Singular perturbations and nonlinear parabolic boundary value problems

JOURNAL

OF MATHEMATICAL

Nonlinear

ANALYSIS

AND

APPLICATIONS

17, 248-261 (1967)

Singular Perturbations and Parabolic Boundary Value Problems BUI AN TON*

University

of Montreal,

Department

of Mathematics,

Canado

Submitted by Norman Lewimon

The purpose of this paper is to prove the solvability of a mixed initial boundary value problem for semilinear parabolic equations of the form

on 52 x [0, T]; with A,(x, t, u ,..., Pu) having at most linear growth, (E, a small positive number), and to show that the weak limit in L2(1, Wm~2(Q)) as E -+ 0, of IL, is a solution of a mixed initial-boundary value problem for the nonlinear parabolic equation

au

z

+

C DaA,(x, I-lSm

t, u ,..., D%)

=f[x,

t)

on

J-2 x [O, T].

Strongly nonlinear parabolic boundary value problems have been considered by Browder [l], Lions [2], and Visik [3], using different methods.

SECTION 1 Let Q be an open subset of Rn with a sufficiently smooth boundary. The points of Q will be denoted by x = (x1 ,..., x,). Set Dj = i-la/ax,; j = l,..., n. For each n-tuple 01= (aI ,..., a,) of nonnegative integers, we write

Da = fi D,q

with

~+=~fY,.

j=l *Partially supported matical Congress.

by the Summer

j-l

Research Institute

248

of the Canadian

Mathe-

NONLINEAR

PARABOLIC BOUNDARY VALUE PROBLEMS

249

The points of El will be denoted by t and differentiation in t by a/at. Let I = [0, T] be a finite, closed interval of El. Let r be an integer, r > 1. By functions, we mean r-vector functions u = (ui ,..,, u,) where each uk is a complex-valued function on D or on Q x I. Thus D-u = (Dw, ,..., D=uJ

and

au

au,

z = at (

af4

,..., z r . )

Let u, v be two elements of La(Q), then we denote by (u, v) their inner product:

(u,v>= k=li 1Q %(x)%(x) dx. We consider systems of partial differential operators on Szwith coefficients depending on t in I, of the following forms: A(t) u =

C

D=A,(x, t, u ,..., D%),

where for each CX,A,(x, t, u ,..., Dmu) is an r-vector function of (x, t) on i2 XI. A,(t) u =

WG,&, c kl.I4<~2

t> D-4

m2 > m.

(1.2)

The coe$icients u&x, t) are assumedto be in L2(Q x El). We assume the following conditions on the smoothness and on the growth of A,: ASSUMPTION (I). The functions A,(x, t, u,...,’ Dmu) are measurable in (x, t) on Q x I, continuous in (u ,..., 0%). We assume there exist a positive constant M and a function g(x, t) in L2(Q x I) such that

I A&

t, u,..., DWI


+M

c ID%1 l=l
for all (x, t) in Sz X I. Let K be a positive integer, we denote by WkJ(sZ) the space Wk*“(12) = (24: u in L2(Q), Doruin L2(Q) 1011< k} (the derivatives are taken in the sense of the theory of distributions). is a Hilbert space with the norm

IIu Ilk.2= 1,& IID”u Ilh,lt a-. and the obvious inner product.

Wkv2(Q)

TON

250

We denote by W,“*“(Q), the closure of C,m(J2), the family of infinitely differentiable functions with compact support in L?, with respect to the II . Ilk,2 norm. It is a Hilbert space. We consider the Hilbert space L2(1, lVr’“(Q)) of equivalence classes of functions u from I to it’s*” which are L2-integrable, with the norm

and the usual inner product. In a similar fashion, we define the Hilbert spaceL2(1, WpS2(Q)) and denote the norm by

Corresponding to the representation (1.1) of A(t), we define the nonlinear Dirichlet form: u(u, v; w) =

c j- (A,(x, I-lGm I

t, u ,..., D”-h,

D%), D=w) dt

for u, v, w inL2(1, Wm~2(Q)). With Assumption (I) on A,; a(u, v; w) is well defined. Similarly to the representation (1.2) of A,(t), we define the linear Dirichlet form:

for 24,v in L2(1, IJP~~~(SZ)). We denote by P(1, IV-“aJ(52)); L2(1, W-“~2(52)) the duals of J~~(~,IV~*~(Q)); L2(1, lVrs”(Q)), respectively, by (,) the pairing between L2(1, We*“) and its dual and by ((,)) the pairing between L2(1, w.a(Q)) and L2(1, W-“2J(Q)). DEFINITION 1.1. (i) Let F = {u : u in L2(I, Wr*“(Q)), u is continuous from I to Wrs2(Q), is continuously dzJKmentiable from I to L2(Q)}.

