Pergamon

NonBnear Analysis, Theory. Methods & Applications, Vol. 29, No. 11, pp. 1287-1301, 1997 © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights rescinded 0362-546X/97 $17.00+000

PII : S0362-546X(96)00173-3

E X T E N S I O N O R B L O W U P O F C L A S S I C A L S O L U T I O N S TO Q U A S I L 1 N E A R

E V O L U T I O N E Q U A T I O N S IN B A N A C H S P A C E S f

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

(Received 20 September 1994; received in revised form 11 September 1996, received for publication 23 October 1996)

Key words and phrases: Abstract quasilinear evolution equation, evolution operator, quasi-dissipative, classical solution, extension or blow up.

1. I N T R O D U C T I O N A N D P R E L I M I N A R I E S

This paper is devoted to the study of the abstract quasilinear evolution equation

u'(t) = A(t, u(t))u(t) for t >- 0 u(0) = u0 (QL)

in a general Banach space X, where {A(t, w): (t, w) E [0, oo) × y} is a family of closed linear operators in X and Y is another Banach space which is densely and continuously imbedded in X.

The theory for abstract quasilinear evolution equations was initiated by Kato [1]. This theory is based on the theory for time dependent linear evolution equations which was studied by Kato [2, 3], and his purpose is to find classical solutions which exist locally in time. In this direction, Sanekata [4] has recently succeeded in proving an existence theorem of local classical solutions to (QL) without assuming the reflexivity of X and Y, and later his result has been improved by Kato [5]. It should be noted that their results have opened the way to deal with problems in spaces of continuous functions.

As stated above, the problem of existence and uniqueness of local classical solutions to abstract quasilinear evolution equations has been extensively studied. However, there seems to be no information about the existence of global classical solutions to abstract quasilinear evolution equations, while there are several works whose purpose is to find global classical solutions of concrete quasilinear hyperbolic differential equations such as the wave equation of Kirchhoff type.

The purpose of this paper is to obtain some results (Theorems 2.1 and 3.1) for the problem of extension or blow up of classical solutions to the abstract quasilinear evolution equation (QL) which will give a sufficient condition (Corollary 3.2) for the existence of global classical solutions to (QL). The proof of our results is essential to the estimate of classical solution by means of I1"11o,~) depending on (t, w) ~ [0, co) × Yunder which Xis a Banach space [we denote it by X(,,w)] and A(t, w) is quasi-dissipative in X(,.,,). Section 3 contains an application of the result obtained here to the problem of existence and uniqueness of global classical solutions to some quasilinear hyperbolic equation which was studied by several authors. (Here refer to the papers by Heard [6], Ikehata and Okazawa [7], and Yamada [8].)

t This work was supported in part by Grant-in-Aid for Encouragement of Young Scientist A-06740118 from the Minista3' of Education.

1287

1288 N, TANAKA

Here is listed the notation used in this paper. Given Banach spaces X, Y, ... the associated norms are denoted by II'llx, I1"11~ and so on. The symbol II'll~,x denotes the norm of the Banach space B(Y,X) which consists o f all bounded linear operators on Y to X. Given a Banach space Z and numbers M >- 1, fl >-- 0 S(Z, M, fl) denotes the set o f all stable families o f infinitesimal generators o f semigroups o f class (Co) on Z with the stability index (M, fl) (see [2]) and A E G(Z, M, fl) is written for the infinitesimal generator A of a semigroup {T(t): t-> 0} of class (Co) on Z satisfying [[T(t)l[z<--Me B' for t - 0. For simplicity in notation a V b = max(a, b) and a A b = min(a, b) i f a and b are real numbers.

In the rest o f this section the generation theorem due to Kobayasi [9, Theorem] o f linear evolution operators and fundamental results on abstract quasilinear evolution equations obtained by Kobayasi and Sanekata [10, Main Theorem] are summarized.

Consider the following conditions (i)-(iii) for a family {A(t): t ~ [0, T]} o f operators in X 'which generates an evolution operator on X.

(i) There exist constants M - - > 1 and fl --> 0 such that {A(t): t E [0, T]} E S(X, M, fl). (ii) There is an isomorphism S of Yonto X a n d a strongly continuous family {B(t): t E [0, T]} in

B(X) such that SA(t)S -~ = A(t) + B(t) for t E [0, T]. (iii) Y C D(A(t)) for each t E [0, T], and A(t) is strongly continuous in B(Y, X) on [0, T].

THEOREM 1.1. Under assumptions (i)-(iii), there exists a unique family {U(t, s): 0 < s -< t < T} in B(X) with the following properties:

(a) U(s, s) = I, and U(t, r)U(r, s) = U(t, s) for 0 -< s -< r <- t --< T. (b) U(t, s) is strongly continuous in B(X) for 0 <-- s --< t < T, and

(c)

where (d) (e) (f)

[[U(t,s)[[x<_MeP('-s) forO<_s<_t<_ T.

U(t, s)Y C Y, U(t, s) is stongly continuous in B(Y) for 0 --< s --< t --< T, and

[[U(t,s)[[r<_ffleB('-s) forO<_s<_t<_ T,

~1 = M[[S[[r,x[[S-lHx, r and/~ = Msup{[[B(t)[[x: t E [0, T]} + ft. (O/Ot)U(t, s)y = A(t)U(t, s)y for y E Y and 0 -< s -< t -< T. (a/as)U(t, s)y = - u(t, s)A(s)y for y E Y and 0 -< s -< t -< T. SU(t, s)y = U(t, s)Sy + ft U(t, r)B(r)SU(r, s)y dr for y G Y and 0 -< s -< t -< T.

