Semilinear equations in the “hyperbolic” case

Pergamon Nonlinear Analysis, Theory, Methods & Applications, Vol. 24, No. 5, pp. 773-788, 1995

Copyright © 1995 Elsevier Science Ltd Printed in Grvat Britain. All rights reserved

0362-54,6X/95 $9.50 + .00

0362-S46X(94)E0089-Y

SEMILINEAR EQUATIONS IN THE "HYPERBOLIC" CASE

NAOKI TANAKA Department of Mathematics, Faculty of Science, Okayama University, Okayama 700, Japan

(Received 26 July 1993; received for publication 30 March 1994)

Key words and phrases: Semilinear equations, variation of constants formula, stability conditions, compatibility conditions.

1. I N T R O D U C T I O N

Let X be a Banach space with norm II" II. We consider the existence and uniqueness of classical solutions to the semilinear initial value problem

(SP) I u'(t) = A(t)u(t) + B(t, u(t))

u(0) = Uo.

for t ~ [0, T]

Here, IA(t): t e [0, T]} is a given family of closed linear operators in X satisfying all conditions which are usually referred to as the "hyperbolic" case except for the density of the common domain D of A(t), and B(t, u) is a nonlinear operator on [0, T] x X.

If [Aft): t e [0, T]I is a stable family of infinitesimal generators of semigroups of class (Co) (and so D is dense in X), then the semilinear problem (SP) is solved by successive approximations defined by

I t

un(t ) = U(t, O)uo + U(t, s)B(s, Un_l(s)) ds, o

where {U(t, s): 0 _< s _< t _< T} is an evolution operator on X generated by [Aft): t e [0, T]]. The variation of constants formula plays an important role in defining successive approximations. (See Goldstein [1] and Pazy [2].)

For this reason, we want to obtain a "generalized variation of constants" formula which will provide a solution in a generalized sense to the following inhomogeneous initial value problem without assuming that the common domain D is dense in X

(CP; x , f ) I u'(t) = A(t)u(t) + f ( t )

u(O) = x

for t e [0, T]

for x ~ X a n d f ~ LI(0, T: X). The problem of this kind, in the special case where A(t) = A is independent of t, was studied by several authors. Among others, Arendt [3], and Kellermann and Hieber [4] obtained an existence result by using a variation of constants formula which is given in terms of integrated semigroups. In a different way from ours, Da Prato and Sinestrari [5] (see also [6]) gave some existence results of the problem (CP; x , f ) under some regularity assumptions on [A(t): t e [0, T]] in addition to three conditions below which we impose throughout this paper.

773

774 N. TANAKA

and

(A0 D(A(t)) = D is independent of t. (A2) There are constants M _> 1 and 09 e R := (-oo, co) such that

(~o, oo) C p(A(t)) for t ~ [0, T]

H R(2: A(tj)) <_ M(2 - o9) -k for 2 > ~o, j=l

and every finite sequence [ti]~= 1 with 0 _< t~ < tz -< ... -< tk -< T and k = 1,2 . . . . . (A3) For y ~ D, A( t )y is continuously differentiable in X. The condition (A2) is referred to as the "hyperbol ic" case (see Kato [7]). Our first purpose is to show that if x e / 3 and f ~ LI(0, T: X) then a solution u to (CP; x, f )

in a generalized sense is given by the "generalized variation of constants" formula

,am( l"' ) u(t) = U×(t, O)x + U×(t, r)f(r) dr (1.1) J0

for t e [0, T], where Ux(t, s) it/x] = I-Ii= [,/×)+1 (I - AA(i2)) -~ for 0 _< s _< t _ T(see theorem 3.1). To do this, we reduce the problem ( C P ; x , f ) to the nonlinear homogeneous problem u'(t) = Ay(t)u( t ) for t e [0, T] with initial condition u(0) = x, where a nonlinear operator A f ( t ) is defined by Af ( t ) y = A( t )y + f ( t ) for y e D. It may be expected that the limit of ux(t) := H} t/x]-= Jx y (i2)x is a solution in a generalized sense, where JYx(t) is the resolvent of Ay(t) . The key to proving the convergence of Ux is an important estimate (lemma 3.3) similar to the estimate given by Crandall and Pazy [8] as well as the stability of {A1(t): t e [0, T]] in the space Y which will be shown in Section 2. Here Y is a Banach space D equipped with a norm 11" II Y defined by

Ilyll~-= Ilyll + Ilh(0)yll for y e D. (1.2)

It will be shown in Section 3 that the limit in (1.1) exists if x e 13 and f e L~(O, T : X ) (see theorem 3.1).

Section 4 is devoted to a fundamental theorem (theorem 4.2) on the existence and uniqueness of classical solutions to (CP; x , f ) which improves [5, theorem 4.2]. The fact that the limit of the right-hand side of (1.1) exists if x e b and f ~ L~(0, T: X) is useful to prove that the limit of u× exists in Y if x e D satisfies the compatibility condition that A(0)x + f(0) e b and f ~ W 1' 1(0, T: X) (see lemma 4.1), and then the existence result will be obtained.