F,={u:uinF,u(O)=O};

F,={u:uinF,u(T)=O}.

NONLINEAR PARABOLIC BOUNDARY VALUE PROBLEMS

(ii)

251

Let L, be the linear mapping from F, into Lz(I, W-“@(Q)) such that

(Lou, v)= j,($- , v) & for all v in L2(I, W~~2(Q)) and for each u in F, . (iii)

Let L, be the linear mapping from F1 into L2(I, W-m12(Q)) such that (L,% v) = - j,($

, v) dt

for all v in L2(I, Wr*“(Q)) andfor each u in F1 . An integration by parts shows that L, CL,* defined adjoint. We may define:

and so Lo has a densely

DEFINITION 1.2. (i) Let L be the closure of Lo as a linear operator with domain in L2(I, Wrm’(sZ)) and range in L2(I, W-+“(Q)).

(ii) Let L, be the analogous of L with Wr*2(Q), W+‘Q+2) replaced respectively by W~*2(Q), W-ma*2(Q). DEFINITION 1.3. For a given f in L2(I, W-“2+2(Q)),u is said to be a solution of the variational Dirichlet boundary value problem of the parabolic system

g + qqu

=f(x, q,

with u(0) = 0 if (i)

u is in D(L),

(ii)

(Lu, v) + a(u, u; v) = (f, v) for all v in L2(I, Wrs2(Q)).

SECTION 2

We have the following equation

existence theorem for the semilinear parabolic

g + eA2(t) u + A(t) El = f. THEOREM 2.1. Let A(t) be a system of partial differential operators on D x I, of the form (I.Z) and satisfying Assumption (I). Suppose that the following conditions are satisfied:

252

TOE

(i) There exists a continuous function c(r) from A+ to R with c(r) -+ - cxz as r---f + co such that

(ii)

u in L2(I, Won*‘(Q)).

fm d

Re a@, u; 4 2 411 u II) II u II

Re{a(u, u; u - v) - a(u, v; u - v)} > Ofor all u, v inL2(I, WFe2(Q)).

Let A,(t) be a system of partial d@rential operators on Q x I of the form (1.2) and such that for all

Re a2(u, 4 2 C III u l/l2

u

in

L2(I, W?*“(Q)),

where C is a positive constant. Then for every f in L2(I, W-mv2(Q)) and for each E, 0 < E < 1; there exists a solution u, of the variational Dirichlet boundary value problem of the parabolic equation 2

+ aA,

u, + A(t) u, = f

on

Qxl

with u,(O) = 0. Moreover, c II I 11,III + II ~1,II + IIL2uc II~mv--w~

G M

M is a constant independent of E, u, . Following Browder [l], we define a global boundary value problem on the infinite cylinder Q x El and use it as a technical device in the proof of Theorem 2.1. LetF# = {u : u in L2(E1, Wp*“(Q)); u is continuous fromElto WoJ*“(Q), is continuously differentiable from El to La(Q) and has compact support in El}. DEFINITION 2.1. Let 9,, L2(E1, W-ap2(Q)) such that

be the linear

mapping from

F#

into

((Lou,VI>= lE1($- , v) dt for all v in L2(E1, WOm,S2(Q)) andfbr 9, is closable.

each u in F#.

DEFINITION 2.2. (i) Let 9 be the closure of Y0 as a linear operator with domain in L2(E1, W?*“(Q)) and range in L2(E1, W-‘Q*~(&?)).