We say that the family { U(t, s): 0 --- s --< t --< T} in B(X) is the euolution operator generated by {A(t): t E [0, T]}. We here state the outline o f the proof o f the last assertion (f). It is shown by Dorroh [11] that the Volterra-type integral equation

W(t, s)x = U(t, s)x + Ii W(t, r)B(r)U(r, s)x dr

has a unique solution W(t, s) E B(X) which is strongly continuous in B(X) for 0 --< s --< t --< T and that

U(t, s)S -l = S-IW(t, s) (1.1)

for 0 -< s -< t -< T. On the other hand, it is known [10, (1.7)] that the solution W(t, s) satisfies the integral equation

W(t, s )x= U(t, s)x + [ ' U(t, r)B(r)W(r, s)x dr Js

Quasilinear evolution equations 1289

for x E X and 0 ~ s <- t <- T. Assertion (f) is therefore obtained by combining this integral equation and (1.1).

Basic hypotheses on the operators appearing in the abstract quasilinear evolution equation (QL) imposed in [10] is stated as follows:

(A1) There exists an open subset W o f Yand To > 0 satisfying the following properties: A(t, w) is a linear operator in X defined for each (t, w) E [0, To] x W. For each p --> 0 there are M -> 1 and fl --> 0 such that

{A(t, v(t)): t E [0, To]} E S(X, M, fl)

for all v E Dp. Here the set Dp is defined by

Op = {o E C([0 , To]: W): IIv(t) - v ( s ) l l x < - - p l t - sl for t , s E [0, To]}.

(A2) There is an isomorphism S of Yonto X a n d a family {B(t, w): (t, w) E [0, To] × W} in B(X) such that

SA(t, w)S -1 = A(t, w) + B(t, w)

for (t, w) E [0, To] × W, where the family {B(t, w): (t, w) E [0, To] X W} satisfies the following properties: for each w G W, B(., w) is strongly continuous in B(X) on [0, To]. There are constants 2B > 0 and/tB > 0 such that

liB(t, w)l lx <- ~-B and liB(t, w) - B(t, z)ll~ - /~81 lw - zll Y

for t E [0, To] and w, z E W. (A3) For each (t, w) E [0, To] × W, D(A(t, w)) D Y. For each w ~ W, A(., w) is strongly con-

tinuous in B(Y, X) on [0, To]. There is a constant p~ > 0 such that

IIA(t, w) - ACt, z)llY.x ~ ~ l l w - zllx

for t E [0, To] and w, z E W.

THEOREM 1.2. If conditions (AI) - (A3) are satisfied then for each initial value Uo E W, there i,; a T E (0, To] such that the (QL) has a unique solution u in the class C([0, T]: W) n c~([o, T]: X).

2. EXTENSION OR BLOW UP OF CLASSICAL SOLUTIONS

In this section we discuss the problem of extension or blow up of classical solutions to the quasilinear evolution equation (QL) , where we mean by a classical solution u to (QL) on an interval J = [0, r] or [0, z) with 0 < r < ~ that u ~ C(J: Y) n C~(J: X) and the (QL) is satisfied for t E J A classical solution u to (QL) on [0, oo) is said to be a global classical solution to (QL). Throughout this section we assume that for each (t, w) G [0, ~) × Y there is a norm II'll~,,w) of X which is equivalent to the original norm I1 IIx of x with the following properties:

(n~) For each • > 0 and r > 0 there is a constant 2x(r, r) --> 1 such that

2x(z, r) -1 Ilxllx-< Ilxll~,.w~ -< .~x(v, r)llxllx

for t E [0, r], w E Br(r) and x ~ X, where By(r) = {w E Y: IlwllY <-- r}; (nz) for each r > 0 and r > 0 there is a constant/ tx(r, r) >-- 0 such that

Ilxll.,w) -< Ilxll~s,z) " exp{/~x(~, r ) ( l lw - zllx + It - s l )}

for t, s E [0, r], w, z E By(r) and x ~ X.

1290 N. TANAKA

It should be noted that Hughes et al. [12] first proposed the equivalent norms satisfying conditions (n~) and (n2), and established the abstract theory which is applied to second-order quas~linear hyperbolic systems on R".

Let {A(t, w): (t, w) ~ [0, oo) X Y} be a family of closed linear operators in X with the following properties:

(al) For each r > 0 and r > 0 there is a constant co(z, r) --> 0 such that

A(t, w) ~ G(X(,,w), 1, co(z, r)) for t E [0, ~] and w ~ Br(r),

where X(,,~ denotes the Banach space X with the norm ]}-II(,.w);

(aD there is an isomorphism S of Y onto X and a family {B(t, w): (t, w) E [0, o~) x Y} in B(X) such that

SA(t, w)S -I = A(t, w) + B(t, w)

for (t, w) E [0, oo) X Y, where the family {B(t, w): (t, w) E [0, 00) X Y} satisfies the following properties: for each w E Y, B(., w) is strongly continuous in B(X) on [0, ~). For each r > 0 and r > 0 there is a constant its(r, r) > 0 such that

l iB(t, w) - B(t , z)llx <-/~a(r, r)llw - z l ly ( 2 . 1 )

for t ~ [0, r] and w, z E By(r); (a3) for each (t, w) ~ [0, oo) x Y, D(A(t, w)) D Y. For each w E Y, A(., w) is strongly continuous in

B(Y, X ) on [0, ~). For each 3 > 0 and r > 0 there is a constant/ tAr, r) > 0 such that

HA(t, w) - A(t, z)[] r.x -----/tA(r, r)llw -- Zllx (2.2)

for t E [0, 3] and w, z E By(r).