Finally, Section 5 contains our main theorem (theorem 5.1) on the existence and uniqueness of classical solutions to the semilinear problem (SP). The "generalized variation of constants" formula (1.1) enables us to construct approximate solutions [u,] of the problem (SP) inductively by Uo(t) = u o on [0, T] and

l imo+ ( l [t/k]X rl u, ( t ) = Ux(t, O)uo + Ux(t, r)B(r, u,_l(r)) d X jO

for t ~ [0, T] and n _> 1. The existence of classical solutions will be established by finding the unique solution v ~ C([0, r]: Y) of the integral equation (see (5.2) and (5.5)) which classical solutions must satisfy, and showing that the sequence [Un] of approximate solutions converges to v in C([0, r]: Y) as n ~ oo.

Semilinear equations 775

2. S T A B I L I T Y IN T H E S P A C E Y

For g e C([0, T]: X) we define Ag(t): D ~ X and Jxg(t): X ~ X by

Ag( t ) y = A ( t ) y + g(t) for y • D

and

Jg(t)z = Jx(t)(z + 2g(t)) for z e X and 2 > 0 with 2090 < 1,

where J×(t) = (I - 2A(t)) -1 for 2 > 0 with 2o9 o < 1 and to o = max(og, 0). This section is devoted to the stability of lAy(t): t ~ [0, T]] in the space Y which will be

needed in the construction of a solution to (CP; x , f ) in a generalized sense. Here, Y is the Banach space D with norm I1" II Y defined by (1.2). In what follows, when X i (i = 1,2) is a Banach space we write B ( X I , X2) for the set of all bounded linear operators on Xm to X2, and the symbol II" Ilx, ~x2 denotes the B(Xm, X2) norm.

LEUUA 2.1. Let 2 > 0 be such that Ao9o < 1, [tilT'=1 be a finite sequence such that 0 <__ tl ___ . . . ___ tm -< T, and g e C([0, TI: X) .

(i) For z ~ X we have

) Jg(ti)z = Jx(ti)Z + Jx(ti) 2g(tg+l). (2.1) i =1 i=1 k = 0 i = k + l

(ii) For y • D we have

Sg(tm) f i Jg(tj)y j = l

= ~I Jx(b)Sg(tl)y + E f i J x ( t , ) ( S s ( t j + j ) - S ~ ( b ) ) P g ( t j ) . S g ( t j) J~(t~)y j = l j = l i = j + l i = l

) - )~o ~ J×(ti) Ag(tj+O, (2.2) j=O i=j+l

where A o > o9, and Sg(t): D ~ X and P~(t): X --, X are defined by

and

S , ( t ) z = (A( t ) - 2o)Z + g(t)

Pg(t)z = (A( t ) - Ao)-l(z - g(t))

for z e D

for z e X.

Proof . Using the definition of Jf,(t) and the linearity of Jx(t) one verifies (2.1) by induction. To prove (2.2), let y e D. By definitions of Sg(t) and Jg(t) we have

Sg(t)J~(t)z = (A( t ) - ;to)Jx(t)(z + Ag(t)) + g(t)

= Jx( t ) (A( t ) - 2o)Z + ( I - (I - 2A(t))J×(t)g(t) - 2oJx(t)2g(t ) + g(t)

for z e D and t e [0, T]. Hence

Sg(t)J~,(t)z = Jx(t)Sg(t)z - 2oJ×(t)Ag(t) (2.3)

776 N. TANAKA

for z e D and t e [0, T]. Here we have used the linearity of J×(t). The case m = 1 of (2.2) is proved by (2.3). We next suppose that (2.2) with m = k holds. By (2.3) and the identity that Pg(t)Sg(t)z = z for z e D and t e [0, T], we have

k+ l

Sg(tk+,) 1-[ J~,(tj)y j = l

k

= Jx(tk+l)S,(tk+l) [I Jg(tj)y - )~oJx(tk+l)Ag(tk+l) j = l

k k

= Jx(te+,)(S,(tk+,) - S,(tk))Pg(tk)'Sg(tk) 1-[ J~,(tj)y + Jx(tk+,)Ss(tk) 1-I J~(tj)y j = l j = l

- 20Jx(tk+l)Ag(tk+l).

Applying the hypothesis of induction to the second term of right-hand side, we see that (2.2) is true for m = k + 1, and so (2.2) is proved by induction. •

For convenience, in what follows we denote constants depending upon p, q . . . . by C(p, q , . , ", .), and we use the notation

Ilhll® -- suplllh(t)l l:t ~ [0, T]I for h ~ C([O, T]: X) .

LEMMA 2.2. Let 2 > 0 be such that 209o ~ 1/2. Then we have: (i) i f y E D and g e C([0, T ] : X ) then there exists a c( l lgL, IlyllY) such that

1 ~ J~,(iA)y - y] <_ )tmC(][g[[.~, [lY[]Y) (2.4) I i = l I

for 0 < m -< [T/A]; (ii) i f y ~ D and g ~ CI([0, T]: X) then there exists a c(llgL, IIg'L, []yIIY) such that

J~(jA)y <_ c(llgll®, Ilg'll®, Ilyllr) (2.5) j = l

for 0 _< m _< IT/A].

