Existence and Regularity of Weak Solutions for Semilinear Second ...

Funkcialaj Ekvacioj, 41 (1998) 1-24

Existence and Regularity of Weak Solutions for Semilinear Second Order Evolution Equations By

Junhong HA and Shin-ichi NAKAGIRI (Kobe University, Japan) Dedicated to professor Junji Kato on the occasion of his sixtieth birthday.

1.

Introduction

This paper is concerned with the Cauchy problem of semilinear second order evolution equations

(1.1)

in

$¥frac{d^{2}y}{dt^{2}}+A_{2}(t)¥frac{dy}{dt}+A_{1}(t)y=f(t, y)$

(1.2)

$y(0)=y_{0}$

,

$¥frac{dy}{dt}(0)=y_{1}$

$(0, T)$

,

,

, $A_{2}(t)$ are time varying operators on appropriate Hilbert spaces and $f(t,y)$ is a nonlinear forcing function. There is a large literature studying this type of equations for the case of being constant operators (cf. [1, 2, 9, , 10-12] and references cited therein). However the researches studying the equation (1.1) having time dependent damping term are rather few compared with the case of constant operators. For such researches we can refer to the books Lions [6], Lions [7] and Dautray-Lions [4]. Particularly in Dautray-Lions [4] the variational method has been developed for the linear equation

where

$A_{1}(t)$

$A_{1}$

$A_{2}$

$A_{¥mathit{2}}(t)dy/dt$

(1.3)

$¥frac{d}{dt}(A_{3}(t)¥frac{dy}{dt})+A_{2}(t)¥frac{dy}{dt}+A_{1}(t)y=f(t)$

in

$(0, T)$

using the structure of a Gelfand triple . In (1.3) it is assumed that is defined via a sesquilinear form on $V$, via a sesquilinear form on or $H$ and is positive and bounded on $H$ , and $f¥in L^{2}(0, T;H)$ . Thus the results on existence, uniqueness and regularity of weak solutions for the problem (1.3), (1.2) with $y_{0}¥in V$, $y_{1}¥in H$ are established in [4, Chapter XVIII, Section 5]. The proofs of the results are complicated and require numerous technical lemmas even for the linear equation (1.3). Further the regularity $C([0, T];V)$ , $dy/dt$ $¥in C([0, T];H)$ is proved for the equation (1.3) with being bounded on $H$ by using parabolic regularization ([4, p. 578]). $V^{c}->H^{e}-¥succ V^{¥prime}$

$A_{1}(t)$

$A_{2}(t)$

$V$

$A_{3}(t)$

$ y¥in$

$A_{2}(t)$

Junhong HA and Shin-ichi NAKAGIRI

In order to avoid the complexity due to the appearance of $A_{3}(t)$ in choosing function spaces and deriving a priori estimates, we suppose that is the identity operator in this paper. For the Cauchy problem (1.1), (1.2) corresponding to the damping term we introduce another Hilbert space , and we present the basic results on existence, uniqueness and under the regularity in the setting of a Gelfand fivefold Lipschitz continuity on the nonlinear term $f(t,y)$ . The introduction of admits an appropriate choice of differential operators according to the types of damping effects, so that our treatment provides a convenient and unified way of obtaining ’weak’ solutions to the nonlinear problem (1.1), (1.2). It should be noted that $V_{2}=V$ or $V_{2}=H$ for (1.3) in [4]. Then the results of DautrayLions [4] do not cover our results even for the linear problem (1.1), (1.2) is not identical with $V$ with $f(t,y)=f(t)¥in L^{2}(0, T;V_{2}^{¥prime})$ in the case where $A_{3}(t)$

$V_{2}$

$A_{¥mathit{2}}(t)dy/dt$

$V¥llcorner_{¥rightarrow V_{2^{¥epsilon}}¥rightarrow H¥llcorner_{¥rightarrow V_{2^{¥mathrm{L}}}^{¥prime}¥rightarrow V}}$

$V_{2}$

$V_{2}$

and

$H$

.

We now explain the content of this paper. In Section 2, after giving the assumptions on operators , $i=1,2$ , and nonlinear term $f(t,y)$ , we state the existence and uniqueness theorem for (1.1), (1.2). Roughly speaking, it is are bilinear forms on $V_{i}(V_{1}=V)$ which satisfy assumed in (1.1) that $f(t,y)$ into is Lipschitz continuous in which maps and coercivity over equations linear the uniqueness result to and . Also we specify the existence and give the continuous dependence result for these equations. In Section 3 we give the proofs of results stated in Section 2. In the existence proof we utilize the Faedo-Galerkin approximation method. A priori estimates of approximate solutions is derived by the energy method modified for the nonlinear equation and Lipschitz continuity of $f(t,y)$ are (1.1), in which the coecivity of essentially used. This leads to the existence of approximate solutions which tend weakly to a function in $L^{2}(0, T;V)¥cap W^{1,2}(0, T;V^{¥prime})$ as in [4, 7, 6]. Thus the most difficult part to show that the limit function is exactly a solution of (1.1) is left. In the reference [1, 2, 9, 13, 14] the monotonicity of nonlinear is utilized to solve the part. In this term on the compact imbedding paper we do not assume such conditions, because of our general settings of (1.1), (1.2). In order to prove this difficult part we shall show that the approximate solutions tend to the solution strongly in $L^{2}(0, T;V)$ . To show this, we adopt the strong convergence argument used in Dautray-Lions [4, pp. 569-570] for the linear system. The continuous dependence result to linear equations is also proved in Section 3. The energy equality and the regularity $y¥in C([0, T];V)$ , $dy/dt$ $¥in C([0, T];H)$ for linear and nonlinear equations, without assuming $A_{2}(t)$ being bounded on $H$ , are established in Section 4. Their proofs are based on the result of Lions and Magenes [8] and do not require parabolic regularization in [4]. Additional continuous dependence result for linear equation is also given in Section 4. Finally in Section 5 we give three applications to $A_{i}(t)$

$A_{i}(t)$

$y$

$V_{i}$

$V_{2}^{¥prime}$

$A_{i}(t)$

$V^{¥mathrm{L}}¥rightarrow H$

$V_{2}$

Semilinear Second Order Evolution Equations

partial differential equations having different types of damping terms including the Sine-Gordon equations.

2. Existence and uniqueness First we explain the notations used in this paper. Let $X$ be a Hilbert denote the inner product and the induced norm on $X$ . and . space. denotes a dual pairing between denotes the dual space of $X$ and $X$ spaces to describe damped second Hilbert underlying and . Let us introduce is denoted order equations. Let $H$ be a real pivot Hilbert space, its norm . be a real separable Hilbert space. Assume . For $i=1,2$ let simply by $(V_{i},H)$ is a Gelfand triple space with a notation, that each pair is dense is continuous and , which means that an embedding is also continuous and the identified in $H$ , so that the embedding . From now on, we write $V_{1}=V$ for notational is dense in convenience. We shall give an exact description of damped second order evolution equations. Let $T>0$ be fixed. , $t¥in[0, 7]$ be a family of bilinear forms on $V¥times V$ satisfying Let $(¥cdot, ¥cdot)_{X}$

$||$

$||_{X}$

$X^{¥prime}$

$X^{¥prime}$

$¥langle¥cdot, ¥cdot¥rangle_{X^{¥prime},X}$

$||_{H}$

$||$

$V_{i}$

$¥cdot|_{H}$

$|$

$ V_{i^{¥mathrm{L}}}¥rightarrow H¥equiv$

$V_{i}¥subset H$

$H^{¥prime_{¥mathrm{L}}}¥rightarrow V_{i}^{¥prime}$

$V_{i}$

$H¥subset V_{i}^{¥prime}$

$H¥equiv H^{¥prime}$

$V_{i}^{¥prime}$

$a_{1}(t;¥phi, ¥varphi)$

(2. 1)

$a_{1}(t;¥phi, ¥psi)=a_{1}(t;¥psi, ¥phi)$

there exists

(2.2)

$c_{11}>0$

$a_{1}>0$

, $¥psi¥in V$ and

and

for all

$t¥rightarrow a_{1}(t;¥phi, ¥varphi)$

$¥phi$

,

$t¥in[0, T]$

for all

and

$¥psi¥in V$

;

and

$¥emptyset¥in V$

is continuously differentiate in

$|a_{1}^{¥prime}(t;¥phi, ¥varphi)|¥leq c_{12}||¥phi||_{V}||¥varphi||_{V}$

for all

$¥phi$

,

$t¥in[0, T]$

such that

$¥lambda_{1}¥in R=(-¥infty, +¥infty)$

$a_{1}(t;¥phi, ¥phi)+¥lambda_{1}|¥phi|_{H}^{2}¥geq a_{1}||¥phi||_{V}^{2}$

the function $c_{12}>0$ such that

(2.4)

$¥phi$

such that

$|a_{1}(t;¥phi, ¥psi)|¥leq c_{11}||¥phi||_{V}||¥varphi||_{V}$

and there exist

(2.3)

for all

$¥psi¥in V$

where $’=d/dt$ . Then we can define the operators $t¥in[0, T]$ deduced by the relations

$t¥in[0, T]$ $[0, ¥mathrm{T}]$

and

$A_{1}(t)$

,

,

$A_{1}^{¥prime}(t)¥in¥ovalbox{¥tt¥small REJECT}(V, V^{¥prime})$

for

,

$a_{1}(t;¥phi, ¥varphi)=¥langle A_{1}(t)¥phi, ¥varphi¥rangle_{V^{l},V}$

for all ,

$¥varphi¥in V$

(2.6)

$a_{1}^{¥prime}(t;¥phi, ¥varphi)=¥langle A_{1}^{¥prime}(t)¥phi, ¥varphi¥rangle_{V^{l},V}$

for all ,

$¥varphi¥in V$

$¥phi$

and there exists

$t¥in[0, T]$

(2.5)

$¥phi$

;

.