Remark. The following (2.3) and (2.4) easily follow from (2.2) and (2.1) together with the strong continuity respectively: For each r > 0 and r > 0 there are constants 2A(r, r) > 0 and 2s(r, r) > 0 such that

[IA(t, w)llr, x <- 2A(~, r) for t ~ [0, r] and w ~ By(r), (2.3)

liB(t, w)[[x ----- 2s(r, r) for t ~ [0, ~] and w ~ By(r). (2.4)

The main result of this paper is given by the following theorem.

THEOREM 2.1. For each initial value Uo E Y the quasilinear evolution equation (QL) has a unique classical solution u on a maximal interval of existence [0, t,~x). If tm~x < co then

lim ]]u(t)l]r = ~. t T tm~

Before proving this theorem, we give some information on the stability condition which may be shown in the proof of [10, Proposition 2.1].

PROr~SnlON 2.2. Let 0 < ro --< r, r > 0 and p > 0. If conditions (n~), (n2) and (a0 are satisfied, then we have

{A(t, o(t)): t U [0, 30]} ~ S(X, M(3, r, p), co(r, r))

for o ~ C([0, r0]: Y) such that u(t) E Br(r) for t ~ [0, Co] and that Ilo(t) - o(s)llx <- p}t - sl for t, s ~ [0, ~0], where

M(3, r, p) = 2x(Z, r) z exp(px(Z, r)(p + 1)~). (2.5)

Quasilinear evolution equations 12!)1

Proof of Theorem 2.1. We begin by considering the existence o f local classical solutions to the quasilinear evolution equation (QL) for each u0 ~ Y. Let Uo E Y and To > 0, and we shall apply Theorem 1.2 to show that the (QL) has a classical solution u on [0, T] for some T E (0, To]. To do so, we take R > 0 such that Iluoll~ < R, and set W = {w G Y: Ilwll~ < R}. Clearly Wis an open subset o f Y and Uo E W. One easily verifies that all the other conditions except for condition (A1) of Theorem 1.2 are satisfied, by conditions (a3) and (a2) combined with (2.4). Condition (A1) follows immediately from Proposition 2.2, in view of conditions (nt), (n2) and (al). Therefore the existence of a local classical solution u to (QL) is assured by Theorem 1.2. We note that the classical solution u to (QL) on a closed interval [0, T] is extendable, by defining u(t + T) = w(t) where w is a local classical solution to the problem

w'(t) = A(t + T, w(t))w(t) for t --> 0

w(O) = u(r )

whose existence is obtained by applying the same argument as above to the family {.~/(t, w): (t, w) ~ [0, oo) x Y} defined by s~(t, w) = A(t + T, w) for (t, w) E [0, ~) x Y; for it is seen that ,~/(t, w) and l" [(,.w) defined below satisfy conditions (ni), (n2) and (a0-(a3), by choosing constants with z replaced by ~ + T, and taking ~(t , w) defined below as an operator B(t, w) appearing in condition (a2):

{ l'l~,,w~ = II'II,+T.~) for(t, w) E [0, oo) × y,

~(t, w) = B(t + T, w) for (t, w) E [0, ~) × Y.

Let [0, tm~x) be the maximal interval to which the classical solution u to (QL) can be extended. For brevity in notation we write A"(t) = A(t, u(t)) and tf'(t) = B(t, u(t)) for t G [0, tm~). By Proposition 2.2 the family {A"(t): t E [0, z]} belongs to the class S(X, M(z, r~,pT), o9(z, tO) for each z E (0, tin, x), where we set r~ = sup{llu(t)ll~: t ~ [0, z]} and p~ = sup{llu'(t)llx: t ~ [0, r]}, and the symbol M(- , . , .) is defined by (2.5). Theorem 1.1 enables us to define a family { U"(t, s): 0 -< s -< t < tmax} in B ( x ) b y

U"(t, s) = U~(t, s) for 0 <- s <-- t -< z and z E (0, tn,~), (2.6)

where {U~(t, s): 0 -< s -< t <- ~} is the evolution operator on X generated by {A"(t): t E [0, T]}. The definition of { U~(t, s): 0 --~ S --< t "< tmax} is unambiguous, since U~(t, s) = U~(t, s) for 0 <-- s --- t <- q A z2 which is obtained by differentiating U~(t, rl)U~(rl, s)y in ~/and then integrating over [s, t]. It is seen that the family {U"(t,s): 0 -< s <- t < t~x} in B(X) satisfies the following properties:

(PO U"(s, s) = L and U"(t, rl)U"(~ I, s) = U"(t, s) for 0 <- s -< r/-< t < tin,x; (P2) U"(t, s) is strongly continuous in B(X) for 0 <-- s --< t < tm~x ; (P3) U"(t, s)Y C Y, U"(t, s) is strongly continuous in B(Y) for 0 <-- s <-- t < t~x, and

SU"(t, s)y = U"(t, s)Sy + Jl U"(t, tl)B"(tl)SU"(r I, s)y drl (2.7)

for y ~ Y and 0 --- s -< t < tr,,x ; (P4) (O/Ot)U"(t, s)y = A"(t)U"(t, s)y for y E Y and 0 -< s -< t < tm~x ; (Ps) (O/Os)U"(t, s)y = - U"(t, s)A"(s)y for y E Y and 0 --- s <- t < tm~x.

We shall prove that if tm~x < oo then Ilu(t)llY--' ~ as tl" tm~x. To this end, we prepare the following lemma in the middle o f the proof.

LEMMA 2.3. I f tm~ < ~ then lim suptr~[lu(t)[[r = oo.

1292 N. TANAKA

Proof: Assume to the contrary that t,,, < ~0 and lim supltr,., Ilu(t)llY < a. Then there exists a constant r,,, > 0 such that

Il~(0ll Y 5 rmax for t E [0, t,,,). Gw

Setting Pmax = &4t,,, rmax)rmax, we have by (2.3)

Il~‘(Ollx 5 II~“~~~llY,xIl~~~)llY 5 PInax for t E [0, t,,,).