Proof. A simple computat ion yields

f i J x ( i A ) y - y = ~ ( f i Jx(iA))AA(kA)y (2.6) i = I k = l i = k

for y e D and 1 _ m _< [T/A]. Combining (2.6) and (2.1) we obtain (2.4) by the stability condition (A2). Here we have used the fact that sup l l lA ( t ) l l r . x : t ~ [0, T]} < ~ and the estimate that (1 - t) -1 _< exp(2t) for 0 _< t _< 1/2. To prove (2.5), l e ty e D, g e C~([0, T]: X) and 1 <_ m <_ [T/X]. For i = 1, 2 . . . . . m, we set t x = iA and define a/x = Ilag(t, x) FIj=I J~(t))yll. Combining (2.2) and the identity

(Sg(t) - Sg(s))Pg(s)z = (A(t) - A(s))(A(s) - Ao)-I(Z - g(s)) + (g(t) - g(s))


for Z e X and s, t e [0, T] , and then applying the stability condi t ion (A2) we obta in the estimate

i - I

a/x _< M(1 - A~Oo)-illsg(tX)yll + ~ M(1 - ;ttOo)-(i-J){ll(A(tX+l ) - A(tX))(A(t x) - ;to)-lll • aj x j = l

+ II(A(bX+l)- m ( t X ) ) ( a ( t ) ) - ;to)-lllilg(tX)ll + I lg( t~+l)- g(tX)lll i - I

+ Ao E M(1 - ;tO~o)-t'-J);tllg(bX+OII; j=O

hence i -1 i -1

(1 - ;tOgo)ia x _< MIISg(tX~)yll + ~, M(1 - ;ttOo)J;tCa x + ~ M(1 - ;tCoo)J;t(Cllgll® + Ilg'll®) j = l j = l

i - 1

+ ;to ~ M(1 - ;t~Oo)J;tllgll® (2.7) j=0

for 1 _< i _< m, where

We now define A i for i = 1, 2 . . . . , rn by the r ight-hand side o f (2.7). It is obvious that ( 1 - ;to%)ia x < AXi for i = 1 . . . . . m, by which we find the inequality

Ai+,x _ AiX < M;tCA x + M(1 - ;t~Oo)';t(Cllgll® + IIg'l[~ + ;tollgll®).

set t ing Kx = 1 + MAC and Lx = M;t(Cllgll~ + IIg'll® + ;tollgll®) we have

Ai+lx < KxA x + L x (2.8)

for 1 <__ i < m, since (1 - ;to%) i _< 1. Solving (2.8) we obtain the estimate

m--2

AXm< r ~ - l A X + ~., KkxLx k = 0

<- exp(MTC)(MIISg( tX~ )Yll + ;toM;tllgll®) + e x p ( M T C ) M T ( CII glI,~ + IIg'll~ + ;tollgll®). (2.9)

Here we have used the fact that 1 + t _< e t for t _> 0. Combin ing (2.9) and the estimate that

Ilsg(t~)yll <- ( sup IlA(t) - ;tollY~x)llyllY + IIgL, X, te[O,T! /

and using the fact that (1 - ;tWo)-" <- exp(2tooT), we obtain the desired estimate (2.5). •

3. "GE NE RAL IZ E D VARIATION OF CONSTANTS" FORMULA

This section is devoted to the construct ion o f a solution to (CP; x , f ) in a generalized sense. To do this, we consider the approximat ing problem

Ux(t) - Ux(t - 2) (CP; x , f ) x 2 = A([t/X]A)ux(t) + f([t/;t];t) for t >_ A

u×(t) = x for 0 <_ t < ;t.

778 N. TANAKA

The solution of (CP; x , f ) x is given by Ux(t) = lli=T'tt/×]l Jf(iA)x. The existence of the limit of u× is provided by the following theorem.

TI-IEORE~ 3.1. If X e D and f e LI(0, T: X) then the limit

([fiX] I[t/X]h( [t/k] ) r) u(t) := lim I-I Jx(iA) x + I I Jx(iA) f ( r ) d (3.1)

k--+O+ i= 1 dO i= [r/k]+l exists uniformly for t e [0, T], and u is a continuous function on [0, T].

Remark 3.1. The special case where x = 0 in theorem 3.1 shows that i f g ~ LI(0, T: X) then the limit

w(t) lim ~1 := Jx(iA) g(r) dr (3.2) k~0+ ,J0 \i=[r/M+l

exists uniformly for t e [0, T], and w is continuous on [0, T].

For A > 0 with Ao9 0 < 1 and g ~ C([0, T]: X), we use the notation

k Px,k(g) = I] Jg(iA)

i=1

for 0 < k < [T/A]. To prove the convergence of (3.1), we define

AX;~(g)z = Px,k(g)z - P~,t(g)z

and

BX;~(g)z = IZ(Ag(kA) - Ag(llz))P~d(g)z

for z e X and g e C([0, T]: X). We shall prove theorem 3.1 after preparing two lemmas.

LEMMA 3.2. Let 2 > / z > 0 be such that 2a~o < 1. Then, for z e X a n d g E C([0, T]: X) we have

AX',~(g)z = Jt~(kA)(otaXk~l,l_l(g)z + pAX;~_l(g)z + BX:~(g)z) (3.3)

for 1 _< k -< [T/A] and 1 <_ l <_ [T/a], where tx =/z/A a n d / / = (A - / z ) /A .

Proof . Let z e X . A ~ ( g ) z is written as

A~'~(g)z = (J~,(kA)Px, k - l (g)z - J~(kA)P~d-l(g)z) + (J~(kA)P~,t-l(g)z - P~d(g)z).

By I t (i = 1, 2) we denote the ith term of the right-hand side. By the definition of J~(t) and the resolvent identity that J x ( t ) = J~(t)(~I + flJx(t)), the first term of I1 is equal to Jr (kA)(oe(Px.k_ l(g)z + 2g(kk)) + flPx, k (g)z); hence

11 = J~(kA)(~AX~l,t_l(g)z + flAX',~_t(g)z).


