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k’onhnearAnal,wrr, Theory. Me/hods&App,rcalrom, “ol.25.Nor 9-10, pp. 105,-,054,,995 Copyright 0 1995 Elsevier Science Ltd

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0362-546X(95)00099-2

PARABOLIC INEQUALITIES IN ORDERED TOPOLOGICAL VECTOR SPACES

ALICE SIMON? and PETER VOLKMANNS t Departement de MathCmatiques, Universitk d’orlkans, 45067 Orleans Cedex 2, France; and

$ Mathematisches Institut I, UniversitPt Karlsruhe, 76128 Karlsruhe, Germany

Key words and phrases: Differential inequalities, parabolic differential equations in topological vector spaces.

1. INTRODUCTION AND NOTATIONS

Let R denote the reals, and let D be a subset of RN x R (N L 1). A well-known theorem of Nagumo [l] gives sufficient conditions such that for functions U, w: D + R the inequality

F(x, t, v, 0x9 v,, u,) -=c F(x, t, w, wx, wxx, w,) ((-6 t) E 0 (1)

(together with some condition “on the boundary of D”) implies

4x, t) < w(x, f) ((x, t) E D).

On the left-hand side of (1) (and similarly for the right-hand side) we mean by U, u,, v,,, u, the values of the function u and of their derivatives at the point (x, t). The first order derivative u, is considered as an element of R", and the second order derivative u, as a symmetric N x N matrix. By u, we only mean the left-sided derivative of u with respect to the variable C. Independently of Nagumo [ 11, a version of this theorem had been rediscovered by Westphal [2]; afterwards Nagumo and Simoda [3] gave a more general result. A discussion of the subject can be found in Walter’s book [4].

In this note we shall present a version of Nagumo’s theorem for functions U, w taking values in an ordered topological vector space. So let E be a real topological vector space, which is assumed to be Hausdorff, denote by 8 the zero of the space and by E* its topological dual. Let K be a closed wedge in E, i.e. K is a closed set containing 0 and such that

aEK,bEK,ArO * Aa + bEK.

For a, b E E we define the relations a 5 b and a 4 b by

a%b u b-aEK, a@b e b-aEIntK,

respectively, Int K denoting the interior of K. The relation I is reflexive and transitive. Let

be the set of symmetric N x N-matrices with entries in the space E. The set S”(R) has the same meaning with E replaced by R, and for (ajk), (/3,,) E S”(R) we put

&k) 5 V/k) e 1 W,k - ql,)5,L 20 ((t,, . . ., t-,v) E RN). (2) /.k = 1

1051

1052 A. SIMON and P. VOLKMANN

For A4 = (Qjk) E S,“(E) and p E E* we define p(M) E S”(R) by

d”) = (d"jk));?jk = 1 and, finally, we denote

K* = {v E E* ) v(a) 2 0 (a E K)].

2. THE COMPARISON THEOREM

Let D be a subset of RY x R, and denote its elements by (x, t) (where x is in RN and t is real). Suppose that u: D + E has second order derivative u, and left-sided derivative u, at every point (x, t) E II. Observe that

U,,(X, t) = (a2U/axj aX,),;l(, = 1 E S”(E)

and u,(x, t) E E. The existence of u, also implies the existence of the first order derivative u,(x, t) = (au/&~, , . , au/a~,~) E E”. From the assumption on 1.4 it follows that to every (x0, to) E D there corresponds an E > 0, such that

i(x, to) I x E R”, /IX - X,11 < El u ((X0, f) 1 t, - E < t 5 to) 5 D, (3)

where 11 . )/ is some fixed norm on R.v.

THEOREM. Consider D G R” x R, and let t,‘, w: D + E be continuous functions having second order derivatives uI1, w,, and left-sided derivatives u,, w, at every (x, t) E D. Suppose

F: D x E x E,’ x S”(E) x E -+ E (4)

to be a function giving

F‘(x, f, u, u, , ~1, t L’, 1 + FM, t, ~‘7 w, , w,, , w,) (6, t) E 0 6)

and satisfying the following hypothesis: (HI if u, P ,r..-,PN,9,U,IS,,... , ph7, 4 E E, M, n;i E SN(E), and p E K* are such that

u 5 0, P(U) = (D(U), CP(P~) = P(P,j) (j = 1, . . ., ,Y), V(Q) % p(q), P(M) 5 I, then (for all 6, f) E D)

W(x, t, u,P,, . . . . Phi 1 M, 4)) 2 y?(F(x, t, a, P, 9 . . ., PN, M, 4)).

Finally, suppose: (B) if (x,, I,) E D is an infinite sequence without accumulation point in D and satisfying

t, > t, > .., then there is an index n,, such that

4x, 7 t,) e w(x, 7 f,) (n L no).

Under these assumptions we get

u(x, t) < w(x, t) ((x, t) E D).