In order to consider a class of damping operators we introduce the second . It is assumed that , $t¥in[0, T]$ on family of bilinear forms $V_{2}¥times V_{2}$

$a_{2}(t;¥phi, ¥varphi)$

(2.7)

$a_{2}(t;¥phi, ¥psi)=a_{2}(t;¥psi,¥phi)$

there exists

$c_{21}>0$

(2.8)

for all

,

$¥psi¥in V_{2}$

and

$t¥in[0, T]$

;

such that

$|a_{2}(t;¥phi, ¥varphi)|¥leq c_{21}||¥phi||_{V_{2}}||¥varphi||_{V_{2}}$

and there exist

$¥phi$

$a_{2}>0$

and

$¥lambda_{2}¥in R$

for all

such that

$¥phi$

,

$¥psi¥in V_{2}$

and

$t¥in[0, T]$


(2.9)

for all

$a_{2}(t;¥phi, ¥phi)+¥lambda_{2}|¥phi|_{H}^{2}¥geq a_{2}||¥phi||_{V_{2}}^{2}$

(2.10)

the function

$t¥rightarrow a_{2}(t;¥phi, ¥varphi)$

Then we have a family of the operators the relation

(2. 11)

is continuous in $A_{2}(t)¥in g(V_{2}, V_{2}^{¥prime})$

$a_{2}(t;¥phi, ¥varphi)=¥langle A_{2}(t)¥phi, ¥varphi¥rangle_{V_{2}^{¥prime},V_{2}}$

We suppose that

$¥emptyset¥in V_{2}$

for all ,

$[0, T]$

,

.

$t¥in[0, 7]$

$¥varphi¥in V_{2}$

$¥phi$

; defined by

.

is continuously embedded in . Then we see that and the equalities for and for , . , We consider the following Cauchy problem for semilinear damped second order evolution equations $V$

$V_{2}$

$V^{¥epsilon}-¥rangle V_{2^{¥epsilon}}-¥succ H¥equiv H^{¥prime_{¥mathrm{C}}}->V_{2^{¥mathrm{e}}}^{¥prime}-¥rangle V^{¥prime}$

$¥emptyset¥in V_{2}^{¥prime}$

$¥varphi¥in V$

$¥langle¥phi, ¥varphi¥rangle_{V^{l},V}=¥langle¥phi, ¥varphi¥rangle_{V_{2}^{l},V_{2}}$

$¥langle¥phi, ¥varphi¥rangle_{V^{l},V}=(¥phi, ¥varphi)_{H}$

(2. 12)

$¥emptyset¥in H$

$¥varphi¥in V$

$¥left¥{¥begin{array}{l}¥frac{d^{2}y}{dt^{2}}+A_{2}(t)¥frac{dy}{dt}+A_{1}(t)y=f(t,y)¥mathrm{i}¥mathrm{n}(0,T),¥¥y(0)=y_{0}¥in V,¥frac{dy}{dt}(0)=y_{¥mathrm{l}}¥in H,¥end{array}¥right.$

where $f$ : . For the nonlinear term in (2.12) we impose the following assumptions on $f(t, ¥xi)$ : : $[0, T]¥times V_{2}¥rightarrow V_{2}^{¥prime}$

$[0, T]¥times V_{2}¥rightarrow V_{2}^{¥prime}$

is (strongly) measurable for all , (A1) (A2) there exists a $¥beta¥in L^{2}(0, T;R^{+})$ such that $t¥rightarrow f(t, ¥xi)$

$¥xi$

$||f(t, ¥xi)-f(t, ¥zeta)||_{V_{2}^{¥prime}}¥leq¥beta(t)||¥xi-¥zeta||_{V_{2}}¥mathrm{a}.¥mathrm{e}$

(A3) there exists a

such that

$¥gamma¥in L^{2}(0, T;R^{+})$

We write $g^{¥prime}=dg/dt$, a space of solutions, by

$g^{¥prime¥prime}=d^{2}g/dt^{2}$

.

$t$

,

$||f(t, 0)||_{V_{2}^{J}}¥leq¥gamma(t)¥mathrm{a}.¥mathrm{e}$

.

$t$

and define a Hilbert space, which will be

$W(0, T)=¥{g|g¥in L^{2}(0, T;V), g^{¥prime}¥in L^{2}(0, T;V_{2}), g^{¥prime¥prime}¥in L^{2}(0, T;V^{¥prime})¥}$

The norm of

$W(0, T)$

$¥ovalbox{¥tt¥small REJECT}^{¥prime}(0, T)$

.

is given by

$||g||_{W(0,T)}=(||g||_{L^{2}(0,T;V)}^{2}+||g^{¥prime}||_{L^{2}(0,T;V_{2})}^{2}+||g^{¥prime¥prime}||_{L^{2}(0,T;V^{¥prime})}^{2})^{1/2}$

We denote by

.

the space of distributions on

Definition 2.1. A function $y¥in W(0, 7)$ and satisfies

$y$

$(0, T)$

.

.

is said to be a weak solution of (2.12) if

$y$

(2. 13)

$¥langle y^{¥prime¥prime}(¥cdot), ¥phi¥rangle_{V^{¥prime},V}+a_{2}(¥cdot;y^{¥prime}(¥cdot), ¥phi)+a_{1}(¥cdot;y(¥cdot), ¥phi)=¥langle f(¥cdot,y(¥cdot)), ¥phi¥rangle_{V_{2}^{l},V_{2}}$

for all

(2. 14)

,

$y(0)=y_{0}¥in V$

in the sense of

$¥emptyset¥in V$

$¥frac{dy}{dt}(0)=y_{1}¥in H$

.


,


Now we can state the existence and uniqueness results of a weak solution

of (2. 12). Theorem 2.1. Assume that respectively and $f$ satisfy solution in $W(0, T)$ .

satisfy $(2.1)-(2.4)$ and (2.7)?(2.10), and . Then the problem (2. 12) has a unique weak

$a_{1}$

$(¥mathrm{A}1)-(¥mathrm{A}3)$

$a_{2}$

$y$

We then specify the above Theorem 2.1 to the linear problem:

(2. 15)

$¥left¥{¥begin{array}{l}¥frac{d^{2}y}{dt^{2}}+A_{2}(t)¥frac{dy}{dt}+A_{¥mathrm{l}}(t)y=f(t)¥mathrm{i}¥mathrm{n}(0,T),¥¥y(0)=y_{0}¥in V,¥frac{dy}{dt}(0)=y_{¥mathrm{l}}¥in H,¥end{array}¥right.$

¥ where ¥ . The definition of a weak solution of (2.15) is same as given in Definition 2.1. $f in L^{2}(0, T;V_{2}^{ prime})$

Corollary 2.1. Assume that and satisfy $(2.1)-(2.4)$ and (2.7)?(2.10), ¥ respectively and ¥ . Then the problem (2.15) has a unique weak in $W(0, T)$ . Moreover, the solution solution depends continuously on the data, that is, the map $a_{1}$

$a_{2}$

$f in L^{2}(0, T;V_{2}^{ prime})$

$y$

$y$

$(f, y_{0},y_{1})¥rightarrow y$

is continuous

3.

from

$L^{2}(0, T;V_{2}^{¥prime})¥times V¥times H$

into

$W(0, T)$ .