NONLINEAR PARABOLIC BOUNDARY VALUE PROBLEMS

253

(ii) u, is said to be a solution of the global variational Dirichlet boundary value problem of

%

+ d,(t)

u, =g

on

Q x El;

g

in

L2(E1, W-m,~2(Q))

for all v in L2(E1, Wp’(sZ)) LEMMA 2.1.

(i)

9 = - Yip*.

(ii) Let g, be in L2(I, W-~S,~(Q)) and g be in L2(E1, WASTE) setting g = g, on I, g = 0 outside of I.

If u, is a solution

obtained by

of the global variational Dirikhlet boundary value problem

2

+ d,(t)

u, = g

on

Sz x El,

then the restriction of II, to I is a solution of the variational Dirichlet problem of 2

+ e&(t) 24,= g,

PROOF. Cf. Browder

on

QXI

with

[l] (L emma 3.1, p. 350; Proposition

PROOF OF THEOREM 2.1. (i) well-known argument, we have

(f, v) = (v, v))

Extend

for all

f

UC(O)= 0. 3.1, p. 353).

to be zero outside

v

of I. By a

L2(E1, W?*‘(Q));

in

f is a uniquely determined element of L2(E1, W-+~+~(f2)). Let w be an element of L2(I, Wan”) and h be a real number with O
a2(u,4 = W2u, 4)

for all

and S,u is in L2(E1, W-ma*2(Q)).

v

in

L2(E1, W?,“(Q))

254

TON

(ii) We show that there exists a unique solution of the variational Dirichlet boundary problem of the parabolic system on

u, = f’ - S(hw)

Q x El;

O
It sufhces to show that the linear mapping 9 + ES, from L2(Ls1,We*“) into L2(E1, W-“~~2(Q)) is I-I and onto. Since 9 = - 9’*, Re ((Zu, u)) = 0. It follows from the hypotheses of the theorem that: & 111u, I112< Re (((3

+ cS2) u, u)) = E Re +(u, U) for all

u in

D(p).

so,

Hence, (9 + ES,) is 1-I and has a closed range for each E. Since 9 = - 9*, the null space of (9 + ES,)* is trivial. But the range of (9 + ES,) is equal to the annihilator of the null space of (9 + es,)*, therefore (3 + ES,) is onto. (iii)

Define the nonlinear mapping Y(X) from D(L,) into itself as follows: O
9-(h) w = 24,

w in

D(L,),

where U, is the restriction to I = [0, T] of the unique solution of the variational Dirichlet boundary value problem for u, = f - S(hw)

on

J-2 x El.

From the preceding part, the mapping Y(h) is well defined. D(L,) is dense in L2(I, Wp*s(Q)), we consider it as a Banach space B with the norm:

IIu lb = III u III + II&P IIL~(r.w-~%?)) . To prove the theorem, it suffices to show that Y(1) has a fixed point. It will be a consequence of the following propositions. PROPOSITION2.1. [0, I] X D(L,)

PROOF. (i)

n4

is

a

completely

continuous

mapping

from

into D(L,).

?(A) is continuous. Indeed, let A, -+ A, A,, h in [0, I] and

NONLINEAR

PARABOLIC

BOUNDARY

VALUE

PROBLEMS

255

let w,, + w in the B-norm. We shall show that u, = F(h,) w, converges to u = F(A) w in the B-norm. By definition, we have uw42 - 4,4) + 4% for all w in L2(1, W,mZ*“(Q)).

- u, 4 = ((fwnw,)

- Ww), 4

In particular, it is true for z1= u, - u. Since Re ((Lsw, w)) > 0; we get E Re a,(~, - u, u, - u) < Re ] a(h,w, , &w,; u,, - U) - a(hw, Aw; Us - U) 1. so,

On the other hand, since &w,, + hw in the B-norm, it follows from Lemma 3.2 of [4] that

ECIII %a- 24III +o

as

n-+ co;

hence,

III %I -ulll-+O

as

n-+ co.

It remains to show that

IIwTI - 4 IILwF%2)) - 0

as

n-+ co.