In the classical case E = R, K = [0, oo), (4) means

(6)

F: D x R x R.v x S”(R) x R -+ R,

and the interpretation of (H) is as follows: if u,p , , . . . , pN, q, 4 E R and M, A? E S”(R) are such that

4 5 4, M<M

Ordered topological vector spaces 1053

(M I II? in the sense of (2)), then

F(x, t, u,p,, . .., P,v,M, 9) 2 W, f, u,P,, . . ..PN.M, 4).

In other words: the function

F(x,t,u,p,,...,4,,M,q)

is (weakly) increasing with respect to the real variable q and (weakly) decreasing with respect to the variable M from S”(R). An interpretation of (H) in the semilinear case will be given later.

3. PROOF OF THE THEOREM

(a) Assume (6) to be false. Then by (B) and the continuity of u, w: D + E, the existence of

t,, = mintt I w(x, r) - u(x, t) $ Int K for som (x, t) E D)

easily follows. So there is some +, E R” such that (x0, to) E D,

w(x,, , to) - tCq,, to) $ Int K, (7) and

u(x, t) 4 w(x, f) ((x, t) E D, t < to). (8)

Take E > 0, such that (3) holds. The continuity of L’, w and (8) imply

W(, 3 f 1 5 w(x,, 9 f ) (to - & < t 5 to), (9) in particular,

L’(X() , 1,)) 5 w(x,, , t,,). (10)

Since every point from ((x, t,,) 1 x E R’, ,/,I- ~ .x,,lI < &I lies in D, the same reasoning gives

Nx, I,,) 5 4x, lo) (11-Y - X”ll < &I. (11)

(b) Owing to (7), the Hahn-Banach separation theorem provides a nontrivial ~1 E K* such that &w(xO, to) - u(x,, , I,,)) 5 0. Inequality (10) implies

We define

d: IQ, f,) I ilx - so I < &I u I(x,,, t) i t,, ~ & < t I to) + R

(9), (12) imply

hence,

(1 l), (12) imply d, b-0 1 10) = cp( M”r (.\‘,, 1 to) - L’, lx,, , to)) 5 0.

4x,, , I,,) = min I I \,,‘J<? 4*~, lo),

(12)

(13)

1054 A. SIMON and P. VOLKMANN

hence,

= 0 (j = 1 , . . ..w. (14)

4A% 9 fo) = lo(wx*(xo, to) - L’,(Xo, to)) 2 0, i.e.

rp(~,(% 7 to)) 5 cp(W,,(% 3 hd) (15)

(in (15) I is taken in the sense of (2)). (c) With cp as above, (IO), (12)-(15) show that we can apply hypothesis (H) for u = v(x,, to),

u = w(XO, to), Pj = (a/aXj)U(X,, to)7 PJ = (a’axj)w(%, to) (j = I, . . ..N). 4 = u[(xo, to),

4 = w,(%, GA Aff = ur,(xOr to), M = w,,(x,, to), to get

W(% Y to 5 07 v,, L’,, 7 u,)) 2 qmx, 1 t, 1 w, w,, w,, , w,)). (16)

On the other hand, v? E K* being nontrivial, (16) contradicts (5), and the theorem is proved.

4. THE SEMILINEAR CASE

The background of our comparison theorem is the general parabolic equation

m, f> u, u, I u,, , u, 1 = Q

for functions U: D + E. A special case is the semilinear equation N a2u ,V u, =

./Sk = I

ax,ax, + c bJtx& +f(x,t,U),

J=l J

(17)

where 0(/k, fiJ : D --t R, ajk = c+, (j, k = I, . . , N). The function (4) associated to (17) is

F(x, t, u,P~, . . . . P,V, @,k), ‘d = ‘3 - i ajk(X, fbjk - f PjCX, f)Pj - f(x, t, U) (18) j,k= I k=l

(k t) E D, u E E, (P , , . . . , P,,!) E E”, (<jk) E sJV(E), 4 E E). Assume IlOw:

(I) the matrices (aJk(x, t).;lik=, are positive semi-definite (i.e. (oljk(x, t)) L 0 in S”(R)); (II) the function f(x, t, u): D x E + E is quasimonotone increasing with respect to the

variable u (cf. [5]), i.e. (x, t) E D, U, U E E, u 5 U, ~7 E K*, q(u) = &ii) imply &f(x, t, u)) I cu(x, t, a). Then it is easily shown that hypothesis (H) is satisfied for the function given by (18).

REFERENCES

1. NAGUMO M., Note in Kansti-horersiki, 15 (1939). (In .lapanese.) 2. WESTPHAL H., Zur Abschatzung der Lbsungen nichtlinearer parabolischer differentialgleichungen, Math. Z. 51,

690-695 (1949). 3. NAGUMO M. & SIMODPI S., Note sur l’inigalite differentielle concernant les equations du type parabolique, Proc.

Japan Acad. 27, 536-539 (1951). 4. WALTER W., Differential and Inregral Inequalitres. Springer, Berlin (1970). (German original (1964).) 5. VOLKMANN P., Gewohnliche differentialungleichungen mit quasimonoton wachsenden Funktionen in topo-

logischen Vektrorraumen, Math. Z. 127, 157-164 (1972).