Proofs of Theorem 2.1 and Corollary 2.1

Existence proof of Theorem 2.1. We divide the existence proof into four steps. Step 1. Approximate solutions We use the Faedo-Galerkin approximation as in [4]. Since $V$ is separable, there exists a basis in $V$ such that is a complete orthonomal system in $H$ , (i) (ii) the set of all finite linear combinations, $¥{¥xi_{j}w_{j}|¥xi_{j}¥in R,m¥in N¥}$ is dense in $V$, where $N$ is the set of natural numbers. For each $m¥in N$ we define an approximate solution of the problem (2.12) by $¥{w_{m}¥}_{m=1}^{¥infty}$

$¥{w_{m}¥}_{m=1}^{¥infty}$

$y_{m}(t)=¥sum_{j=1}^{m}g_{jm}(t)w_{j}$

where

$y_{m}(t)$

satisfies

,


(3.1)

$¥left¥{¥begin{array}{l}¥frac{d^{2}}{dt^{2}}(y_{m}(t),w_{j})_{H}+a_{2}(t¥cdot.y_{m}^{¥prime}(t),w_{j})+a_{1}(t¥cdot.y_{m}(t),w_{j})¥¥=¥langle f(t,y_{m}(t)),w_{j}¥rangle_{V_{2}^{¥prime},V_{2}},t¥in[0,T],1¥leq j¥leq m,¥¥y_{m}(0)=y_{0m},¥¥¥frac{d}{dt}y_{m}(0)=y_{1m}.¥end{array}¥right.$

By (i) and (ii) we can find real numbers that

$¥xi_{im}^{0}$

(3.2)

$y_{0m}=¥sum_{i=1}^{m}¥xi_{im}^{0}w_{i}¥rightarrow y0$

(3.2)

$y_{1m}=¥sum_{i=1}^{m}¥xi_{im}^{1}w_{i}¥rightarrow y_{1}$

Then the equation (3.1) can be written as

and

$¥xi_{im}^{1}$

,

$i=1,2$ ,

as

$ m¥rightarrow¥infty$

,

in $H$ as

$ m¥rightarrow¥infty$

.

in

$m$

$V$

$¥ldots,m$

,

$m¥in N$

such

vector differential equation

$¥frac{d^{2}}{dt^{2}}¥overline{g}_{m}+¥tilde{A}_{2}(t)¥frac{d}{dt}¥tilde{g}_{m}+¥tilde{A}_{1}(t)¥tilde{g}_{m}=¥tilde{f}(t,¥tilde{g}_{m})$

with

initial

values . Here

and

$[¥xi_{1m}^{0}, ¥xi_{2m^{ }}^{0},¥ldots, ¥xi_{mm}^{0}]^{t}$

$¥tilde{g}_{m}(0)=$

$(d¥tilde{g}_{m}/dt)(0)=$

, , and . Then denotes the transpose of , where is vector forcing function and the of class are elements of , it follows Lipschitz continuous. Indeed, for by the assumption (A2) that $[¥xi_{1m}^{1}, ¥xi_{2m}^{1}, ¥ldots, ¥xi_{mm}^{1}]^{t}$

$¥tilde{g}_{m}=[g_{1m}, ¥ldots, g_{mm}]^{t}$

$¥overline{f}(t,¥tilde{g}_{m})=$

$¥tilde{A}_{2}(t)=(a_{2}(t;w_{i}, w_{j})_{j=1,,m}^{i=1,m}¥dot{.}¥ldots¥ldots’)$

$¥tilde{A}_{1}(t)=(a_{1}(t;w_{i}, w_{j})_{j=1,,m}^{i=1,m}¥dot{.}¥ldots¥ldots’)$

$[¥langle f(t, ¥sum_{j=1}^{m}g_{jm}w_{j}), w_{1}¥rangle_{V_{2}^{¥prime},V_{2}},$

$¥ldots$

$[¥cdots]$

$[¥cdots]^{t}$

$¥langle f(t, ¥sum_{j=1}^{m}g_{jm}w_{j}), w_{m}¥rangle_{V_{2}^{¥prime},V_{2}}]^{t}$

$¥tilde{A}_{1}(t),¥tilde{A}_{2}(t)$

,

$C^{1_{-}}$

$¥overline{f}$

$¥vec{g}_{m}=¥sum_{j=1}^{m}g_{jm}w_{j},¥overline{h}_{m}=¥sum_{j=1}^{m}h_{jm}w_{j}$

$|¥overline{f}(t,¥overline{g}_{m})-¥overline{f}(t,¥vec{h}_{m})|^{2}=¥sum_{i=1}^{m}|¥{f(t,$

$¥sum_{j=1}^{m}g_{jm}w_{j})-f(t$

,

$¥sum_{j=1}^{m}h_{jm}w_{j})$

,

$w_{i}¥}_{V_{2}^{¥prime},V_{2}}|^{2}$

$¥leq¥beta(t)^{2}(.$ $¥sum_{i=1}^{m}||w_{i}||_{V_{2}}^{2})^{3}¥sum_{j=1}^{m}|g_{jm}-h_{jm}|^{2}$

.

$=¥beta(t)^{2}(¥sum_{i=1}^{m}||w_{i}||_{V_{2}}^{2})^{3}|¥overline{g}_{m}-¥tilde{h}_{m}|^{2}$

Therefore this second order vector differential equation admits a unique solution on $[0, T]$ , by reducing this to a first order system and applying Caratheodory type existence theorem. Hence we can construct the approximate solutions of (3.1). $¥overline{g}_{m}$

$y_{m}(t)$


Step 2. A priori estimates In this step we shall derive a priori estimates of $y_{m}(t)$ . We multiply both sides and sum over to have of the equation (3.1) by $j$

$g_{jm}^{¥prime}(t)$

(3.4)

$(y_{m}^{¥prime¥prime}(t), y_{m}^{¥prime}(t))_{H}+a_{2}(t;y_{m}^{¥prime}(t), y_{m}^{¥prime}(t))+a_{1}(t;y_{m}(t), y_{m}^{¥prime}(t))$

$=¥langle f(t,y_{m}(t)),y_{m}^{¥prime}(t)¥rangle_{V_{2}^{l},V_{2}}$

.

It is easily verified by the differentiation of

(3.5)

$a_{1}$

and symmetry (2.1) that

$a_{1}(t;y_{m}^{¥prime}(t),y_{m}(t))=¥frac{1}{2}¥frac{d}{dt}a_{1}(t;y_{m}(t), y_{m}(t))-¥frac{1}{2}a_{1}^{¥prime}(t;y_{m}(t),y_{m}(t))$

and

(3.6)

$(y_{m}^{¥prime¥prime}(t),y_{m}^{¥prime}(t))_{H}=¥frac{1}{2}¥frac{d}{dt}|y_{m}^{¥prime}(t)|_{H}^{2}$

.

, $i=1,2$ in all estimations in what follows. For simplifying notations let Then by substituting (3.5) and (3.6) to (3.4), we have $¥lambda_{i}=|¥lambda_{i}|$

(3.7)

$¥frac{1}{2}¥frac{d}{dt}[a_{1}(t;y_{m}(t),y_{m}(t))+|y_{m}^{¥prime}(t)|_{H}^{2}]-¥frac{1}{2}a_{1}^{¥prime}(t;y_{m}(t), y_{m}(t))+a_{2}(t;y_{m}^{¥prime}(t), y_{m}^{¥prime}(t))$

$=¥langle f(t,y_{m}(t)),y_{m}^{¥prime}(t)¥rangle_{V_{2}^{¥prime},V_{2}}$

Let

$¥epsilon>0$

that

(3.8)

.

be the imbedding constant such be an arbitrary real number and for all . From (A2) and (A3) we also obtain $k_{2}$

$¥emptyset¥in V$

$||¥phi||_{V_{2}}¥leq k_{2}||¥phi||_{V}$

$2|¥int_{0}^{t}¥langle f(¥sigma,y_{m}(¥sigma)),y_{m}^{¥prime}(¥sigma)¥rangle_{V_{2}^{l},V_{2}}d¥sigma|$

$=2|¥int_{0}^{t}¥langle f(¥sigma,y_{m}(¥sigma))-f(¥sigma, 0)+f(¥sigma, 0),y_{m}^{¥prime}(¥sigma)¥rangle_{V_{2}^{l},V_{2}}d¥sigma|$

$¥leq 2¥int_{0}^{t}¥beta(¥sigma)||y_{m}(¥sigma)||_{V_{2}}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}d¥sigma+2¥int_{0}^{t}¥gamma(¥sigma)||y_{m}^{¥prime}(¥sigma)||_{V_{2}}d¥sigma$

$¥leq¥frac{1}{¥epsilon}||¥gamma||_{L^{2}(0,T;R^{+})}^{2}+¥frac{k_{2}^{2}}{¥epsilon}¥int_{0}^{t}¥beta(¥sigma)^{2}||y_{m}(¥sigma)||_{V}^{2}d¥sigma+2¥epsilon¥int_{0}^{t}||y_{m}^{¥prime}(t)||_{V_{2}}^{2}d¥sigma$

.