We have

- A,(-, t, h -9X gmw) llh) 4) + M III %I--u Ill/ III fI III for all w inP(1,

We*“),

IIL(u, - 4 lI~wv-vn~~ G IM III us - u III

256

TON

Therefore, IIL2(un - n) ~ILvz.w-~~~)) - 0

n--t co.

as

(ii) Y(h) is compact. Let {run> be such that I/ w, lIB < M, where M is a constant independent of n. Since m2 > m and Q has a smooth boundary, the injection mapping from lP@(sZ) into W”@(Q) is compact (Sobolev imbedding theorem). From a theorem1 of Aubin [5], and the hypotheses that II eo, i/B G M, we get wnj +

w

in

L’(I, W~*2(Q))

as

nj--+ co.

The first part of the proof gives II unr - u llB--f 0 as nj + co. PROPOSITION

2.2. I - F(0) is a homeomorphismofD(L,). If u, = F(h) u,

then,

E III II, III +

IIL2u< ll~vz.w-~w~

< M,

whereMisindependentof~,h~O
3 u,)) + ea,(u,,

UC>+ +u,

, h;

4

= (f, 4.

Since Re ((L,u, , u,)) 2 0, we get

ECIII u, Ill2 + II4 IIco IIu, II) d II4 II If IIL~~z.w-~%)) ’ so, 4 IIus II) B Ilf llL2~z.Fi-%?)) * On the other hand, by hypothesis c(r) --+ co as Y-+ co, therefore, there exists a constant M, independent of ;\, E such that

h IIu, II < M. 1 More precisely, we have {wn*} in D(L,,.J; w,,~ + w, inL2(I, inL2(I,

WO~***(Q),L,-,2w,k +L,w,

W-m~*a(12)) as I2 -b co. Since I/ w, 11~< M, we get

From [5], there exists a Cauchy sequence {wt,> in L*(l, follows easily.

W”+l*z(Q)).

The conclusion

NONLINEAR PARABOLIC BOUNDARY VALUE PROBLEMS

257

We have

ECIII fh III G IlfllLvz.I+J-@(n)) + I c@IIUCII) I 7 so We also have I ((L2uc 3fJNI ~NfIlL~ca(zN--%2))+ M + IIUCII + III 11,III>III 7JIII * Hence

PROOF OF THEOREM 2.1 (continued). The nonlinear mapping F(h) satisfies the conditions of the Leray Schauder theorem [6], so Y(1) has a fixed point, i.e., F(1) u, = u, . Moreover from the proof of Proposition 2.2, we get

E III us III + IIu, II + IIbe ll~uv--~w

Q.E.D.

G M.

REMARK. As observed by Browder, the uniform continuity condition of the Leray-Schauder theorem is not necessary. PROPOSITION 2.3.

If u E D(L,) then u E D(L), and

((L,u, v)) = (Lu, w)

v in L2(1, Wp2(i2)).

fur all

PROOF. Let u be in D(L,). Since L, is the closure of L,, as an operator with domain in L2(I, FVpV2(52))and range in L2(I, W-m~J(S2)) there exists {un} in D(L,,) with u, -+ u in L2(1, W,m1*2(f2))and L,u, -+L2u in Lz(I, W-maJ(sZ)). By definition, we have

for all

L2(I, Wp*2(J2)).

in

21

Let J be the injection mapping of L2(I, W-“@(@) into L2(I, W-%92(Q)) with ((JLw, w)) = (Lw, v) for all w in L2(I, W~*2(12)) and w in D(L). We get UL2%

9n)) = (LUG, w) = ((JLun , v))

for all

0

in

L2(1, W~*2(i2)),

80

JLu,, = L,u,, -+L,u

in

L2(I, WmqQ))

as

n-co.

258

TON

Since J is bounded, JL is closed in L’(I, W-V~z92(Q)). Therefore u is in D(L) and

((Jh Q>)=

((L,u,

for all

4) = Vu, v)

L”(I, W?*“(Q)).

in

V

THEOREM 2.2. Let A(t) be a system of partial d@eerential operators on Q x I, of theform (1.1) and satisfying Assumption (I). Suppose: (i) There exists a continuous function C(Y)from Rf to R with c(r) ----f$ co as y---f + 03 such that

Rea(u,u;u)>c(ljuIj)IIuI/

forall

u

in

L2(I, wy(sz)).