By the definition of J~(t) and the linearity of J~(t) we have

I2 = J~(k2)((I - #Ag(l#))Pu,l(g)z + #g(k2) - (I - gA(k2))P~, t (g)z) = J~(k2)BX:~(g)z.

Combining these equalities we obtain the desired relation (3.3). •

Let 2 > # > 0 be such that 2090 < 1 and let m be an integer such that 0 ___ m _< [T/2]. We denote by H ( m , k) the set of all operators Q obtained by multiplying k operators J~(ti) (i = 1 . . . . . k) in the family {J~(i2): i = 1 . . . . . m} such that Q = 1-I~= 1J~(ti) for 0 _ tl -< "" -< tk --< T; note that H ( m , 0) = H(0, k) = {I1. By H ( m , k , j ) we denote the set of all sums o f j operators Qi (i = 1 . . . . . j ) in H ( m , k), where in j operators Q1 . . . . . Qj, same operators are allowed to appear repeatedly.

Using the relation (3.3), and then taking account of the definition of H( - , -, ") together with the linearity of J~(t), we obtain the following crucial estimate to show the convergence of (3.1), by double induction on (m, n) (see also [8, Appendix 11).

LEMMA 3.3. Let 2 > # > 0 be such

(m-1)An / ax,~ ¢~)Z e ~, etiB"-iH m, l q t piI , t l ~, o

that 2090 < 1. Then, for z ~ X a n d g ~ C([0, T]: X) we have

i=0

i=m ~t-1

+E j=o

n AX n , ( i ) ) ~-i ,o(g)z

{ i - 1 \ \ x~ c~ 'n f l i - 'H(m, i, ~m - 1 ) ) A° :n- i (g ) z

~, a i f lJ- iH m , j + 1, BX'~-i n- j (g)z (3.4) i=0 \ t / / '

for 0 <_ m <_ [T/21 and 0 <_ n <_ [T/g], where o~ = #12, p = (2 - #)12, l A k = min(l, k) and

( ~ ) is the binomial coefficient, and we use the conventions 0 ° = 1 , ( - 1 1 ) = 1 and ( _ ~ ) = 0

if j _ 0.

P r o o f o f theorem 3.1. Let 2 > # > 0 be such that 2too -< 1/2, and let m and n be integers such that 0 _< m2, n# <__ T. Applying the stability condition (A2) to (3.4) we have

I I P x , m ( g ) z - - P~,,(g)zll <- E=o ~,~-i M ( 1 - #~Oo)-'llA~_i,o(g)zll

m Aim I / i - 1 \ ' Ilmo:~-,(g)zU +

i = r a ° l

n-l (m-1)^j (J i ) + E E ~'B j-~ M(1 - #O~o)-¢J+'lln~,~_,,~_Ag)zll ( 3 . 5 ) j=O i=0

for z ~ X and g ~ C([0, T] : X), where o~ = g /2 , fl = (2 - #) /2 . Since

(Ag(t) - As(s ) )z = (Ag(t) - Ag(S))Pg(S)Sg(s)z

= (A( t ) - A(s) ) (A(s) - 2o)-X(Sg(s)z - g(s)) + (g( t) - g(s))

780 N. T A N A K A

for z e D and t, s e [0, T], we find by (2.5)

I lB~:f(g)yll <- u l k ~ - t~ , l c ( l lg l l . . I Ig' l l®, I ly l lO (3 .6 )

for y e D and g e C~([0, T]: X). Here we have used the fact that A( . ) is Lipschitz continuous in the B(Y, X) norm (by the condition (A3)) and g is a Lipschitzian with Lipschitz constant [Ig'll®. Substituting (2.4) and (3.6) in (3.5), and then estimating it just as in the proof of [8, theorem 2.1 ], we have

Ilex,m(g)Y - P, ,n(g)YH

-< c(llgll®, IIg'll®, Ilyllr)l((n~ - m 2 ) 2 + n/z(2 - / z ) ) ' /2 + ((n/~ - mA) 2 + m 2 ( 2 - l / ) ) 1 / 2

+ nl~(lnlz - mAI + ~ + (T/ j2 )n l~(2 - / z ) ) } (3.7)

for y ~ D, g ~ C~([0, T]: X) and J > 0. Now, let x ~ b and f ~ LI(0, T: X), and define an X-valued function u ~ by

u' ( t ) = I ] J~(ie)x + J,(ie) f ( r ) dr i = 1 ,JO \ i = [ r l e l + 1

for t e [0, T] and e > 0 with eo9 o _< 1/2. Then we have by (2.1),

( r) I l u " ( t ) - P~,t..j(g)yll <- Mexp(2eJoT) I [ x - yll + I[f(r) - g(([r/e] + 1)e)ll d (3.8) J 0

for t ~ [0, T] and e > 0 with ew0 --- 1/2. Combining (3.8) and (3.7) we have for t, s ~ [0, T],

IluX(t) - u~(s)l[ _< M e x p ( 2 ~ o T ) ( 2 l [ x - yll + Ex(g) + E.(g)) + c ( l l g L , IIg'll®, Hylly)12((It - sl + A +/~)2 + T(2 - / z ) ) 1/2