Integrating (3.7) on and using (2.2)?(2.4), (2.8)?(2.10) and (3.8), we have the following inequality $[0, t]$


(3.9)

$ a_{1}||y_{m}(t)||_{V}^{2}+|y_{m}^{¥prime}(t)|_{H}^{2}+2a_{2}¥int_{0}^{t}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma$

$¥leq c_{11}||y_{0m}||_{V}^{2}+|y_{1m}|_{H}^{2}+¥lambda_{1}|y_{m}(t)|_{H}^{2}$

$+c_{12}¥int_{0}^{t}||y_{m}(¥sigma)||_{V}^{2}d¥sigma+2¥lambda_{2}¥int_{0}^{t}|y_{m}^{¥prime}(¥sigma)|_{H}^{2}d¥sigma$

$+¥frac{1}{¥epsilon}||¥gamma||_{L^{2}(0,T;R^{+})}^{2}+¥frac{k_{2}^{2}}{¥epsilon}¥int_{0}^{t}¥beta(¥sigma)^{2}||y_{m}(t)||_{V}^{2}d¥sigma+2¥epsilon¥int_{0}^{t}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma$

Since the equality,

$y_{m}(t)=y_{0m}+¥int_{0}^{t}y_{m}^{¥prime}(s)ds$

.

implies

$|y_{m}(t)|_{H}^{2}¥leq 2|y_{m}(0)|_{H}^{2}+2T¥int_{0}^{t}|y_{m}^{¥prime}(s)|_{H}^{2}ds$

and since and (3.9) , it follows from (3.9) that $||y_{0m}||_{V}¥leq c_{1}||y_{0}||_{V}$

$|y_{1m}|_{H}¥leq c_{2}|y_{1}|_{H}$

for some

$c_{1}$

, $c_{2}>0$ (see (3.2),

$)$

(3.10)

$ a_{1}||y_{m}(t)||_{V}^{2}+|y_{m}^{¥prime}(t)|_{H}^{2}+2(a_{2}-¥epsilon)¥int_{0}^{t}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma$

$¥leq c_{1}^{2}(c_{11}+2¥lambda_{1}k_{1}^{2})||y_{0}||_{V}^{2}+c_{2}^{2}|y_{1}|_{H}^{2}+¥frac{1}{¥epsilon}||¥gamma||_{L^{2}(0,T;¥mathrm{R}^{+})}^{2}$

,

$+¥int_{0}^{t}(c_{12}+¥frac{k_{2}^{2}}{¥epsilon}¥beta(¥sigma)^{2})||y_{m}(¥sigma)||_{V}^{2}d¥sigma+2(¥lambda_{1}T+¥lambda_{2})¥int_{0}^{t}|2y_{m}^{¥prime}(¥sigma)|_{H}^{2}d¥sigma$

is the embedding positive constant such that . Let us divide (3.10) by $a=$ $¥min¥{¥alpha_{1},1¥}>0$ . We choose small such that $¥eta=2¥alpha^{-1}(a_{2}-¥epsilon)>0$ and set

where

for all sufficiently

$|¥phi|_{H}¥leq k_{1}||¥phi||_{V}$

$k_{1}$

$¥emptyset¥in V$

$C=¥frac{1}{¥alpha}[c_{1}^{2}(c_{11}+2¥lambda_{1}k_{1}^{2})||y_{0}||_{V}^{2}+c_{2}^{2}|y_{1}|_{H}^{2}+¥epsilon^{-1}||¥gamma||_{L^{2}(0,T;R^{+})}]$

$¥epsilon$

.

Then (3.10) implies

(3.10)

$||y_{m}(t)||_{V}^{2}+|y_{m}^{¥prime}(t)|_{H}^{2}+¥eta¥int_{0}^{t}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma$

,

$¥leq C+¥int_{0}^{t}¥tilde{¥beta}(¥sigma)(||y_{m}(¥sigma)||_{V}^{2}+|y_{m}^{¥prime}(¥sigma)|_{H}^{2})d¥sigma$

where Gronwall’s inequality that

$¥tilde{¥beta}(¥sigma)=c_{12}+2(¥lambda_{1}T+¥lambda_{2})+¥epsilon^{-1}k_{2}^{2}¥beta(¥sigma)^{2}$

(3.12)

$||y_{m}(t)||_{V}^{2}+|y_{m}^{¥prime}(t)|_{H}^{2}¥leq$

Cexp

.

Thus it follows by Bellman-

$(¥int_{0}^{t}¥tilde{¥beta}(¥sigma)d¥sigma)¥leq$

Cexp(B),


where

$¥mathrm{B}=||¥tilde{¥beta}||_{L^{1}(0,T;R^{+})}$

(3.13)

By substituting (3.12) to (3.11), we also obtain

.

Cexp(B)B

$||y_{m}(t)||_{V}^{2}+|y_{m}^{¥prime}(t)|_{H}^{2}+¥eta¥int_{0}^{t}||y_{m}^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma¥leq C+$

Step 3.

$j$

and using inte-

$¥int_{0}^{T}[-(y_{m_{l}}^{¥prime}(t), ¥phi_{j}^{¥prime}(t))_{H}+a_{2}(t;y_{m_{l}}^{¥prime}(t), ¥phi_{j}(t))+a_{1}(t;y_{m_{l}}(t), ¥phi_{j}(t))]dt$

$=¥int_{0}^{T}¥langle f(t, y_{m_{l}}(t)), ¥phi_{j}(t)¥rangle_{V_{2}^{l},V_{2}}dt-(y_{1_{m_{l}}}, ¥phi_{j}(0))_{H}$

If we take

(3.24)

$¥mathit{1}¥rightarrow¥infty$

.

in (3.23) and use (3.15)?(3.17), (3.19), then we have

$¥int_{0}^{T}[-(z^{¥prime}(t), ¥phi_{j}^{¥prime}(t))_{H}+a_{2}(t;z^{¥prime}(t), ¥phi_{j}(t))+a_{1}(t;z(t), ¥phi_{j}(t))]dt$

,

$=¥int_{0}^{T}¥langle ¥mathrm{Y}(t), ¥phi_{j}(t)¥rangle_{V_{2}^{J},V_{2}}dt-(y_{1}, ¥phi_{j}(0))_{H}$

so that

(3.25)

$¥int_{0}^{T}¥zeta^{¥prime}(t)(-z^{¥prime}(t), w_{j})_{H}dt+¥int_{0}^{T}¥zeta(t)¥{a_{2}(t;z^{¥prime}(t), w_{j})+a_{1}(t;z(t), w_{j})$

$-¥langle ¥mathrm{Y}(t), w_{j}¥rangle_{V_{2}^{¥prime},V_{2}}¥}dt=-¥zeta(0)(y_{1}, w_{j})_{H}$

It we take

(3.26)

$¥zeta¥in¥ovalbox{¥tt¥small REJECT}(0, T)$

.

in (3.25), then

$¥frac{d}{dt}(z^{¥prime}(¥cdot), w_{j})_{H}+a_{2}(¥cdot;z^{¥prime}(¥cdot), w_{j})+a_{1}(¥cdot;z(¥cdot), w_{j})=¥langle ¥mathrm{Y}(¥cdot), w_{j}¥rangle_{V_{2}^{r},V_{2}}$

in the sense of distribution in , we conclude by (3.26) that for all


.

Since

$¥{¥sum_{j=1}^{m}¥xi_{j}w_{j}|¥xi_{j}¥in R,m¥in N¥}$

is dense and

$z^{¥prime}=-A_{1}(t)z-A_{2}(t)z^{¥prime}-¥mathrm{Y}¥in L^{2}(0, T;V^{¥prime})$

$¥mathrm{V}$

$¥emptyset¥in V$

(3.27)

$¥langle z^{¥prime}(¥cdot), ¥phi¥rangle_{V^{¥prime},V}+a_{2}(¥cdot;z^{¥prime}(¥cdot), ¥phi)+a_{1}(¥cdot;z(¥cdot), ¥phi)=¥langle ¥mathrm{Y}(¥cdot), ¥phi¥rangle_{V_{2}^{l},V_{2}}$

in the sense of . Multiplying both sides of (3.26) by using integration by parts, we have from (3.24) $¥ovalbox{¥tt¥small REJECT}^{¥prime}(0, T)$

$(z^{¥prime}(0), w_{j})_{H}¥zeta(0)=(y_{1}, w_{j})_{H}¥zeta(0)$

$¥zeta$

in (3.22) and

,

Since is dense in , we obtain . This also proves that is a weak solution of the linear problem $f(t)$ in which replaced is by . (2.15)

and that

$z^{¥prime}(0)=y_{1}$

$(z^{¥prime}(0), w_{j})_{H}=(y_{1}, w_{j})_{H}$

.