It follows from Proposition 2.2 that the family (A”(t): t E [0, t]} belongs to the class WG ~(t,,, , rm, p,d, Nt-, r,,, )) for each r E (0, t,,,,,), where A4(-, ., .) is defined by (2.5). By (2.4) we have

Il~“Wll~ 5 Mtmaxr rmax) for t E [0, tmax). (2.9)

For each r E (0, t,,,) we have by properties (b) and (c) of Theorem 1.1,

{

II C(t, s)IIx 5 Mtmx, r,,, , pm,,) exp(4tmax, rmax)(t - 41,

II Wt, dll Y 5 fib,, rmax7 pmax) exp(Wmm, r,,, , h& - 4)

for 0 I s 5 t 5 t. Here the symbols iii<., ., .) and O(., ., .) are defined by

i

Mt,,,, r,, , p,d = Mt,,, , rma, pmax)ll~II Y,X IP -’ lb. Y,

Wmax, r,,, , p,,J = MtmaxI +-max)Wmax, r,, , pm4 + dt,,, , rmax>

respectively. Taking account of the definition of U”(t, s) [see (2.6)], we deduce that there exist constants A4,, > 0 and I@,,,,, > 0 satisfying

II U”(c s)llx 55 Mnax and II U”(4 s)ll Y 5 Mnax forOSsIt<t,,,. (2.10)

LetxEXandOSsSt,t^<t,,.By(&)wehave

l l U”(t, s)x - U”<i, s)xll~ 5 2M,,llx - YIIX + A4(fmax, ~m~x)~m~llrll Y It - 4

for all y E Y. Passing to the limit as t, t^t tmax, and noting that Y is dense in X we see that the limit lim, t fmu U”(t, s)x exists in X for x E X and 0 5 s < t,,,,x. By (pS) we have (d/@)U”(t, ~)u(v) = 0 for OSgzSt<t,,; hence u(t) = U”(t, 0)~~ for t E [0, t,,,). Property (p,) implies u(t) = U”(t, s)U”(s, O)u, = U”(t, s)u(s) for 0 5 s 5 t < t,,,. Noting this and the fact that u(s) E Y for s E [0, tmax), we find by (2.7)

S(t) = uyt, s)Su(s) + I

t lqt, ~)B”(tjqh(~) dq s

for OIsSt<t,,,. We represent the difference Su(t) - S(i) by this equality, and estimate the resultant equality by (2.8)-(2.10). This yields

IIS(u(t) - u(i))llx 5 Il(U”(t, s) - U”(i, s))SuWll~ + 2Mma,Mtmaxr r~ax)llSIIY,Xrmax(tmax - s> (2.11)

forOSsIt,i<t,,. The first term of the right-hand side of (2.11) vanishes as t, t^ f t,,, , by the fact which has been proved above. Taking the lim sup of (2.11) as t, f t tmax we have

lim sup IIS(u(t) - 24fl)ll~ 5 2M,.,&(tmax, rmax)llSII~,xrmax(t,, - s), t.itr,,

and the right-hand side tends to zero as s t t,,,,x. Consequently, we see that the limit lim, 7 I, u(t) (exists in Y, since S is an isomorphism of Y onto X. This fact shows that the classical solution u to (QL) on [0, t,,,& is extendable by the first part of the proof of Theorem 2.1, which is a contradiction to the maximality of t,,,. n

Quasilinear evolution equations 12'93

End o f the p r o o f o f Theorem 2.1. To conclude the proof we shall show that lim, T,~ [lu(t)llr = oo. I f this is false then there are a sequence {t,} and a constant K > 0 such that t. ~ tm~, as n -~ oo and

llu(t.)llr < - K for all n -> 1. (2.12)

We now set

gmax = ~.x(tmax. K)llS-'llx. rO'x(tm,,. K)IISIIr.xK + 1)

and then define the set H. by

g , = {h E [0. tmax -- t,): I lu(t, + h) l l , >- Kin,x}

for each n - 1. Since lim supt~,m, llu(t)llr = oo by Lemma 2.3, all set H. are nonempty. Defining h. = i n fH . for each n -> 1, we see by (2.12) that 0 < h. < tm.x -- t., since Km~ > K and u is continuous on [0, tin.x) in Y. Taking account o f the definition o f h. , we obtain by the continuity o f u

on [0, tm~x) in Y,

Ilu(t. + h.)llr = gmax, (2.13) Ilu(t. + h)ll r < gmax for h E [0, h.).

Let { U"(t, s): 0 -< s -< t < tm~x} be the family in B ( X ) defined by (2.6). We shall show the estimate

J(,-S),x,, (2.14) II u"(t. s ) x l l . . ~ o ~ - ~ I, , ,~..~.~

for x E X and t, ~ s ~ t ~ t, + h, . where the constant N is determined by

N = o~(tmax, Kin.x) + p x ( t . . . . Kmax)(l + )],A(tmax, Kmax)Kmax).