+ T(l t - sl + 2 + tt + J + (T/j)2(A -/z))} (3.9)

for y e D, g e C~([0, T]: X) and J > 0, where we set

t ' [ T l ~ l t

E,(g) = IIf(r) - g(( t r / t ] + 1)c)ll dr. J0

Setting s = t in (3.9) and taking the limit as A,/z --, 0+ we have

l i m s u p ( sup ]]uX(t)- u~(t)ll) k,#--* 0+ \ t ~ [0,T]

<- 2Mexp(2tooT) IIx - yll + I l l ( t ) - g(t)[I dt (3.10) 0

for y ~ D and g ~ C1([0, T ] : X ) . Since CI( [0, T] : X ) is dense in LI(0, T: X ) i t fo l lows f rom (3.10) that the l imi t u ( t ) : = l i m x ~ o + uX(t) exists un i fo rmly on [0, T ] . To prove u e C([0, T ] : X ) , let t ,s ~ [0, T] . Passing to the l imi t in (3.9) as 2,/~ ~ 0+ we have

( f" ,) I lu(t) - . (s) l l -< 2 M e x p ( 2 t o o T ) IIx - Yll + I I f ( t ) - g(t) l l d 0

+ c ( l l g l l® , I Ig ' l l®, I ly l lY) l t - sl


for y E D and g ~ C1([0, T]: X). Therefore, we conclude that u is a continuous function on [0, T], by a standard argument. Finally, we note that the representation

[tim

u(t) = lim H J1x(i~)x X~0+ i=1

holds for t e [0, T], if x e / 3 a n d f e C([0, T]: X). •

4. C L A S S I C A L S O L U T I O N S TO T H E P R O B L E M ( C P ; x , f )

This section is devoted to the existence and uniqueness of classical solutions to the problem (CP; x , f ) . We start with the convergence of ux in the space Y.

L~.ra~ 4.1. Let f ~ W 1,1(0 , T" X) and suppose that x e D satisfies the compatibility condition that A(O)x + f(O) ~ D. Then, the limit

[fiX]

v(t) := lim Sf([t/;~]A) H Jf(iA)x k~0+ i=1

exists uniformly for t e [0, T], and v is a continuous function on [0, T]. Here, w l ' l (0 , T: X) is the set of all X-valued functions f satisfying the property that there exists h e L~(0, T: X) such that l'

f(¢) - f ( O ) = h(s) ds for t e [o, r l . (4.1) 0

Proof. We define a nonlinear operator H(t): X ~ X by

H(t)z = A(t)(A(t) - ~.o)-l(z - f(t)) + h(t)

for z e X and t e [0, T], where h is an X-valued integrable function satisfying (4.1). Then the function r-~ H(r)v(r) is an X-valued integrable function if v e C([0, T ] : X ) . By the compatibility condition we have Sf(O)x = A(0)x + f(0) - 3.0x e / ) . By theorem 3.1 and remark 3.1, the above facts enable us to define an operator W: C([0, T]: X) ~ C([0, T]: X) by

(Wv)(t) = U(t, O)Sf(O)x + lira U×(t, r)(H(r)v(r) - 2of(r)) dr X~0+

for v e C([0, T]: X), where we define U(t, 0)z = lim×-.0+ Ux(t, O)z for z e / ) and Ux(t, s) = [t/X] Hi= ts/xj+l Jx(o!.) for 0 _ s _< t <_ T. Using the definition of H(t) one verifies by induction

I t ( t - r ) n - 1 l i v e ( r ) - v2(r)ll dr ] l ( w n v O ( t ) - (wnvg(t)ll <_ c ~ 30 (n---- ~.

for n = 1, 2 . . . . where

C1 = (t~lo,rlsup [IJt(t)(A(t)- 2o)-lll)Mexp(tooT).

782 N. TANAKA

The fixed point theorem asserts that W has a unique fixed point v, namely W has a unique element v e C([0, T]: X) such that

l [t/XlX v(t) -- U(t, 0)Sf(0)x + lira Ux(t, r)(H(r)v(r) - A0f(r)) dr (4.2)

X-*0+ J0

for t e [0, T]. On the other hand, we find by (2.2)

r t /×jx S:(([r/A] + 1),,l) - S:([r/212) vx(t) = Ux(t, O)S:(2)x + Ux(t, r) , P:([r/;q2)vx(r) dr

jx ~,

[tiM), - 20 Ux(t, r)f(([r/)t] + 1)2) dr (4.3)

do

for t e [2, T]. Here vx is defined by vx(t) = S1([t/Z]2) Iltt/=Xl 1 JYx(j2)x for t e [0, r l . From the fact that h e LI(0, T: X ) we deduce that

f ttmx IlA-l(f(([ r /z] + D2) - f ( [ r / Z ] 2 ) ) - h(r)ll dr k

s Ilh(s + [ r / ~ l~ ) - h(r)ll ds d r ,)x o

<-,~- o x . x IIh(s + [r/21,~) - h(r)ll d (is .1, 0 (4.4)

as 2 ~ 0 + , since h can be approx imated by cont inuous funct ions o n [0, T] . Subtracting (4.3) from (4.2), and using (4.4) we find the integral inequality

IIv(t) - vx(t)ll -< ax + c 2 IIv(r) - vx(r)ll dr o

for t e [0, T], where {6x: A > 0} is a null sequence and C2 is a positive constant. The Gronwall inequality shows that IIv(t) - vx(t)ll -< Ox exp(CET) ~ 0 as 2 ~ 0+, and so the desired claim is proved. •

The following theorem improves [5, theorem 4.2].