$¥{w_{j}¥}_{j=1}^{¥infty}$

$z$

$¥mathrm{Y}(t)$

$¥mathrm{H}$

11


Step 4. Strong convergence of approximate solutions The most difficult part of existence proof is prove that strongly in (3.27). In order to prove this we shall show In what follows we write $y_{m_{l}}=y_{m}$ for simplicity. , we have Integrating (3.7) on

$¥mathrm{Y}(¥cdot)=f(¥cdot,z(¥cdot))$

$y_{m_{l}}¥rightarrow z$

in

$L^{2}(0, T;V)$ .

$[0, t]$

(3.28)

$a_{1}(t;y_{m}(t),y_{m}(t))+|y_{m}^{¥prime}(t)|_{H}^{2}+2¥int_{0}^{t}$

$¥mathrm{a}_{2}(¥sigma;y_{m}^{¥prime}(¥sigma), y_{m}^{¥prime}(¥sigma))d¥sigma$

$=a_{1}(0;y_{m0},y_{m0})+|y_{m1}|_{H}^{2}+¥int_{0}^{t}a_{1}^{¥prime}(¥sigma;y_{m}(¥sigma),y_{m}(¥sigma))d¥sigma$

$+2¥int_{0}^{t}¥langle f(¥sigma,y_{m}(¥sigma)),y_{m}^{¥prime}(¥sigma)¥rangle_{V_{2}^{l},V_{2}}d¥sigma$

.

Since is a weak solution of (2.15) with $f(t)=¥mathrm{Y}(t)$ , we can prove the following energy equality (see Lemma 4.3 below and Ha [5] for more detailed proof) $z$

(3.29)

$a_{1}(t;z(t),z(t))+|z^{¥prime}(t)|_{H}^{2}+2¥int_{0}^{t}$

$¥mathrm{a}_{2}(¥sigma; z^{¥prime}(¥sigma),z^{¥prime}(¥sigma))d¥sigma$

$=a_{1}(0;y_{0},y_{0})+|y_{1}|_{H}^{2}+¥int_{0}^{t}a_{1}^{¥prime}(¥sigma;z(¥sigma),z(¥sigma))d¥sigma$

$+2¥int_{0}^{t}$

For each

$t¥in[0, T]$

$¥langle ¥mathrm{Y}(¥sigma),z^{¥prime}(¥sigma)¥rangle_{V_{2}^{¥prime},V_{2}}d¥sigma$

.

, the following equalities hold:

$a_{1}(t;y_{m}y_{m})+a_{1}(t;z,z)=a_{1}(t;y_{m}-z,y_{m}-z)+2a_{1}(t;y_{m},z)$

; ;

$a_{2}(t;y_{m}^{¥prime},y_{m}^{¥prime})+a_{2}(t;z^{¥prime},z^{¥prime})=a_{2}(t;y_{m}^{¥prime}-z^{¥prime},y_{m}^{¥prime}-z^{¥prime})+2a_{2}(t;y_{m}^{¥prime},z^{¥prime})$

$a_{1}^{¥prime}(t;y_{m},y_{m})+a_{1}^{¥prime}(t;z,z)=a_{1}^{¥prime}(t;y_{m}-z,y_{m}-z)+2a_{1}^{¥prime}(t;y_{m},z)$

;

;

$|y_{m}^{¥prime}(t)|_{H}^{2}+|z^{¥prime}(t)|_{H}^{2}=|y_{m}^{¥prime}(t)-z^{¥prime}(t)|_{H}^{2}+2(y_{m}^{¥prime}(t),z^{¥prime}(t))_{H}$

$¥langle f(t,y_{m}),y_{m}^{¥prime}¥rangle_{V_{2}^{t},V_{2}}+¥langle ¥mathrm{Y}(t),z^{¥prime}(t)¥rangle_{V_{2}^{l},V_{2}}$

$=¥langle f(t,y_{m})-f(t,z),y_{m}^{¥prime}-z^{¥prime}¥rangle_{V_{2}^{J},V_{2}}+¥langle f(t,z)-¥mathrm{Y}(t),y_{m}^{¥prime}-z^{¥prime}¥rangle_{V_{2}^{l},V_{2}}$

$+¥langle f(t,y_{m}),z^{¥prime}¥rangle_{V_{2}^{l},V_{2}}+¥langle ¥mathrm{Y}(t),y_{m}^{¥prime}¥rangle_{V_{2}^{l},V_{2}}$

.

Adding (3.28) to (3.29) and using the above equalities, we have

(3.30)

$a_{1}(t;y_{m}(t)-z(t),y_{m}(t)-z(t))+|y_{m}^{¥prime}(t)-z^{¥prime}(t)|_{H}^{2}$

$+2¥int_{0}^{t}a_{2}(¥sigma;y_{m}^{¥prime}-z^{¥prime},y_{m}^{¥prime}-z^{¥prime})d¥sigma$

12


$=¥mathrm{Y}_{m}^{0}+¥sum_{i=1}^{3}¥mathrm{Y}_{m}^{i}(t)+¥int_{0}^{t}a_{1}^{¥prime}(¥sigma;y_{m}-z,y_{m}-z)d¥sigma$

$+2¥int_{0}^{f}¥langle f(¥sigma,y_{m})-f(¥sigma,z),y_{m}^{¥prime}-z^{¥prime}¥rangle_{V_{2}^{¥prime},V_{2}}d¥sigma$

,

where

(3.31)

$¥mathrm{Y}_{m}^{0}=a_{1}(0;y_{m0}, y_{m0})+|y_{m1}|_{H}^{2}+a_{1}(0;y_{0},y_{0})+|y_{1}|_{H}^{2}$

(3.32)

$¥mathrm{Y}_{m}^{1}(t)=-2a_{1}(t;y_{m}(t),z(t))-2(y_{m}^{¥prime}(t),z^{¥prime}(t))_{H}$

(3.33)

$¥mathrm{Y}_{m}^{2}(t)=-4¥int_{0}^{t}a_{2}(¥sigma;y_{m}^{¥prime},z^{¥prime})d¥sigma+2¥int_{0}^{t}a_{1}^{¥prime}$

(3.34)

,

,

$(¥sigma;y_{m},z)d¥sigma$

,

$¥mathrm{Y}_{m}^{3}(t)=2¥int_{0}^{t}[¥langle f(¥sigma,y_{m}),z^{¥prime}¥rangle_{V_{2}^{¥prime},V_{2}}+¥langle ¥mathrm{Y}(¥sigma),y_{m}^{¥prime}¥rangle_{V_{2}^{J},V_{2}}]$$ d¥sigma$

.

$+2¥int_{0}^{t}¥langle f(¥sigma,z)- ¥mathrm{Y}(¥sigma),y_{m}^{¥prime}-z^{¥prime}¥rangle_{V_{2}^{¥prime},V_{2}}d¥sigma$

We set

(3.35)

$¥mathrm{Y}_{m}(t)=¥mathrm{Y}_{m}^{0}+¥sum_{i=1}^{3}¥mathrm{Y}_{m}^{i}(t)$

.

By similar calculations as in the step 2, the equalities (3.30) and (3.35) imply

(3.36)

$ a_{1}||y_{m}(t)-z(t)||_{V}^{2}+|y_{m}^{¥prime}(t)-y^{¥prime}(t)|_{H}^{2}+(2a_{2}-¥epsilon)¥int_{0}^{t}||y_{m}^{¥prime}(¥overline{¥sigma})-z^{¥prime}(¥sigma)||_{V_{2}}^{2}d¥sigma$

$¥leq ¥mathrm{Y}_{m}(t)+2¥lambda_{1}k_{1}^{2}||y_{m0}-y_{0}||_{V}^{2}+2¥lambda_{1}T¥int_{0}^{t}|y_{m}^{¥prime}(¥sigma)-z^{¥prime}(¥sigma)|_{H}^{2}d¥sigma$

$+¥int_{0}^{t}[c_{11}+¥epsilon^{-1}k_{2}^{2}¥beta^{2}(¥sigma)]||y_{m}(¥sigma)-z(¥sigma)||_{V}^{2}d¥sigma$

. We devide (3.36) by $a=¥min¥{a_{1},1¥}>0$ and choose for any small such that $¥gamma=a^{-1}(2a_{2}-¥epsilon)>0$ . If we set $¥epsilon>0$

(3.37)

$¥Phi_{m}(t)=||y_{m}(t)-z(t)||_{V}^{2}+|y_{m}^{¥prime}(t)-z^{¥prime}(t)|_{H}^{2}$

(3.38)

$Z_{m}(t)=a^{-1}(¥mathrm{Y}_{m}(t)+2¥lambda_{1}k_{1}^{2}||y_{m0}-y_{0}||_{V}^{2})$

and

(3.39)

$h(t)=a^{-1}(c_{11}+2¥lambda_{1}T+¥epsilon^{-1}k_{2}^{2}¥beta^{2}(t))$

,

,

,

$¥epsilon$

sufficiently


13

then the inequality (3.36) implies

(3.40)

.