Re (a(u, u; u - v) - a(u, v; u - v)} > Ofor all u, v inL2(I, Wr*“(Q)). Then for every f in L2(I, W-m*2(Q)), there exists a solution of the variational Dirichlet boundary value problem of (ii)

$+A(t)u=f

ou

QXI

with u(O) = 0. It is the weak limit in L2(I, W:*“(Q)), as E+ 0 of a solution u, of Theorem 2.1. The solution is unique if (ii) is replaced by : (ii)’

Re(a(u,

u; u - v) - a(v, v; u - v)> > Ofor allu, v inL2(I, W:*“(Q))

PROOF. First, we note that there exist linear elliptic

differential

operators

AZ(t), of the form (1.2) satisfying the hypotheses of Theorem 2.1. (i)

Let uE be a solution of Theorem

E III u, III + IIu, II + IIL,u,

2.1, then we have

Il~~,w--m~.sm d M.

M is a constant independent

of l . It follows from the weak compactness of the unit ball in a Hilbert space that there exists a subsequence, which we may assume to be the original one such that u, + u weakly in L2(I, W?*“(Q)), EU, + 0 weakly in L2(I, W,“tp*“(Q)) as E -+ 0, and L2u, -+g weakly in L2(I, W-m~~2(sZ))as E + 0. (ii) We show that u is in D(L). From Proposition 2.3, u, E D(L). Since u, + u weakly in L2(I, W2*2(sZ)) as E + 0 and JLis a closed mapping from L2(I, Wp*“(Q)) into L2(I, W+s~~(1;2)), hence, weakly closed, we get L,u, = JLu, -+ JLu = g as Q--+ 0. (iii) Let v be an element of D(L,) and let vn = jn*cn”, where j,(x) is a sequence of C,“(Rn) functions, of radius 1/2n, converging to the Dirac delta measure as n -+ co; en(x) is a smooth function on Q, vanishing in a l/n-

NONLINEAR

neighborhood of nuous from I to hence belongs to By hypothesis, Re {((L,u, , 4)

PARABOLIC

BOUNDARY

VALUE

259

PROBLEMS

the boundary of Q. Then ~Jx, 0) = 0, where v,, is contiWe*“, continuously differentiable from I to L2(Q) and D(L,,). the following expression is non-negative:

+ l a(%

, 113+ 4% 9UC;4 +

-

3us>- ~a,(% P%z) -

Ea,(%

79%))+ l 2h

((~2%2

9%a -

(CL,%

((L2Pn

, Pn) P4)

- a@,, u,; %a) - 4% , p”n;u, - T&J). Taking into account the definition of u, , we obtain Re{(f, uc - ~4 + (W2~n , vn - uJ) + EU,(S y ~2 - ~a2h ,uJ -- a@,, 7%;u, - %a,>2 0. By a standard argument, we may write

where S(u, , p),J is in La(1, W+@(Q)). It follows from Proposition 2.3 that Re ((f, u, - s) + (Ls , pa - u.) + cu2(pn , p)J-

EU,(S , u,)

- (S(%9%A%- w,>2 0. Since u, + u weakly inL2(1, W~*2(Q)) and EU+ 0 weakly inL2(1, Wp*‘(Q) as E+ 0, we get (f, ur - %J - (h u - 4; Ea,(%

94 + 0;

@?b >%a- 4 + WPn , 9% - 4; Ea,(Pn

94 -+0

as E+ 0.

The injection mapping of Wp*2(Q) into Wr-112(9) is compact (Sobolev imbedding theorem). On the other hand, from Theorem 2.1; we have l

/IL~(z.w-w)< M.

III % III + II% II + IIL,u,

It follows from a theorem of Aubin [5] that u, + u in L2(1, W~-1*2(Q)) as E+ 0. From Lemma 3.2 of [4] ; we get S(u, ,A

+ S(u, R>

in

L2(1, W-yQ)).

Therefore, Re 6%

P),J + LB, -h

ya - 4 2 0.

(iv) Let 7t -+ co; then by a standard argument we have v,, + v in L2(I, We*“); Lv,, --+ Lv weakly in L2(1, W-*2(Q)) v is in D(L,).