THEOREM 4.2. Let f e WI'I(0, T;X) and suppose that x e D condition that A(O)x + f ( O ) E D . Then, the ( C P ; x , f ) has a u e C([0, T]: Y) f3 C1([0, T ] :X) given by (3.1).

satisfies the compatibility unique classical solution

Proof . Since Pf ( t ) S f ( t ) y = y for y e D, we have

I - [ J ~ ( i , l ) x = ( A ( [ t / , q , l ) - ,~o) -~ s:([t/;~];,) ,v~(i~)x - f( [ t / ;~] ;O i=1 i ~ l


for t E [0, T] and ). > 0 with Ao90 < 1. By lemma 4.1, the right-hand side tends to ( A f t ) - 20)-1(o(0 - f ( t ) ) as 2 --* 0+; hence u( t ) = ( A ( t ) - A0)-l(v(t) - f ( t ) ) for t • [0, TI, by theorem 3.1. This shows that u • C([0, T]: Y) and

v( t ) = ( A ( t ) - ).o)u(t) + f ( t ) (4.5)

for t • [0, T]. Since Sy( t ) y = A y ( t ) y - ).oY for y • D, we have

JYx(j).)x - x = ~ (I - ( I - ) .Af(i) .))) [ I J f ( j ) . ) x = ).(Sy(i).) + ).01) H J f ( j ) . ) x j = l i=1 j = l i=1 j = l

for 2 > 0 with ).o90 < 1 and 0 <_ m <_ [T/).], by which we find

[t/h] ~ ([t/X] + 1)k [r/k]

I-I Jfx(jA)x - x = I (s f([r/) .]A) + ).0 I ) 1-I J fx ( j ) . )xdr j = l ,JX j = l

for t • [0, T] and ). > 0 with ).o9o < 1. Taking the limit as ). ~ 0+ we have

u( t ) - x = (v(r) + ).0u(r))dr 0

for t • [0, T]. This together with (4.5) implies that u • C1([0, TI: X) and (CP; x , f ) is satisfied. We have only to prove the uniqueness of classical solutions to (CP; x , f ) . To this end, let ui (i --- 1, 2) be classical solutions to (CP; x , f ) , and set w = ua - u2. Let ). > 0 and define ex(t) = ) . - l ( w ( t ) - w( t - ).)) - w' ( t ) for ). _< t _< T. Since w' ( t ) = A ( t ) w ( t ) for t • [0, TI we find

w(t) = Jx( t ) (w( t - ,,1.) + ).ex(t))

for ). < t < T. Noting w(O) = 0 we have by induction

w ( i ) . ) : k=,~ ( j = ~ J x ( J ) O ) ).ex(k).) (4.6)

for 0 <_ i <_ IT~Z]. Since Ilex(t)ll -< p(A) for t • [0, T], where p denotes the modulus of continuity of w', we obtain by (4.6) the estimate that Ilw([t/A]).)ll <-- T M e x p ( 2 o g o T ) p ( 2 ) for t • [0, T]. Taking the limit as A ---, 0+ yields w(t ) = 0 for t • [0, T]. This shows the uniqueness of classical solutions to (CP; x , f ) . •

5. S E M I L I N E A R I N I T I A L V A L U E P R O B L E M S

This section is devoted to the existence and uniqueness of classical solutions to the semilinear initial value problem

I u ' ( t ) = A ( t ) u ( t ) + B( t , u( t ) ) for t • [0, TI

(SP) (u(O) = Uo.

784 N. TANAKA

Here, the nonlinear operator B(t, u) defined on [0, T] × X satisfies two conditions: (BI) for each R > 0 there exists LB(R) >- 0 such that

liB(t, nO - B ( t , u2)ll --- LB(R)IIu~ - uzll

for t ~ [0, T] and Ilu, II -< R (i = 1,2); (B2) for u e C1([0, T]: X) , the function B(-, u(-)) is represented as

B(t, u(t)) - a(o, u(O)) = t'i F(s, u(s), u'(s)) ds

for t ~ [0, T], where F(t, u, v) is an X-valued function on [0, T] × X × X satisfying the following three conditions:

(F1) for each R > 0,

lim (suplllE(t, u, v) - E(t, w, v)ll: t ~ [0, T] and Ilvll -< RI) = 0; Uu-wll-.o

Ilul[, Ilwll < R

(F2) for each R > 0 there exists LF(R) > 0 such that

liE(t, u, vO - E(t, u, vz)ll --- LF(R)IIvl - vzll

for t e [0, TI, ]lull < R and vi e X (i = 1, 2); (F3) u, v ~ C([0, T ] :X) , the function F( . , u(.), v(.)) is integrable on [0, T]. Our main result of this paper is given by the following theorem.

TrmoaE~ 5.1. If uo ~ D satisfies the compatibility condition that A(O)uo + B(O, Uo) ~ D then there exists a r > 0 such that the semilinear problem (SP) has a unique classical solution u e CI([0, r]: X) CI C([0, r]: Y).