$¥Phi_{m}(t)¥leq Z_{m}(t)+¥int_{0}^{t}h(s)¥Phi_{m}(s)ds$

Since $Z_{m}(t)$ is continuous and Gronwall inequality to have

(3.41)

$h(s)¥geq 0$

, we can apply the extended Bellman-

$¥Phi_{m}(t)¥leq Z_{m}(t)+¥int_{0}^{t}¥exp(¥int_{s}^{t}h(¥tau)d¥tau)h(s)Z_{m}(s)ds$

We let $K(t,s)=$ see easily that

and

$¥exp(¥int_{s}^{t}h(¥tau)d¥tau)h(s)$

.

$M_{m}(t)=¥int_{0}^{t}K(t,s)Z_{m}(s)ds$

.

Then we

$|K(t,s)|¥leq$ $¥exp(||h||_{L^{1}(0,T;R^{+})})h(s)$

and

$M_{m}(t)$

is uniformly bounded on

(3.42)

$[0, T]$

$¥lim_{m¥rightarrow¥infty}¥int_{0}^{t}Z_{m}(s)ds=0$

for each $t¥in[0, T]$ . prove that

(3.43)

By (3.38) and

$¥lim_{m¥rightarrow¥infty}¥int_{0}^{t}¥mathrm{Y}_{m}(s)ds=0$

.

and

$y_{m0}¥rightarrow y_{0}$

and

We shall show that $¥lim_{m¥rightarrow¥infty}M_{m}(t)=0$

strongly in

, it is sufficient to

$V$

.

$¥lim_{m¥rightarrow¥infty}¥int_{0}^{t}K(t,s)¥mathrm{Y}_{m}(s)ds=0$

Now we consider the integral $I_{m}=¥int_{0}^{t}K(t,s)¥mathrm{Y}_{m}(s)ds=(¥int_{0}^{t}K(t,s)ds)¥mathrm{Y}_{m}^{0}+¥sum_{i=1}^{3}¥int_{0}^{t}K(t,s)¥mathrm{Y}_{m}^{i}(s)ds$

Since


strongly in

(3.44) For each

$V$

and


strongly in

$¥mathrm{Y}_{m}^{0}¥rightarrow 2a_{1}(0;y_{0},y_{0})+2|y_{1}|_{H}^{2}$

$t¥in[0, T]$

(3.45)

, we see

.

we have by (3.17), (3.18), (3.19) and (3.15) that

$¥mathrm{Y}_{m}^{2}(t)¥rightarrow-4¥int_{0}^{t}a_{2}(¥sigma;z^{¥prime},z^{¥prime})d¥sigma+2¥int_{0}^{t}a_{1}^{¥prime}(¥sigma;z,z)d¥sigma$

(3.36) We note that

$¥mathrm{V}$

.

$¥mathrm{Y}_{m}^{3}(t)¥rightarrow 4¥int_{0}^{t}$

$¥mathrm{Y}_{m}^{2}(t)$

and

$¥mathrm{Y}_{m}^{3}(t)$

$¥langle ¥mathrm{Y}(¥sigma),z^{¥prime}¥rangle_{V_{2}^{¥prime},V_{2}}d¥sigma$

,

.

are uniformly bounded on

$[0, T]$

.

Since

14


$K(t, ¥cdot)¥in L^{1}(0, t;R^{+})$

(3.47)

, then it follows from (3.16) and (3.15) that

$¥int_{0}^{t}K(t,¥sigma)¥mathrm{Y}_{m}^{1}(¥sigma)ds$

$=-2¥int_{0}^{t}¥langle A_{1} (¥sigma)y_{m}(¥sigma),K(t, ¥sigma)z(¥sigma)¥rangle_{V_{2}^{l},V_{2}}d¥sigma$

?

$2$

$¥int_{0}^{t}¥langle y_{m}^{¥prime}(¥sigma),K(t, ¥sigma)z^{¥prime}(¥sigma)¥rangle_{V_{2}^{l},V_{2}}d¥sigma$

$¥rightarrow-2¥int_{0}^{t}K(t,¥sigma)¥{a_{1} (¥sigma; z(¥sigma),z(¥sigma))+|z^{¥prime}(¥sigma)|_{H}^{2}¥}d¥sigma$

.

We also note that the integrals in (3.47) are uniformly bounded on $[0, T]$ . Hence, by using (3.44)?(3.47) and the Lebesgue dominated convergence theorem, we have

(3.48)

$I_{m}¥rightarrow 2¥int_{0}^{t}K(t,s)¥{a_{1}(0;y_{0},y_{0})+|y_{1}|_{H}^{2}¥}ds$

$+2¥int_{0}^{t}K(t,s)¥{-a_{1}(s;z(s),z(s))-|¥mathrm{z}^{¥prime}(s)|_{H}^{2}¥}ds$

+2

$¥int_{0}^{t}K(t,s)¥{-2¥int_{0}^{s}a_{2}(¥sigma; z^{¥prime},z^{¥prime})d¥sigma+¥int^{ss}a_{1}^{¥prime}$

+2

$¥int_{0}^{t}K(t,s)¥{2¥int^{ss}$

$(¥sigma; z,z)d¥sigma¥}ds$

$¥langle ¥mathrm{Y}(¥sigma),z^{¥prime}¥rangle_{V_{2}^{l},V_{2}}d¥sigma¥}ds=0$

,

because of (3.29). This shows the second part of (3.43). Similarly we can prove the first part of (3.43). Thus (3.42) is shown. By integrating (3.41) on $[0, T]$ and using (3.42) and the Lebesgue dominated convergence theorem, we verify that

(3.49)

$¥lim_{m¥rightarrow¥infty}¥int_{0}^{T}¥Phi_{m}(s)ds=0$

.

converges strongly to This implies that in $L^{2}(0, T;V)$ , and hence in . Then, it follows from (A2) and (3.19) that in . Therefore, we prove the existence of a weak solution of $z$

$y_{m}$

$L^{2}(0, T;V_{2})$

$¥mathrm{Y}(¥cdot)=f(¥cdot,z(¥cdot))$

$L^{2}(0, T;V_{2}^{¥prime})$

$z$

(2. 12). Uniqueness proof of Theorem 2.1. The proof of uniqueness easily follows from the step 4 of that of proof. Indeed, let and be the solutions of (2. 12) and let $z=y_{1}-y_{2}$ . Then by the energy equality we have $y_{1}$

$y_{2}$


(3.50)

15

$ a_{1}(t;z(t),z(t))+|z^{¥prime}(t)|_{H}^{2}+2¥int_{0}^{t}a_{2}(¥sigma; z^{¥prime}(¥sigma),z^{f}(¥sigma))d¥sigma$

$=¥int_{0}^{t}a_{1}^{¥prime}(¥sigma;z(¥sigma),z(¥sigma))d¥sigma$

$+¥int_{0}^{t}¥langle f(¥sigma,y_{1}(¥sigma))-f(¥sigma,y_{2}(¥sigma)),z^{¥prime}(¥sigma)¥rangle_{V_{2}^{J},V_{2}}d¥sigma$

.

Now by the similar calculations as in step 4 (see (3.30) and note that in this case), we have

(3.51)

for all

$||z(t)||_{V}^{2}+|z^{¥prime}(t)|_{H}^{2}=0$

$t¥in[0, T]$

$¥mathrm{Y}_{m}(t)¥equiv 0$

.

Therefore the uniqueness is proved.

Proof of

Corollary 2.1. For the linear problem (2.15) the step 4 is unnecessary. Now it is enough to prove the continuous dependence on the data. By setting in (A2), we can derive the estimate (3.13) for linear equation (2.15). Integrating (3.13) on $[0, T]$ we have for some $C>0$ $¥beta(t)¥equiv 0$

(3.50)

$||y_{m}||_{L^{2}(0,T;V)}^{2}+¥eta T||y_{m}^{¥prime}||_{L^{2}(0,T;V_{2})}^{2}¥leq C$

.

Take some subsequence such that tends to the solution as . Then by the fact (3.14) and (3.15) (2.15) $¥{y_{m_{k}}¥}¥subset¥{y_{m}¥}$

$y_{m_{k}}$

$y$

of

$ k¥rightarrow¥infty$

(3.53) Since

(3.54)

.