260

TON

Taking into account Lemma 3.2 of [4] again, we obtain Re(S(u,u)+Lv--f,v---)30

for all

v

in

WCJ.

for all

0

in

D(L)-

By a similar argument, we get Re(S(u,u)

+Lv

-f,v

-u)

30

On the other hand, the operator TV = Lv + S(u, a) from D(L) into L2(I, W-m>2(Q)), is continuous from lines in D(L) into the weak topology of L2(I, W-@(Q)). A proof by contradiction, e.g., as in Lemma 2.1 of [l], then gives Tu = S(u, u) + Lu = f. (v) It remains to show that under certain conditions on a(u, v; w) the solution is unique. The proof is trivial, we will not reproduce. The theorem is proved. Let I’ be a given closed subspace of Wm~2(Q)with wp”(Q)

c v c wm*2(Q).

With certain hypotheses on V, which we shall state below; we may prove the existence of V-variational solution of the parabolic system g +

c DaAA,(x, t, u,..., D%) = f (x, t) lal
on

QXI

with u(O) = 0. ASSUMPTION (II). Let V be a given closed subspace of Wm*2(Q) with Wrs2(sZ) C V C Wn”12(Q). We assume that there etits a closed subspace V2 of Wm812(Q)such that the following conditions are satskjied:

(i) V, C V, m2 > 171. (ii) There exists a set X, XC V, , X densein V. For each v in D(L,) there exists p”nin L2(I, X); ~~(0) = 0, vn , continuous from I to V, , avn/atcontinuous from I to L2(Q), and T-%&-v

in

L2(I, V);

%+$f

in

L2(Q)

as

n-+ oc).

We have the following theorem: THEOREM 2.3. Let A(t) be a system of partial

differential operators on

NONLINEAR

PARABOLIC BOUNDARY VALUE PROBLEMS

261

Sz x I, of the form (1.1) and satisfying Assumption (I). Let V be a closed subspace of W’Q(Q). Suppose that Assumption (II) is satisfied. Suppose further that (i)

Re{a(u,

u; u - v) - a(u, o; u - v)> > 0 for all u, v in L2(I, V).

(ii) There exists a continuous function r + 0~) such that Re a(u, u; u) 2 ~(11u 11)11u 11

c(r) from R+ to R with c(r) -+ 00 as

for all

u

in

L2(I, V).

Then for every f in L2(I, V*), there exists a solution u of the variational boundary vatue problem of the nonlinear parabolic equation $+A(t)u=f

on

Q x I,

corresponding to V with u(O) = 0. The solution is unique if (i) is replaced by (i)’

Re{a(u, u; u - v) - a(w, v; u - 7~)) > 0 for all u, v in P(1,

V).

PROOF. The proof is the same as that of Theorem 2.2. It suffices to replace W?*‘(Q); W:*‘(Q); W-a12(J2); W-R”*2(s2) respectively by V, , V, Q.E.D. v2*, v*.

Note added in proof. In [8], by an argument similar to the one of this paper (it is even simpler), we show the solvability of a variational elliptic boundary value problem for an elliptic operator A written in the generalized divergence form and A, having at most a polynomial growth in u,..., Dmu. REFBFULNCES

1. F. E. BROWDER. Strongly nonlinear parabolic boundary value problems. Amer. J. Moth. 86 (1964), 339-357. 2. J. L. LIONS. Sur certaines equations paraboliques nonlineaires. Bull. Sot. Moth. France 93 (1965), 155-175. 3. M. I. VISIK. On the solvability of boundary value problems for quasi-linear parabolic equations of arbitrary order. Mat. Sbornik. 59 (lOl), (1962), 289-325. 4. F. E. BROWIXR. Nonlinear elliptic boundary value problems. II. Trans. Amer. Math. Sot. 117 (1965), 530-550. 5. J. P. AUBIN. Un theoreme de compacitt. Coopt. Rend. Acad. Sci. 256 (1963), 5042-5044. 6. J. LERAY AND J. SCHAUDER. Topologie et equations fonctionnelles. Ann. &Cole Normule Sup. 51 (1934), 45-78. 7. F. E. BROWDER. On the spectral theory of elliptic differential operators. I. Math. Ann. 142 (1961), 22-130. 8. B. A. TON. On strongly nonlinear elliptic boundary value problems. To appear.