I’onhneor
Ano,.vx,s.
Theory.
Mefhodr
& Applicnrionr,
Vol 25, Nor 9-10, PP. 1063-1069, 1995 Copyright 0 1995 Elsevier Science Lid Printed in Great Britain. All rights reserved 0362-546X/95 $9.50+ .oO
0362-546X(95)00101-8
DIFFERENTIAL
INEQUALITIES
FOR EVOLUTION
GIOVANNI S.I.S.S.A.-Via
EQUATIONS
VIDOSSICH
Beirut 2-4, 34013 Trieste, Italy
Key words and phrases: Differential inequalities, global existence, stability, parabolic equations.
generator
of contraction
semigroups,
uniqueness,
In the 1950s-and 196Os, the technique of differential inequalities led to a great improvement in the fundamental theory of ordinary differential equations, cf. Lakshmikantham and Leela [l] and Szarski [2]. In this paper we develop the same theory for mild solutions to nonlinear evolution equations u’ = Au + f(t, u),
44
= %I,
(El
where A is the generator of a contraction semigroup and f is continuous. The main achievement is that problems related to uniqueness, global existence and asymptotic behaviour of mild solutions to (E) can be handled by comparing (E) with a scalar ordinary differential equation v’ = u(t, u) associated to (E). Applications
to parabolic equations are given.
0. STANDING
ASSUMPTIONS
AND
NOTATIONS
We shall denote by: l
X a Banach space;
. [*,*I a semi-inner product on X according to Lumer and Philips [3]; a A a generator of a contraction semigroup; l T(t), t 2 0, the contraction semigroup generated by A; l U an open subset of X; l f a continuous mapping [a, 6[ x U + X, a 2 0; 0 0 a continuous function [a, 6[ x R+ - Y?; l D- the Dini derivative o-v(t) = lim suph,,((v(t + h) - v(t))/h). 1. THE
FUNDAMENTAL
The whole paper is based on the following LEMMA.
LEMMA
lemma.
Let IIuOll -< t+, and let U, be the maximal solution on [a, b[ of u’ = w(f, u),
u(a) = ug.
If
V(f, u),ul 5 dt, lI4)ll4 1063
(all t, 4,
G. VIDOSSICH
1064
then every mild solution u of (E) satisfies Il4f)ll
5 Qo(t)
for all t in the domain of U. Proof. For every f > a and every small h > 0, we have on the one hand ” I
‘I-h
u(t) = T(h)T(t
- h - a)~,, + T(h)
T(t - h - s)f(s, u(s)) ds +
I .u
3t-h
T(t - s)f(s, u(s)) ds
II
= T(h)u(r - h) +
~I 1-h
(1)
T(t - s)f(s, u(s)) ds
and, on the other hand T(t - s)f(s, u(s)) ds - f(t, u(t)) IT(t - s)f(s, u(s)) - f(t, u(t))) d.s + ; I’; T(t - s)f(t, u(t)) ds .I h 1 8I T(r ~ dlfb, h , r-h
=II-1
u(s)) - f(t, u(t))) ds + Lh
[T(t - s)f(t, u(t)) - f(t, u(t))) ds (I
lifts, U(S)) - f(t, u(t))11 dr;
5 r-hrsat
ITU
From the continuity
-
s)f(r,
w))
~
fU,
uU))l
of f(. , u( *)) we get #f lim L llfh hi-0 h II r-h
while from the continuity
ds
w) -
fct,
w)ll b = 0,
of T and T(0) = I we get
,I I
ITU - s)f(t, u(t)) - f(t, u(t))1 ds = 0.
, r-h
Therefore,
from the above inequality
‘I I
we deduce T(r - s)f(s, U(S)) ds = f(t, 2.4)).
, r-h
Combining
with (1) we obtain u(r) = T(h)u(t
- h) + hf(r, u(r)) + he(t, h),
where lim e(t, h) = 0. h10
(2)
Evolution
Multiplying
1065
equations
(2) by u(t) in the semi-inner product we obtain
IlW)l12 = [T(h)u(t - h), u(t)1 + hLf(t, u(f)), u(t)1 + h[EG,h), u(f)1
5 llw - ~)ll/lW)ll + hot?, Ilw)ll)llw)ll + mf, ~)llIl~(~)ll by virtue of II~(t)ll
5 1. Dividing
by IJu(~)ll we get
Ilu(t - Ilu(t - h,ll h
5 Ml,
llW)li)
+ II&, h)ll,
5 o(t, llu(t)ll)
+ IlO, h)tl.
i.e.