$||y||_{L^{2}(0,T;V)}^{2}+¥eta T||y^{¥prime}||_{L^{2}(0,T;V_{2})}^{2}¥leq C$

$y^{¥prime¥prime}=f-A_{2}(t)y^{¥prime}-A_{1}(t)y$

, from (2.2) and (2.8) we have

$||y^{¥prime¥prime}||_{L^{2}(0,T;V^{¥prime})}¥leq||f||_{L^{2}(0,T;V_{2}^{t})}+c_{12}||y||_{L^{2}(0,T;V)}+c_{22}||y^{¥prime}||_{L^{2}(0,T;V_{2})}$

.

From (3.53) and (3.54), we conclude that $||y||_{W(0,T)}^{2}=||y||_{L^{2}(0,T;V)}^{2}+||y^{¥prime}||_{L^{2}(0,T;V_{2})}^{2}+||y^{¥prime¥prime}||_{L^{2}(0,T;V^{¥prime})}^{2}$

$¥leq K(||y_{0}||_{V}^{2}+||y_{1}||_{H}^{2}+||f||_{L^{2}(0,T;V_{2}^{l})}^{2})$

where $K>0$ is a proper constant. is continuous from $(f,y_{0},y_{1})¥rightarrow y$

,

This proves that the affine map

$L^{2}(0, T;V_{2}^{¥prime})¥times V¥times H$

into

$W(0, T)$ .

4. Energy equah.ty and regularity The main purpose of this section is to prove the energy equality for the linear problem (2.15) used in Section 3 and the basic regularity result of solutions of (2.12). It is well known that the solution of linear equation has the regularity $y¥in C([0, T];V)$ , $y^{¥prime}¥in C([0, T];H)$ in the case $A_{2}(t)¥equiv 0$ (cf. Lions $y$

16


and Magenes [8] . Further in [4] this type of regularity result is proved for bounded $A_{2}(t)$ on $H$ by using ’parabolic’ regularization technique. We shall show this regularity result for the solution of nonlinear problem (2.12) by following the line of the proof in Lions and Magenes [8]. In proving the regularity it is required that the energy equality holds for the linear problem (2.15). For it we prepare the following lemmas. Lemma 4.1 is proved in [8]. $)$

Lemma 4.1. Let being reflexive. Set

,

$X$

be two Banach spaces,

$¥mathrm{Y}$

$C_{s}([0, T];¥mathrm{Y})=¥{f$

$¥in L^{¥infty}(0, T;¥mathrm{Y})|¥forall¥phi¥in ¥mathrm{Y}^{¥prime}$

is continuous of

with dense, and

$X¥subset ¥mathrm{Y}$

,

$X$

$t¥rightarrow¥langle f, ¥phi¥rangle_{Y,Y^{¥prime}}$

$[0, T]¥rightarrow R¥}$

.

Then .

$L^{¥infty}(0, T;X)¥cap C_{s}([0, T];¥mathrm{Y})=C_{s}([0, T];X)$

Lemma 4.2. Assume that is a weak solution of (2.12) or (2. 15). Then we can assert {after possibly a modification on a set of measure zero) that $y$

$y¥in C_{s}([0, T];V)$

that

and

$y^{¥prime}¥in C_{s}([0, T];H)$

.

Let us recall that $y¥in L^{¥infty}(0, T;V)$ and ¥ ¥ ¥ . It is clear $y ¥ in C_{s}([0, T];V)$ We shall show that . Let and $ ¥ psi ¥in H$ be fixed. Since is dense in $H$ , for any there exists a

Proof.

$y^{ prime} in L^{ infty}(0, T;H)$

$y¥in C([0, T];H)$ .

$t¥in(0, T)$

$¥emptyset¥in V^{¥prime}$

$¥epsilon>0$

$V^{¥prime}$

such that $||¥psi-¥phi||_{V^{¥prime}}¥leq¥epsilon(4||y||_{L^{¥infty}(0,T;V)})^{-1}$

We fix this $¥psi¥in H$ . $¥delta>0$ such that

Further since

$y¥in C([0, T];H)$ ,

for any

$¥epsilon>0$

there exists a

$|y(f)-y(t)|_{H}¥leq¥epsilon(2|¥psi|_{H})^{-1}$

if

$|f$

$-t|0$ and $f¥in L^{2}(Q)=L^{2}(0, T;L^{2}(¥Omega))$ (cf. Temam [13]). We consider the Dirichlet problem for (5.1). So that we take $V=H_{0}^{1}(¥Omega)$ and $V_{2}=H=$

$Semih¥dot{n}ear$

$L^{2}(¥Omega)$

.

21

Second Order Evolution Equations

To do a variational formulation let us introduce two bilinear forms $a_{1}(¥phi, ¥varphi)=¥int_{¥Omega}V¥phi¥cdot V¥varphi dx$

,

$¥forall¥phi$

,

$¥varphi¥in H_{0}^{1}(¥Omega)$

and $a_{2}(¥phi, ¥varphi)=¥int_{¥Omega}a¥phi¥varphi dx$

,

$¥forall¥phi$

,

$¥varphi¥in L^{2}(¥Omega)$

.

satisfy $(2.1)-(2.4)$ and Then we can easily check that the bilinear forms (2.7)?(2. 10). Also we can easily verify that $ f(t,y)=-¥beta$ siny $+f(t)$ satisfies the . For example, for (A2) it follows from that assumptions $a_{1},a_{2}$

$(¥mathrm{A}1)-(¥mathrm{A}3)$

$¥int_{¥Omega}|¥mathrm{s}¥mathrm{i}¥mathrm{n}¥mathrm{u}(¥mathrm{x})$

$-$ $¥sin$

$¥mathrm{v}(¥mathrm{x})|^{2}dx$

$¥leq¥int_{¥Omega}|u(x)-v(x)|^{2}dx$

,

$¥forall u$

,

$v¥in L^{2}(¥Omega)$

.

Now we define the nonlinear forcing terms as $(f(t,y(t)),¥phi)_{L^{2}(¥Omega)}=¥int_{¥Omega}(-¥beta¥sin y(t,x)+f(t,x))¥phi(x)dx$

Hence by Theorem 2.1, for solution satisfying

$y0¥in H_{0}^{1}$

$(¥Omega)$

,

$y_{1}¥in L^{2}(¥Omega)$

for all

$¥emptyset¥in H_{0}^{1}$ $(¥Omega)$

.

there exists a unique weak

$y$

(5.2)

$¥{y¥frac{¥partial^{2}y}{y¥partial t^{2}=}+a¥frac{¥partial y}{¥partial ¥mathrm{n}t}-¥Delta y+¥mathrm{i}¥mathrm{n}¥Omega ¥mathrm{a}¥mathrm{n}¥mathrm{d}¥frac{¥partial y}{¥partial t}(0,x’)¥beta¥sin y=f¥mathrm{i}¥mathrm{n}Q(0,x)=y¥mathrm{o}(x)0¥mathrm{o}¥Sigma,=y_{1}(x)$

in

$¥Omega$

and $y$

,

$¥frac{¥partial y}{¥partial t}$

,

$¥frac{¥partial y}{¥partial x}¥in L^{2}(Q)$

.

Note that (5.2) is the formal form written by (2.1).

Example 5.2 (Sine-Gordon equation : Neumann Problem). We consider a viscous damped Sine-Gordon equation having a Neumann boundary condition

(5.3)

$¥{^{¥frac{¥partial^{2}y}{ya¥partial t^{2}¥frac{¥partial}{¥partial 0,¥mathrm{n}}}-a¥Delta¥frac{¥partial y}{¥frac{¥partial¥partial yt}{¥partial ¥mathrm{n}}}-¥Delta y+¥beta¥sin y=f}(x¥frac{¥partial y}{¥partial t}+=g¥mathrm{o}¥mathrm{n}¥Sigma)=y_{0}(x)¥mathrm{i}¥mathrm{n}¥Omega ¥mathrm{a}’ ¥mathrm{n}¥mathrm{d}¥frac{¥partial y}{¥partial t}(¥mathrm{i}¥mathrm{n}Q0,x)’=y_{1}(x)$

in

$¥Omega$

,

Junhong HA arid Shin-ichi NAKAGIRI

22

denotes the outer where $a,¥beta>0$ , $f¥in L^{2}(Q)$ , $g¥in L^{2}(0, T;H^{-1/2}(¥Gamma))$ and $y0 ¥ in H^{1}$ be the . Let us take , normal derivative, and $¥varphi¥in V=H^{1}(¥Omega)$ in Example 5.1 and same bilinear form for , $¥partial/¥partial ¥mathrm{n}$

$(¥Omega)$

$y_{1}¥in L^{2}(¥Omega)$

$a_{1}(¥phi, ¥varphi)$

$¥emptyset$

$a_{2}(¥phi, ¥varphi)=¥int_{¥Omega}aV¥phi¥cdot V¥varphi dx$

,

$¥forall¥phi$

,

$¥varphi¥in V_{2}=H^{1}$

We define the dual form of the nonlinear term

(5.4)

$(¥Omega)$

.

as

$f(t,y)$

$¥langle f(t,y(t)), ¥phi¥rangle=¥int_{¥Omega}(-¥beta¥sin(y(t,x))+f(t,x))¥phi(x)dx+¥int_{¥Gamma}g(t,x)¥phi(x)dx$

. We note that the second term of (5.4) has a meaning by for all theorem. Hence by Theorem 2.1 there is a unique weak solution of (5.3) satisfying $¥emptyset¥in H^{1}(¥Omega)$

$y$

$¥mathrm{t}¥mathrm{r}¥mathrm{a}¥mathrm{c}¥mathrm{e}$

$y$

,


,

$¥frac{¥partial y}{¥partial x}$

,

$¥frac{¥partial^{2}y}{¥partial t¥partial x}¥in L^{2}(Q)$

.