Proof . Set uo(t) = Uo on [0, T]. By the compatibility condition and the condition (B2), theorem 4.2 enables us to define un e C1([0, T]: X) f'l C([0, T]: Y) inductively by the unique classical solution to the problem (CP; uo, B(. , u,_l( '))) . Then each u, is represented as

l im+( l lt/x]x r ) u , ( t ) = Ux(t, O)uo + Ux(t, r)B(r, u,_l(r)) d (5.1) jo

for t e [0, T], where we set Ux(t,s) = 1-I[~]~/x]+~ Jx(i2) for 0 _< s _< t ___ T. Using the integral equation on the unique classical solution to (CP; x , f ) obtained by combining (4.5) and (4.2), we find

(A( t ) - Ao)Un(t) + B(t, un_~(t))

= U(t, 0)((A(0) - Ao)Uo + B(0, Uo)) I [t/hlX

+ lira Ux(t, r)(Jt(r)u,(r) + Gn(r) - ;t0B(r, Un_l(r))) dr (5.2) ~.~0+ j0

for t e [0, T], where U(t, 0)z = lim×~o+ Ux(t, O)z for z ~ 13 and t ~ [0, T], and

Gn(r) = F(r, u,_~(r), A(r)un_l(r ) + B(r, Un_z(r)) )

for r e [0, T] and n ___ 2.


We first show that there exists a r > 0 such that suplJlu,(t)H: t e [0, r]} is bounded as n --, oo. By the condition (Bz) we have

N : = sup IIu(t,O)uo - uoll + TMexp(a~oT) sup IIe(t, Uo)ll < oo. t e [O,T] t E [O,T]

Let K > N and choose r > 0 such that

r M e x p ( m o r ) L e ( K + IlUoll)g + N < K.

Then it suffices to prove

suPlJlu,(t) - uoll: t e [0, r] and n _ 11 _< K. (5 .3 )

Subtracting u0 f rom both sides of (5.1), and estimating it we find

I l u . ( t ) - uoll <- N + Mexp(~ooT)llB(r, un_l(r)) - - B(r, uo)ll dr o

(5.4)

for t ~ [0, r] and n > 1. The case n = 1 of (5.3) follows f rom (5.4) with n = 1. We thus inductively obtain (5.3) by applying the condition (BI) to (5.4).

We next show that the sequence {u,} converges in the space C([0, r]: X) as n --* ao. By (5.1) we have for n >_ 1,

l [t/klX U,+l(t) - u~(t) = lim U×(t, r)(B(r, u~(r)) - B(r, un- l ( r ) ) )dx

h~O+ ,J0

for t e [0, T]. By virtue of (5.3) we use the condition (B 0 to obtain

I lu .+~( t ) - u . ( t ) l l --- M e x p ( o % T ) L n ( K + Ilu011) I lu . ( r ) - u . _ l ( r ) l l dr 0

for t e [0, r] and n _ 1. A standard argument shows that there exists u e C([0, r]: X) such that limn~® un(t) = u(t) uniformly on [0, z], and [lu(t) - uol[ <- K for t e [0, 1:].

We shall prove that u, converges to u in the space C([0, r]: Y) as n ~ oo. From the fact shown above and the condition (B0 we deduce that B(t, u~ (t)) converges to B(t, u(t)) uniformly on [0, z]. By the condition (B2) the function B( . , u~(-)) is continuous on [0, r], and so the uniform limit B(-, u(.)) of continuous functions is continuous on [0, r]. The condition (F3) guarantees that the function F( . , u ( . ) , A ( . ) v ( . ) + B ( ' , u( '))) is integrable on [0, r] if v e C([0, z]: Y). By the compatibility condition, we have (A(0) - Ao)U 0 + B(0, Uo) ~ / ) . In view of these facts, theorem 3.1 and remark 3.1 enable us to define a mapping ~ : C([0, r]: Y) -* C([O, r]: Y) by

(~v ) ( t ) = (A( t ) - Ao) -~ IU(t , O)((A(O) - Ao)Uo + B(O, Uo))

[t/xlx 1 + lim U×(t, r)(,it(r)v(r) + G(r, v(r)) - AoB(r, u(r))) dr - B(t , u(t)) k~O+ JO

786 N. TANAKA

for v E C([0, r]: Y). Here the funct ion G ( . , .) on [0, r] × Y is def ined by G ( t , v ) = F(t, u(t) , A ( t ) v + B(t, u(t))). By the condi t ion (F2), a rout ine calculat ion shows tha t there exists K1 > 0 such tha t

I t(t - - s ) n - 1 IIo~(s) - vz(s) l ly d s I I ( ~ % ) ( t ) - (~%2)( t )11~ ~ KF 30 -(~----

for n _ 1 and t e [0, r]. Therefore , ~ has a unique fixed point v, namely there exists a unique element v e C([0, r]: Y) such that

(A( t ) - ).o)V(t) + B(t, u(t))

= U(t, 0)((A(0) - 2o)U o + B(0, Uo))

[t/x]x

+ lim U×(t, r)(A(r)v(r) + G(r, v(r)) - 2oB(r, u(r))) dr X~O+ dO

(5 .5 )

for t e [0, r]. By (F1) and (F2), the n o r m o f the difference between G . (r) and G(r, v(r)) is major ized by K2l[un_l(r) - v(r)l[ r + t~. for some posit ive constant K 2 and a null sequence [5.1. Subtract ing (5.5) f rom (5.2) and using this fact , we find a posit ive c o n s t a n t / ( 3 and a null sequence {e.} such that