IlW - h)ll - IlW)il -h
Setting k = -h and taking lim sup,,, we obtain
W4f)ll
5 Mr, II4f)ll)
for all t > a. Then the conclusion follows from a well-known theorem on differential inequalities, namely [l, theorem 1.4.11 or [4, theorem 4.1, p. 261. H 2. UNIQUENESS
AND
LOCAL
EXISTENCE
A simple example of functions f satisfying the assumptions of the theorem below is
f = fl + fi7 where f, is Lipschitz and -f2 is accretive (= monotone when X is a Hilbert space). 1. If the following conditions (i) [f(t, x) - f(t, y), x - y] I w(t, IIx - yll)llx - yll for all a 5 t < b and x, y E U; (ii) v = 0 is the only solution to c” = w(r, v), u(a) = 0 are satisfied, then (E) has a unique local mild solution. THEOREM
Proof. Uniqueness. Assume there are two different mild solutions x and y to (E). We have to show that x(c) = y(c) for every point c in the common domain of x and y. Fix such a c. Let a = sup@ 5 t 5 c / x = v in [a, t]]. We have X(Q) = y(a) by continuity. be so small that
llzll 5 E
Assume cy < c and argue for a contradiction.
and
a I f I a! + 6 *
llw - x(cY)l~ i & * 11x(r) - x(a)ll I F,
z + w E u;
IIY@)- xb)II 5 E.
Now define g: [CY,CY+ 61 x (Z I I]z/] I E] + X by &?(f,z) = f(t,Y(f)
+ z) - f(t,y(t)).
From (i) we derive
[s(t, a, zl = UIt, Y(f) + z) - “of, Y(f)), zl 5 4t, llzllMl.
Let E, 6 > 0
1066
G. VIDOSSICH
The mapping w = x - Y is a solution to 1’ = AZ + g(t, z),
z(a) = 0.
(3)
Applying the lemma in Section 1 to (3) we obtain I(w(t)jl I 0 for (Y 5 t 5 (Y + 6. This means x = y in [CY,CY+ 61 and we have contradicted the definition of CY. Existence. By a lemma in Lasota and Yorke [5], there is a sequence of locally Lipschitz mappings f,,: [a, b[ x U + X such that lim,f, = f uniformly. The continuity off and the uniform convergence guarantee the existence of 6, > 0 and of E > 0 such that all f, as well as f are uniformly bounded in [a, a + S,] x {z 1 11~- ~“11I a]. We take E so small that and
IIX - %ll 5 &
llyll P & *
x + y E u.
Then from standard methods (i.e. at first we obtain a local solution by applying the contraction fixed point theorem and next we extend it up to a + 6 by the argument used in the proof of theorem 2 below) we derive the existence of 0 < 6 5 6, independent of n such that all Cauchy problems
u' = Au + f,(r, u),
u(a) = ug
have a unique mild solution U, defined on [a, a + 61 with Ilu,(t) - uOll 5 E. Now we fix n and m, and define g”,,,,: [a, a + 61 x lz I lIzI/ I E] + X by
&,,,,(f~4 = fr?(t, U,,,(f) + z) - fm(tt &n(t)). We have
kL,,A~?ZL zl = [f&7 4,,(t) + z) - f,K &A~)), zl
5 IIf&> 4,,(r) + 2) - f(t, &n(f) + z)llllzll + [f(r, &n(t) + 4 - f(t, &n(t)), zl
+ IIfUT&n(t)) - fm(t, 4n(t))llllzll 5
E”
,m + [f(f, u,(t) + z) - f(?, %Jf)), zl,
where lim a,,,,, = 0 Pl.!?? since llz]l I E and lim, fk = f uniformly.
Combining
(4) with (i) we obtain
k,.&~ -3,zl 5 E”,,, + Mf> lIzll)llzll~ By (ii), by (4) and by the convergence theorem at p. 14 of [4], for n and m large enough the Cauchy problems VI = En,,,* + o(t, u),
u(a) = 0
have a maximal solution cl,,, on [a, a + 61 and (5) uniformly.
Since w = u, - u, is a solution to w’ = Aw + g,,,,(t, w), w(u) = 0,
Evolution
1067
equations
we are in position to apply the lemma in Section 1 and derive that II%(t) - 4n(t)ll
(a I t I a + 6).
5 4l,,r,m
Combining with (5) we obtain that (u,), is a Cauchy sequence in C([a, a + 61, X) and, hence, converges to a u E C([a, a + 61, X). Taking limits in sI u,(t) = T(t - a)u, + vt - s)f& 4k9) ds ,I 0 we get ,I u(t) = 7-(t - a)u, + T(t - s)f(s, 44) d.s I ,u for a 5 t 5 CI + 6.
n
3. GLOBAL
EXISTENCE
AND
ASYMPTOTIC
BEHAVIOUR
We say that (E) satisfies the local existence condition if for every u0 E U, the Cauchy problem (E) has at least one local mild solution. Simple examples are the following: f satisfies the assumptions of theorem 1 or, alternatively, the semigroup T(t) generated by A is compact (cf. Pazy [6]). and let f be bounded on bounded sets. If the following conditions: (i) [f(t, u), U] I o(t, IlulI)~lull for all t and U; (ii) u’ = w(t, v), u(a) = u0 has a maximal solution u, on [a, b[ and the closed ball B in X with center the origin and radius IIL~,I~~- is contained in U are satisfied, then for every u,, E U with llu,J I Q,, the Cauchy problem (E) has a global mild solution u in [a, b[ satisfying the estimate
THEOREM 2. Let (E) satisfy the local existence condition
llW)ll 5 &o(t)
(a i t < b)
and all mild solutions are global and satisfy the same estimate. A direct application
yields the following
corollary.
