there exists a constant C4 such that
(87)∥u∥L2((s,s+T)×Ω)+∫ss+T‍∫Ω‍|ut(t,x)|2dxdt+ess.sups<t≤s+TJ(u(t,·))≤C4(u0,v0,f0,u(0),ε,T,δ).
Repeating this process, we have uniform estimates for ∥u∥L2((0,T~)×Ω), ∥u∥L∞((0,T~);BV(Ω)∩L2(Ω)), ∥ut∥L2((0,T~)×Ω) for each T~>0. However, their upper bounds depend on ∥f0∥L2(Ω).



---

*Source: 936915-2013-09-18.xml*