Example 5.3 (General higher order equation : Dirichlet problem). Let us define a function $g:[0, T]¥times R¥times R^{n}¥times R¥rightarrow R$ satisfying the following conditions: , and is measurable on $[0, T]$ for all (i) (ii) there is a $¥beta_{i}¥in L^{2}(0, T;R^{+})$ , $i=1,2$ , 3 such that for each $t¥in[0, T]$ $¥xi,¥zeta¥in R$

$g(¥cdot, ¥xi,¥eta,¥zeta)$

$¥eta¥in R^{n}$

$|g(t, ¥xi,¥eta,¥zeta)-g(t, ¥xi^{¥prime},¥eta^{¥prime},¥zeta^{¥prime})|¥leq¥beta_{1}(t)|¥xi-¥xi^{¥prime}|+¥beta_{2}(t)|¥eta-¥eta^{¥prime}|+¥beta_{3}(t)|¥zeta-¥zeta^{¥prime}|$

$¥forall¥xi$

(iii) there is a

$¥gamma¥in L^{2}(0, T;R^{+})$

, , , $¥xi^{¥prime}$

$¥zeta$

$¥zeta^{¥prime}¥in R$

,

$¥forall¥eta,¥eta^{¥prime}¥in R^{n}$

,

,

such that

$|g(t,0,0,¥mathrm{O})|¥leq¥gamma(t)$

,

$¥forall t¥in[0, T]$

.

We consider an equation with the nonlinear term given by

in

(5.5)

$Q$

,

$¥left¥{¥begin{array}{l}¥frac{¥partial^{2}y}{¥partial t^{2}}-V¥cdot(bV¥frac{¥partial y}{¥partial t})+¥Delta(a¥Delta y)=g(t,y,Vy,¥Delta y)¥¥y=¥frac{¥partial y}{¥partial ¥mathrm{n}}=0¥mathrm{o}¥mathrm{n}¥Sigma,¥¥y(0,x)=y_{0}(x),¥frac{¥partial}{¥partial t}y(0,x)=y_{¥mathrm{l}}(x),¥mathrm{i}¥mathrm{n}¥Omega.¥end{array}¥right.$

¥ ¥ satisfying $a(t,x),b(t,x)>0$ for all $(t, x)¥in Q$ . where , ¥ and $H=L^{2}(¥Omega)$ . We To solve this equation we take $V=H_{0}^{2}(¥Omega)$ , , respecand as define norms on $V$ and tively. It is well known that these norms are equivalent to the norms induced and by , respectively. Let us introduce two bilinear forms defined $a$

$b in C^{1}$ $([0, T];L^{ infty}( Omega))$

$V_{2}=H_{0}^{1}$

$V_{2}$

$H_{0}^{2}(¥Omega)$

$H_{0}^{1}$

$(¥Omega)$

$||¥phi||_{V}=||¥Delta¥phi||_{L^{2}(¥Omega)}$

$(¥Omega)$

$||¥phi||_{V_{2}}=||V¥phi||_{L^{2}(¥Omega)}$


23

by $a_{1}(t;¥phi, ¥varphi)=¥int_{¥Omega}a(t,x)¥Delta¥phi¥Delta¥varphi dx$

,

$¥forall¥phi$

,

$¥varphi¥in V$

and

,

$a_{2}(t;¥phi, ¥varphi)=¥int_{¥Omega}b(t,x)V¥phi¥cdot V¥varphi dx$

$¥forall¥phi$

,

$¥varphi¥in V_{2}$

.

Then these bilinear forms satisfy $(2.1)-(2.4)$ and (2.7)?(2.10). Define a function by $[f(t,y)](x)=g(t,y(x), Vy(x),¥Delta y(x))$ , . Let us check the conditions (A2), (A3). (A3) is obvious from (iii). It is clear that is a bounded operator from to of operator norm 1. Let $k_{1},k_{2}>0$ be imbedding constants such that , . Then by using (ii), we have

$f(t,y):[0, T]¥times V_{2}¥rightarrow V_{2}^{¥prime}$

$¥forall x¥in¥Omega$

$H^{-1}(¥Omega)$

$H_{0}^{1}(¥Omega)$

$¥Delta$

$||¥emptyset||_{H^{¥_}(¥Omega)}1¥leq k_{1}||¥phi||_{L^{2}(¥Omega)}$

$||¥phi||_{L^{2}(¥Omega)}¥leq$

$k_{2}||¥phi||_{H_{0}^{1}(¥Omega)}$

$||f(t, ¥phi)-f(t, ¥psi)||_{H^{-1}(¥Omega)}¥leq¥beta(t)||¥phi-¥psi||_{H_{0}^{1}(¥Omega)}$

with $¥beta(t)=k_{1}k_{2}¥beta_{1}(t)+k_{1}¥beta_{2}(t)+¥beta_{3}(t)¥in L^{2}(0, T;R^{+})$ , and hence (A2) follows. Therefore, for given there is a unique solution satisfying (5.5) and $y_{0}¥in H_{0}^{2}(¥Omega),y_{1}¥in L^{2}(¥Omega)$

$y$

,


,

$¥frac{¥partial^{2}y}{¥partial x^{2}}$

,

$y$

$¥frac{¥partial^{3}y}{¥partial tx^{2}}¥in L^{2}(Q)$

.

Finally we note that the above equations can be treated by another methods used in [1,2,9,13,14], but our treatment is straightforward and rather easy by the appropriate choice of function spaces. Acknowledgment. The authors express their sincere thanks to the referee for pointing out the errors in the proof of Theorem 2.1 and giving valuable suggestions which improve the presentation of this paper. References [1] Ball, J. M., Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418. [2] Carghey, T. K. and Ellison, J., Existence, uniqueness and stability of solutions of a class of nonlinear partial differential equations, J. Math. Anal. Appl., 51 (1973), 1-32. [3] Chen, S. and Triggiani, R., Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,

J. Differential Equations, 88 (1990),

279-293.

[4] Dautary, R. and Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems L Springer-Verlag, 1992. [5] Ha, J., Optimal control problems for hyperbolic distributed parameter systems with damping term, Doctoral thesis, The graduate school of science and technology, Kobe University, 1996.

24


[6] Lions, J. L., Equations Differentielles, Operationnelles et problemes aux limits, SpringerVerlag, Berlin, 1961. [7] Lions, J. L., Quelques method des resolution des problemes aus limites non lineaires, Dunod, Paris, 1969. [8] Lions, J. L., and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications L Springer-Verlag, Berlin-Heidelberg-New York, 1972. [9] Nakano, M., Decay of solutions of some nonlinear evolution equations, J. Math. Anal. AppL, 60 (1977), 542-549. [10] Showalter, R. E., Hilbert Space Method for Partial Differential Equations, Pitman, London, 1977.

[11] Strauss, W. A., On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. Tanabe, H., Equations of evolution, Pitman, London, 1979. [12] [13] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Math. Sci. 68, Springer-Verlag, 1988. [14] Tsutsumi, M., Some nonlinear evolution equations of second order, Proc. Japan Acad., 47 (1971), 950-955.

nuna adoreso: Junhong Ha Department of Mathematics Pusan National University Kumjung, Pusan 609-735 Republic of Korea

Shin-ichi Nakagiri Department of Applied Mathematics Faculty of Engineering Kobe University Rokko, Nada, Kobe 657 Japan E-mail: nakagiricgodel.seg.kobe-u.ac.jp

(Ricevita la 4-an de junio, 1996) (Reviziita la 26-an de marto, 1997)