To prove (2.14) it suffices to show that (2.14) holds for x G Y, since Y is dense in X. To do so, we first note that for every x E Y, the function t --* [1U"(t, s)xll(,, .~,)) is continuous on Is, t. + h.] where s >- t.. This fact follows from the inequality

III u"(i, s)xll(; , .( i)) - II u"(t, s ) x l l , , ~ , . I -< II u"(i. s)x - u"(t, s)xll(i . .( i))

+ III f"(t , s)xll~i,.~,~) - II u"(t, s)xll(i,.~,))l

, g m , ~ ) l l U (t, s )x - U"(t , s)xllx < ~ ).x(tmax u "

+ (exp{ / tx( tm,x. Km~x)(lt - i l + I l u ( t ) - u ( / ) l l x ) } - 1)

× Z~(/m.x, Km.x)ll U"(t, s)xll~

for s --< t, i < t. + h.. Here we have used the fact that u(t) E Br(Km~x) for t ~ [t., t. + h.] I b y (2.13)], and two conditions (n~) and (n2). Now, let x E Y and s > t . , and we shall compute

D-II U"(t, s)xll(,,.,,

for t ~ (s, t. + h.], where D - denotes Dini 's derivative defined by D - w ( t ) = lim suph~o(W(t) - w(t - h))/h. Condition (a,) together with (2.13) implies

lim (11 u"(t, s ) x l l . . . ( , . - II U"(t - h, s)xll(,,.~,.)/h = [U"(t, s)x, A"(t)U"(t , s)x]u,.~o ) h~O

<- O~(tm.x, g..x)llU"(t, s)xll~,..(,)) (2.15)

for t ~ (s, t. + h.], where the inner-product [', "](~,.(o) is defined by

[P, q]u,.u)> = l im( l l p l l ( , , . . ) ) - l ip - ; tqll( , , .(o))/~. 2J, 0

1294 N. TANAKA

for everyp, q E X. Since u is the classical solution to (QL), we have Ilu'(t)llx -- 11,4"(t)ll r, xllu(t)llr for t E [0, tm~). Estimating it by (2.3) and (2.13) yields

Ilu'(t)llx <- ~.A(tmax, gmax)gmax for t • [t., t. + h.]. (2.16)

We find by condition (n2) the inequality

(11U"(t - h, s)xll~,,.~,, - II U"( t - h, s)xll ,-h, , t , -h)))/h

<-- h- l (exp{t~x( tm~x, Kin.x)(1 + 2A(tm~x, Km~x)K~x)h } - 1)11 u" ( t - h, s)xll<,-h,.~,-h,

for t ~ (s, t, + h,] and h E (0, t - s). Combination of this and (2.15) gives

D-II U"(t, s)xll<,,.<,, <- Nil U"(t, s)xll(t,.tt))

for t E (s, t, + h,]. Solving this differential inequality we obtain the desired estimate (2.14). (See Lakshmikantham and Leela [13, Theorem 1.2.1 or 1.4.1] for comparison theorems.)

Setting y = u(t,) , s = t, and t = t, + h. in (2.7), and then noting that u is represented as u(t) = U"(t , O)uo for t ~ [0, tm~x) we find by (P0

If . + h .

Su( t . + h . ) = U"( t . + h . , t . )Su( t . ) + U"( t . + h . , rl)B"(rl)Su(rl) drl n

for n -- 1. Using (nO and (2.4) we have by (2.13)

IIB"(~)Su(~)II¢~,.(.)) <- 2x( t . . . . K,~x)2B(tm~x, Km~,)llSllr, x Kmax

for r /~ It,, t, + h.] and n -> 1. We apply (2.14) and this estimate to the above equality. This yields

IlSu(t. + h.)ll~,. + h.,.¢,. + h.)) <- eNh"llSu(t.)ll¢,.,.<,.)) + eNh"h.J'x(tmax, gm~x))~n(t . . . . gm~x)llSllr, x g . ~

for n --> 1. By condition (nO combined with (2.12), the first term of the right-hand side is majorized e "2X(tm~,, K)IISIly.xK. Since h, ~ 0 as n ~ 0% the second term of the right-hand side vanishes as by Nh

n --* w, and hence

lim supllSu(tn + h.)ll<,. + h.,.t,. + h.)) - ),x(tmax, K)IISIIr, x K . 112.17) n ---~oo

But, we have by condition (n/)

IISu(t.+h.)llt~.+h.,~+h.))-->exp{#x(tm~x, Km~x)(1 + 2A(tm~x, Km~x)Km~x)h.}-'llSu(t. + h.)lt<,.,.~,.~,),

where we have used the inequality (2.16). The inequality

IISu(t. + h.)ll..,.(,.)>-> 2x(t ,~x, g)-~llS-~llx,~gm~x

is obtained by using condition (nO together with (2.12), and the estimate that Km~x-- IIS -~llx,r IISu(t. + h.)llx which follows from the first part of (2.13). Combining these inequalities we find

K) IIS II~,~gm~x, l im infllSu(t . + h.)llc,. + ~.,.~,. + h., >- J.X(tmax, - t --1 --1

and the right-hand side is equal to 2X(tm~x, K)IISIIr, x K + 1 by the definition of Km~x. This is a contradiction to (2.17). Therefore we conclude that limttt~ I lu(t) l lr-- o~ if tm~ < ~.

Finally we shall prove the uniqueness of classical solutions to the quasilinear evolution equation (QL). To this end, let v be another classical solution to (QL) and [0, rm.~) the largest interval in which o exists. Let t ~ (0, tma~ A rm~x) be arbitrarily fixed, Differentiating U"(s, r/)(u(q) - o(q)) in q and

Quasilinear evolution equations 1295

integrating over [0, s] we obtain by (Ps) the integral equation

u(s) - o(s) = Ii UU(s' ,1)(A"(,# - A°(,7))0(,7)

for 0 --< s <-- t. The integral term is estimated by

;o MdtA(t,R,)R, Ilu(~) - o(tt)llx ~ ,

where we set

Mr = sup{ l lU"(s , q)l lx: 0 - - t /< - s <- t},

R, -- sup{l lu( t / ) l l r : ,7 e [0, t]} V sup{llo(~)llY: t / ~ [0, t]).