COROLLARY. If b = +oc, and f(t, 0) = 0 = o(t, 0), then conditions (i) and (ii) of theorem 2 ensure that the stability, the asymptotic stability, etc. of the null solution of the scalar ordinary differential equation
1’) = implies the stability,
w(t,
v)
the asymptotic stability, etc. of the null solution to the evolution equation
24’= Au + f(t, u). Proof of theorem 2. Since (E) satisfies the local existence condition, (E) has local mild solutions. Let u be any of them. Let I be the maximal domain of existence of U. The estimate
Ilu(t 5 kc(t)
for t E 1
G. VIDOSSICH
1068
follows from the lemma in Section 1. It remains to show that u is global, i.e. I = [a, b[. Assume that c = sup I is less than b and argue for a contradiction. By the uniform continuity of T on [a, c] and by the boundedness off on [a, c] x B, B being the closed ball with center the origin and radius llu,llLm, it follows from
u(t,) - u(t,) = IT(t, ~ a) - T(r, - a)& +
IfI
II f2
nt, - 4f(s, W) ds
j f2 + I I T(t, - s) - T(f, ~ s)lf(s, 4s)) ds ,u
(f2 5 t,)
that lim u(t) rtc exists by vitue of the Cauchy criterion. Then we extend u to the right of c by solving locally a Cauchy problem and we contradict the maximality of I. We conclude that c = 6. n 4. APPLICATIONS
TO PARABOLIC
EQUATIONS
In this section we apply the results of Sections 2 and 3 to the case of parabolic equations whose nonlinear term satisfies one-sided conditions. Consider the following problem u, = Au + g(t, X, U)
Bu(r;) = 0
(P)
u(a, .I = ul), where a L 0, g: [a, + co[ x fi x R + R is continuous, R G IR” is a bounded domain with smooth boundary and B is a boundary operator satisfying the two conditions (B,), (B2) below. Let X be the Banach space C(a) of continuous functions d + R with the sup norm 11.llLm. Define
D(A)= (u E C2(k?lBu = 0) and A The (B,) (B,)
= A on D(A). assumptions on the boundary operator are the following: A is a closed operator; for every u E D(A), lu(.)l has a least one point of maximum X, such that Au(x,,) * u(x,) 2 0.
Therefore, B can stand for the Dirichlet m = 1 and Q = ]a, /3[, for a Sturm-Liouville
problem, or for a periodic condition condition
or, in the case
ol,u(t, 01) - P,u,U, a) = 0 CY,U(f,8) + PzU,(f, P) = 0 with Cyi,pi 2 0 and CY,+ j3, # 0 for i = 1, 2. In order to apply the previous theorems, we define on X an ad hoc semi-inner product in the following way. For each u E x let x,, be a point of maximum for lu(*)l such that (B2) holds
Evolution
1069
equations
whenever u E D(A). We set for U, v E X
[u, 01 = 4X”M-%). It is readily seen that this defines a semi-inner product on X with [Au, u] 5 0 Therefore, A is the generator of a contraction
(u E WA)). semigroup. Define
f: [a, +w[ x x - x by f(f, u)(x) = at, x, 4x)). After these preliminaries, it is easily seen that theorem 1 implies theorem 3, while theorem 2 and its corollary imply theorem 4. THEOREM
3.If g(f, x, u) - g(f, x, w) 5 w(f, u - w)
for 242 w,
and if the only solution to u’ = o(t, v), u(a) = 0 is ~1= 0, then (P) has a unique local mild solution, THEOREM
4.
If
(all t, x, U) k(f,X, u), ul 5 dr, l4)lul then (a) problem (P) has a global mild solution whenever U’ = o(t, u), v(a) = IIu~II~- does; (b) for u0 = 0 and g(t, x, 0) E 0 = o(t, 0), the null solution to (P) is stable, asymptotically stable, etc. whenever the null solution to u’ = o(l, v) has the same property. REFERENCES I. LAKSHMIKANTHAM V. & LEELA S., Diffeerenf& and lnfegral Inequafifies, Vol. I. Academic Press, New York (1969). 2. SZARSKI J., Differential Inequalities. PWN, Warsaw (1965). 3. LUMER G. & PHILLIPS R. S., Dissipative operators in a Banach space, Pacif. J. Math. 11, 679-698 (1961). 4. HARTMAN P., Ordinary Differenrial Equations. Wiley, New York (1964). 5. LASOTA A. & YORKE J. A., The generic property of existence of solutions of differential equations in Banach spaces, J. diff. Eqns 13, 1-12 (1973). 6. PAZY A., A class of semi-linear equations of evolution, Israel J. Math. 20, 23-36 (1975).