I t

I lu . ( t ) - v ( t ) l l y - ~ ~ . + K3 ( l lu . ( r ) - v(r) l l~ + I lu ._ , (r ) - v ( r ) l l 0 d r o

for t e [0, r] and n _ 2. Put t ing 4~,(t) = [[u,(t) - v(t)[lr, we have by the Gronwal l inequali ty

I t

dp.(t) < eX3ten + K 3 eg3(t-s)dP._l(s) (Is o

for t e [0, r] and n _> 2. Induct ively we have

.-1 g~t ~ I' (t - s)"-' e-KVc~n.m(t) <--l~=O--~-.en.m_ I + K; -(n --- ~ . e-K'Sdpra(S) ds'

= * 0

which is reduced to

~ / \ Knr n s u p e-K3tdpn+m(t) <_ e g3~ sup 8k + [ s u p e-K3t~m(t)] ~3~

t ~ [0,'rl \ k ~ m + l / ~ t ~ [0,¢] / H!

for n _> 1 and m _> 2; hence

/ \ lim sup(sup[e-K3t~b.(t): t e [0, r]]) _< eK3~{ sup ek) ~ 0

n ~ \ k > m + l /

as m --, oo. Thus it is p roved that u . converges to v in C([0, r]: Y) as n ~ oo. Consequent ly , we have u = v e C([0, z]: Y) and

lim(sup{llu~(t) - u(t)llv: t ~ [0, r]l) = o. n~oo

I


From this fact it follows that u ' ( t ) ( = A ( t ) u , ( t ) + B(t, u,_l(t))) converges to A ( t ) u ( t ) + B(t, u(t)) uniformly on [0, r] as n ~ co, and so we see that u is a classical solution to the semilinear problem (SP).

Finally, we shall prove the uniqueness of classical solutions to (SP). To this end, let ui (i = 1,2) be classical solutions to (SP) and set w = u~ - u2. Then w is a classical solution to the problem (CP; 0, B( . , Ul(')) - B(. , Uz('))). By theorem 4.2 we have

[t/XlX

w(t) = lim Ux(t, r)(B(r, ul(r)) - B(r, uE(r))) dr X--*O+ ,)0

for t ~ [0, r]. Using the condition (B0 we find the integral inequality of Gronwall type

f IIw(t)ll -< Mexp(cooT)Ln(maxlHui[l~:i -- 1, 21) IIw(r)ll d r o

for t e [0, r] and, hence, w --: 0. •

The rest of this section is devoted to an application of the theory treated above to the first order semilinear problem

I ut + a(t, x)ux = b(t, x, u) + f ( t , x)

u(O, x) = Uo(X) (5.6)

u(t, O) = u(t, 1)

Hence, a e C~([0, T]: C[0, 1]) satisfies a > 0 on [0, T] x [0, 1] and the function b( t , x ,p ) , defined on [0, T] x [0, 1] x R, satisfies b e C~([0, T] x R: C[0, I]), and let

f 6 wl ' l (0 , T: C[0, 1]).

For convenience, we use the notation Zo = [u e Z: u(0) = u(1)} if Z is a space of functions defined on [0, 1].

for (t, x) e [0, T] x [0, 11,

for x e [0, 11,

for t e [0, TI.

THEOREM 5.2. If UO e C~[0, 1]~ satisfies the compatibility condition

-a(0 , -)u~(.) + b (0 , . , Uo(')) + f(0, .) e C[0, 1]0

then the problem (5.6) has a unique local classical solution.

Proof. Let X -- C[0, 1] and D = C1[0 , 1]~. It is seen by a modification of the proof of [5, theorem 6.1] that the family [A(t): t e [0, T]I of closed linear operators from D into X, defined by (A(t)u)(x) = -a(t , x)u'(x) for (t, x) e [0, T] × [0, 1], satisfies three conditions (A0 through (A3). For each t e [0, T] we define B(t, u) and F(t, u, v) by

B(t, u)(x) = b(t, x, u(x)) + f ( t , x),

F(t, u, v)(x) = (O/Ot)b(t, x, u(x)) + (O/Op)b(t, x, u(x)) " v(x) + h(t, x)

for x ~ [0, 11. Here, h is an element in LI(0, T: C[0, 1]) such that f ( t , x) - f (0 , x) = j~ h(s, x) ds for (t, x) e [0, T] × [0, 1]. Then, we see by a simple computation that all conditions on B and F are satisfied. Since/3 = C[0, 1]~, this theorem follows from theorem 5.1. •

788 N. TANAKA

R E F E R E N C E S

1. GOLDSTEIN J. A., Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985). 2. PAZY A., Semigroups o f Linear Operators and Applications to PartiaI Differential Equations. Springer, New York

(1983). 3. ARENDT W., Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59, 327-352 (1987). 4. KELLERMAN H. & HIEBER M., Integrated semigroups, J. Funct. Analysis 84, 160-180 (1989). 5. DA PRATO G. & SINESTRARI E., Non autonomous evolution operators of hyperbolic type, Semigroup Forum

45,302-321 (1992). 6. DA PRATO G. & SINESTRARI E., Differential operators with non dense domain, Annal. Sci. norm. sup. Pisa 14,

285-344 (1987). 7. KATO T., Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Tokyo, Sec. I 17, 241-258 (1970). 8. CRANDALL M. G. & PAZY A., Nonlinear evolution equations in Banach spaces, IsraelJ. Math. 11, 57-94 (1972).

Documents

Semilinear equations in the “hyperbolic” case