Application of Gronwall 's inequality gives u = o on [0, t]. Since t ~ (0, tmax A Zm~) is arbitrary, we have u = o on [0, tm~x A Zm~x). Therefore, both u and o must have the same maximal interval of existence; for if tm~x < Zm~x then the fact shown above together with the continuity of o in Y on [0, tm~x] shows that the limit lim, r,o~ u(t) = o(tm~x) exists in Y, and so u can be extended beyond t,,ax contradicting the definition of tm~x. Consequently we conclude that u = u on [0, tro~x). •

3. CONCLUDING REMARKS AND APPLICATIONS

In this section we study the following quasilinear evolution equation with the nonlinear t e r m f b y reducing it to the case where f = 0 and using the result in the previous section:

u'(t) = A(t, u(t))u(t) + f( t , u(t)) for t >-- 0 u(0) = Uo. ( Q L ; f )

Here {A(t, w): (t, w) ~ [0, co) × y} is a family of closed linear operators in X satisfying conditions (al)-(a3) in the previous section, andf( t , w) is a nonlinear operator with the following condition:

( f ) For each (t, w) E [0, oo) x Y, f( t , w) is defined and belongs to Y. For each w E Y, f ( . , w) is continuous in Yon [0, oo). For each r > 0 and r > 0 there are constants ffi(z, r) > 0 andfif(z, r) > 0 such that

IIf(t, w) - f(t , z)llx --< ,u,(z, r)llw - zllx,

Ilf(t, w) - f ( t , z)[lr <-- fif(~, r)llw - zllr

for t E [0, 3] and w, z E Br(r). According to the device due to Kato [14, Subsection 1.3] we shall reduce the problem ( Q L ; f ) to

the problem (QL) studied in the previous section. Consider the following system in the product spaces

. ( .= ( X ) a n d ~ '= ( Y ) w i t h the norms defined by I(i)1 =[[ ' [[x + ~ I'l and IC)I ,,,,+, , ,

respectively:

a'(t) = .4(t, a(t))~(t) for t >- 0

5(0)__ ( 1 ) . (3.1)

1296 N. T A N A K A

,K,,, ,

# = ( ~ ) E ~ ' . It is obvious that ~ ' is densely and continuously imbedded in 2 and that the

corresponding ones to all the other conditions in (a3) except for (2.2) hold. The corresponding inequality to (2.2) is obtained by an easy computation; in fact we have

IA(t, ~) - .4(t, z')l ~,.t --< (/l~(r, r) V/~f(z, r))l~ - ;?l~c

fortE[O,z]andff,~.EB~(r).Wedefineanorm](2) I o f ) ( b y (t, ~)

(t, a,)

t E [0, o o ) a n d S = ( ~ ) E ~'. One easily verifies that the corresponding conditions to (hi) for each

and (n2) are satisfied. Condition (a,) implies

fort~[O,~]and~,=(~)~.B,(r).Byasimplecomputationwehave

0 /1(,,~)

fortE[O,r]and~=(w~)EB~(r),where2F(r,r)istheconstantsatisfying IIf(t, w)llx -< 2s(r, r) for t ~ [0, z] and w E Br(r).

From the perturbation theorem (see Pazy [ 15, Theorem 3.1.1 ]) we deduce that

A(t, ~) E G(2(t,~,), I, co(z, r) + 2x(Z, r)Af(z, r)).

The corresponding condition to (a2) is satisfied by choosing the isomorphism S of ~" onto .~" and /}(t, ~) defined by

fortE[O,~)and#=(~)E:_Y. Allconditionsimposedintheprevioussectionarechecked, andso

Theorem'2.1 is applicable to the problem (3.1) with Uo E Y. Noting that if ~ = (~) is the classical

solution to (3.1) then ~(t) = 1 and u is the classical solution to the problem (QL;f) , we obtain the following theorem which is an extension of the result [15, the Remark after Theorem 6.1.7] in the semilinear case.

Quasilinear evolution equations 1297

THEOREM 3.1. If conditions (n~), (n2), (al}-(a3) and (f) are assumed then for each initial value u0 ~ Y, the problem ( Q L ; f ) has a unique classical solution u on a maximal interval of existence [0, tin,x). If trnax "( oo then

lim Ilu(t)ll v--- oo. t T t ~

We give a sufficient condition for the existence of global classical solutions to ( Q L ; f ) under assumptions of Theorem 3.1.

COROLLARY 3.2. Assume conditions (nO, (n2), (a~)-(a3) and (f). Then for each initial value u0 ~ Y, the problem ( Q L ; f ) has a unique global classical solution u if there exists a real-valued continuous function ko defined on [0, oo) such that [lu(t)l[r <- ko(t) for every t in the interval of existence of u.

We conclude this section with an application of our abstract theory. Let us consider the following quasilinear hyperbolic equation in a real Hilbert space H with inner product (., .) and the associated n o r m I'1:

u"(t) + tr(ls~/2u(t)[2)s~u(t) + fiu'(t) = 0 for t >- 0 (3.2)

with initial condition

u ( O ) = ~ , u ' ( O ) = 6 . (3.3)

Here ~ is a nonnegative selfadjoint operator in H, fi --- 0 is a constant and a is a positive function of class C a on [0, 0% (The assumption that there is a or0 > 0 such that a(t/) --- tro for t/-> 0 is imposed in [6-8].) We note that the square root ~ / 2 of ~ is well-defined and s~ ~12 is also a nonnegative selfadjoint operator in H.

By D and V we denote real Hilbert spaces D(.~) and D(s~ ~/2) equipped with the following inner products (., ")o and (., ")v respectively.

(u, o)o = (u, o) + (sgu, s~v) for u, o E D(s~),

(u, o)v = (u, v) + (s~l/2u, s~1/2o) for u,/) E D(.5~1/2).

We convert the differential equation (3.2) into the first order system in a pair of Hilbert spaces

where a(t) = [ ~/u(t)/and A(ff) = \ft(t)]

X and Y are defined by

a'(t) = A(a(t))~(t)

(-a(ls2 '2wl2)sa

(27, fi)x = (x, y) v + (2,)~)

(e, ~ ) , = (z, w),, + (i, ~ )v

for t >-- 0,

I i) f o r ~ = - 5

(3.4)

(WlEY. The inner products in \1,}

(z) (w) for :~ = ~ , ~ , = E Y.

1298 N. TANAKA

We shall study the problem of not only existence and uniqueness but also extension or blow up of classical solutions to (3.4) by using Theorem 2.1. We first note that

--<r for rb = (wW.)~ Br(r), (3.5) 1~"2wl

since [s~mw[ z = (,~w, w) --< [W[2D/2 for w E D. Now, we introduce the following inner product (., .)~ in X which depends on d, G Y:

(a, ~)~ = a(lsa'/~ wl2)(.~'/~u, ~ ' % ) + (u, o) + (a, 0)

for 5, 6 E X. To verify condition (nO, set __2#(r) = inf{~r(t/): 0 ---< r/<- r 2} and ,~(r) = sup{a(q): 0 ~ ~/-< r 2 } for r -- 0. It should be noted that the two functions are positive, since a is positive on [0, co). From (3.5) it follows that condition (nl) is satisfied with 2 x ( r ) = (2~(r) A 1) -vz V (•(r) V 1)m; namely

2x(r)- 'll~llx -< IPlP -< 2x(r)ll~llx (3.6)

for ;f E X and ff E Br(r). We have by (3.5) again

a(l~"2wl ~) -a(la~,2zl ~) _< Zo(r)(l~"~wl + I~t%l)ll~"Zwl- I~'Zzll <- 2Z,(r)rl[~, - ellx (3.7)

for if,, ~ ~ Br(r), where L,(r) denotes the Lipschitz constant of a on the interval [0, rZ]. We use (3.7) to find the inequality

I1~11~ - IPlI~ -< 2L,(r)rll~ - ellxll~ll~

for ~ E X and if, :f E Br(r). Application of (3.6) and the fact that 1 + a -< ¢ for a --> 0 gives

I1~11~ -< exp(2J.x(r)ZL~(r)rll ~ - ellx)lPll~

for ~ E X and if, :f G By(r), which implies that condition (nz) is satisfied with px(r) = 2x(r)ZL,,(r)r. We shall prove that condition (a0 is satisfied. Since ,~ is a nonnegative selfadjoint operator in a Hilbert space H, it is a densely defined maximal monotone operator in H. It follows by an easy computation that R ( 2 / - A(ff)) = X for 2 > 0. The fact A(ff,) E G(X~, 1, 1) is shown by the range condition and the inequality (A(ff)tL a),~ -< (u, u) - (lul ~ + lal2)/2 -< Ilall~ for w e Y. Choosing

+ ~/)v2 0 S = ( ( I 0 (i+.~t)1/z)

as an isomorphism of Y onto X, we see that condition (a~) is satisfied with B(ff) = 0. The desired inequality (2.2) is easily obtained by using (3.7). Therefore, we deduce from Theorem 2.1 that if

~b ~ D and ~ ~ V then the problem (3.4) with initial condition ~(0) = ( : ) has a unique classical

solution ~ on a maximal interval of existence [0, t~,x), and that l imm,,lla(t)l lv = ~ if tr,~ < ~. In other words, the problem (3.2)-(3.3) has a unique solution u in the class C([0, t ,~): D) fq C~([0, tm~,): V) ~ C~([0, tin,x): H) satisfying either tm,x = ~ or tm~, < ~ and l im ,~ ( lu ( t ) [ 2 + [u'(t)lZv) ~/z = oo.

We shall prove that the problem (3.2)-(3.3) has a unique global solution if 5 > 0 and small initial data is given. We now define #(r) = f[~ a(r/) dr / for r -> 0. Since ~r is positive on [0, 0% # is strictly increasing on [0, 0% It is obvious that #(oo) = limt-~oo 6(t) exists and 6(oo) ~ (0, ~]. This fact enables

Quasilinear evolution equations 1299

us to choose ro > 0 such that r 2 + t~(r g) < t~(o~). We then set Ro = (t~-~(r 2 + #(r2))) ~/~ and define C(e) = ;.¢(Ro)l/2(1 + 1/(A~(Ro)))~/:e for each e ~ (0, ro). Since 6 > 0 and__2~(Ro) > 0, it is possible to choose eo ~ (0, ro) such that

~o(R0) > ~,(~0)~0 C(~0), (3.8) where we set 2~,(r) = sup{ la'('~)l: 0 --: ,I ~ r% for r -> 0. We now define the set Do of initial data by

Do = { ( ~ ) ~ ( D ) : ]~blo-< Co, i , [v<e0}, and let ( ~ ) ~ D 0 . To attain our object, it suffices to

show that [u(t)l 2 + lu'(t)[ 2 is dominated on [0, tm~x) by a continuous function defined on [0, o~), by Corollary 3.2. This fact will be proved as follows, by a standard argument used in [7] and [8]. Taking the inner product of (3.2) with 2u'(t) we have

d(lu '( t) l 2 + ~(l~/~/2u(t)l:)) = -261u'(t)12 ~ 0

for t ~ [0, tm~x). Integrating this from 0 to t we obtain the inequality

lu'(t)l 2 + t~(]~'/2u(t)]2) --< I~l 2 + ~(l~/~J2~bl 2) - e 2 + O(e~)

for t ~ [0, tr~x). We thus have for t ~ [0, t~x),

lu'(t)[ --- (e~ + ~(e~))v%

lu(t)l -< t(e~ + ~(eo2)) '~2 + I¢'1, (3.9) 1~'/2u(t)l <- Ro.

By offa and ~l~ we denote the resolvent and the Yosida approximation of ~ respectively. It is well known that .~x = ~ / ~ for 2 > 0 and that limato ~ u = u for u ~ H. Taking the inner product of 2~zu ' ( t ) and each term of the equation obtained by operating with offa from the left on (3.2), we find

dtd 1~/,/2~au,(t)l z + a(l~/2u(t)12)d i ~ u ( t ) l z + 261~,2~au,(t)12 = o,

from which it follows that

d 2 a(l~'2u(/)l 2) ~" (I.~x u(t)l

for t E [0, tm~). If r ~ (0, tmax] and

[d a(l~¢/2u(t)[z)l<_2&r(l~tl,/2u(t)]2 )

then we have

[,~/V2u'(t)l 2 < I~/u(t)l 2 + a(l~,/2u(t)[2) - I~bl 2 +

Indeed, the differential inequality

d ( ]~l/2~;u'(t)12~ < dt [~xu(t)12 + trfl~Zu(t)12)] - o

l~/l/2~ u'(t)12 / [(d/clt)tr(l~/2u(t)[2) ) + ~r(l~L%(t)12)j = --\ ~ + 26 l~'/2offxu'(t)l 2 (3.10)

a(la'J2¢12 )

for t ~ [0, t)

for t E [0, ~). (3.12)

for t E [0, r) (3.11)

1300 N. TANAKA

is obtained by combining (3.10) and (3.11). Integrating this over [0, t] and letting 2 $ 0 we ,obtain (3.12).

We have only to show that (3.11) is true with r = tmax; for i f this is done then the fact above implies that (3.12) is true with r = tmax and a combination o f this fact and (3.9) proves the desired claim that lu(t)l~ + lu'(t)12v is dominated on [0,/max) by a continuous function defined on [0, oc.). We now show that (3.11) is true with z - tmax. For this purpose, it suffices to prove

2~,(Ro)Rol.~mu'(t)[ <- O._2~(Ro) (3.13)

for t E [0, tmax), since we have by (3.9)

[(d/dt)a(lalt/Zu(t)12)l = 21a'(l,~11/2u(t) 12)(s~/2u(t), .711/2u'(t))[

<_ 2~,(Ro)Rol,~l~/Zu' (t)l

for t E [0, tr.~x). Assume to the contrary that the set

A = {t E [0, tm~x): L'(Ro)RoI~V2U'(t)I > ~ ( R o ) }

is nonempty, and then set t* = in fA . Since I¢~lz -< eo and C(eo) >- Co, we have t* ~ 0 by the choice o f eo [see (3.8)]. From the definition o f t* we deduce that

{ fw(Ro)RolNl/2u'(t*)l = ~_.2o(Ro), (3.14)

f~,(Ro)RolM.~/2u'(t)] <-- 6__2,(Ro) for t E [0, t*).

The second part o f (3.14) implies that (3.11) is true with r = t*, and so is (3.12) by what has been already proved above. Using (3.121) with r = t* we have by the definition o f C(eo)

1~"2u'(t)l <- a(l~'/2u(t)12) ''2 I~/~l 2 + a ~ 2 ) } -< C(eo)

for t U [0, t*); hence I~/'%'(t*)l -- C(e0) by the continuity o f u' in V on [0, tm~x). Combining this and the first part o f (3.14) we obtain 6_2~(Ro) -< 2~,(Ro)RoC(eo), which contradicts the choice o f Co.

R E F E R E N C E S

1. Kato, T., Quasi-linear Equations of Evolution, with Applications to Partial Differential Equations. Lecture Notes in Mathematics, Vol. 448. Springer, Berlin, 1975, pp. 25-70.

2. Kato, T., Linear evolution equations of "hyperbolic" type. J Fac. Sei. Univ. Tokyo, 1970, 17, 241-258. 3. Kato, T., Linear evolution equations of "hyperbolic" type II. J Math. Soc. Japan, 1973, 25, 648-666. 4. Sanekata, N., Abstract quasi-linear equations of evolution in nonreflexive Banach spaces. Hiroshima Math. J., 1989, 19,

109-139. 5. Kato, T., Abstract Evolution Equations, Linear and Quasilinear, Revisited. Lecture Notes in Mathematics, Vol. 1540.

Springer, Berlin, 1991, pp. 103-125. 6. Heard, M. L., A quasilinear hyperbolic integrodifferential equation related to a nonlinear string. Trans. Amer. Math. Sot.,

1984, 285, 805-823. 7. Ikehata, R. and Okazawa, N., Yosida approximation and nonlinear hyperbolic equation. Nonlinear Analysis, 1990, 15,

479-495. 8. Yamada, Y., On some qtmsilinear wave equations with dissipative terms. Nagoya Math. J, 1982, 87, 17-39. 9. Kobayasi, K., On a theorem for linear evolution equations of hyperbolic type. J Math. Soc. Japan, 1979, 31,647--654.

10. Kobayasi, K. and Sanekata, N., A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces. Hiroshima Math. 2, 1989, 19, 52t-540.

11. Dorroh, J. R., A simplified proof of a theorem of Kato on linear evolution equations. J Math. Soe. Japan, 1975, 27, 474-478.

Quasilinear evolution equations 1301

12. Hughes, T. R., Kato, T. and Marsden, J. E., Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodypamics and general relativity. Arch. Rational Mech. Anal., 1977, 63, 273-294.

13. Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities. Academic Press, New York, 1969. 14. Kato, T., Abstract Differential Equations and Nonlinear Mixed Problems. Lezioni Fermiane, Accademia Nazionale dei

Lincei, Scuola Normale Superiore, 1985. 15. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983.