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Singularly perturbed superquadratic boundary value problems
Nonlrneor Analysis, Theory, Methods CJPergamon Press Ltd. 1979. Printed
SINGULARLY
&Applications. Vol. 3. No in Great Britain.
2. pp
PERTURBED VALUE
0362.546x/79/0301-Ol7OSO2.00/0
175-192.
SUPERQUADRATIC PROBLEMS*
BOUNDARY
F. A. HOWEST School of Mathematics,
University
of Minnesota,
Minneapolis,
MN 55455, U.S.A
f Received in revised form 7 March 1978) Key Words: Singular perturbation, greater than quadratic.
reduced problem,
differential
inequality,
rate of growth in the derivative
1. INTRODUCTION IN THIS
paper we study the existence
of solutions
of the two singularly
perturbed
boundary
value
problems o
s2Y’ = Mt, L’)j-(4 L‘,Y’) = F(4 L’,Y’), (P,)
b,~‘(l, F) + b&(1, E) = B,
a,L’(O, E) - Q&(0,&) = A,
and s2Y’ = F(4 L’,v’), (Pz)
o
U,L‘(O,&)- U,L”(O,&) = A,
for small positive values of the parameter s and for nonnegative values of the constants a,, a,, b, and b,. Our principal assumptions are that the function h has certain zeros which are stable in a sense to be described below and that the function f is strictly positive and grows faster than JJ” as /JJ’/ + co (that is, f is superquadratic). In the course of proving the existence of solutions of (Pi) and (P,) we will also derive rather sharp estimates on the behavior of these solutions which improve as F decreases to zero. As will become apparent below the assumption that f is strictly positive implies that the existence and the asymptotic behavior of solutions of (P,) and (P,) are governed by the nature of the solutions of the equation h(t, U) = 0. However the assumption that f is superquadratic has an equally significant effect on the types of solutions which these problems possess. Our interest in such problems was generated by the lack of any coherent theory for singularly perturbed superquadratic boundary value problems and also by the occurrence ofproblems ofthis type in several applied areas. For example (cf. [ 1; Chap. 11) the elevation y of the free surface of a liquid meeting a plane, vertical, rigid wall (t = 0) is described by a problem of the form (P,) with h(t, L‘) = L‘,f (t, y,y')= (1 + y’2)3!2, s2 = T/(pg), a, = O,a, = 1 andA = -tan @for0 < 8 < 42. Here p is the density of the liquid, g is the gravitational constant, T is the coefficient of surface
* Dedicated + Supported
to my father on the occasion of his seventieth birthday. in part by the National Science Foundation under grant 175
No. MCS 76-05979.
176
F. A. HOWES
tension and 8 is the contact angle, that is, the angle which the surface makes with the wall. If T is small then E = [T/(pg)] “’ is small and the problem for y is singularly perturbed. As a second example consider a liquid confined between two plane vertical walls at t = 0 and t = 1 or equivalently, a liquid contained in an open capillary tube. Then (cf. [2,3]) the elevation y of the free surface of the liquid satisfies a problem of the form (P,) with E and 0 as before and with a, = b, = 0, a2 = b, = 1 and -A = tan 8 = B for 0 < 0 < 71/2. Here however the right-hand side F is equal to F(t, y, y’, F) = (L. - &y’/[t(l + L.“)~‘*]) (1 + L.“)~‘*. Once again if the coefficient of surface tension T is small then E is small and the problem is singularly perturbed. (We will not consider functions such as F” in this paper per se although the results we obtain for the simpler problem (P,) can be extended to this more general setting with little difficulty.) Before commencing our study of the problems (P,) and (P,) we review briefly some of the literature in both the perturbed (0 < E 6 1) and the unperturbed (e = 1) cases. For the perturbed problem (P,) (and for its system analogs) Vasil’eva [4; Chap. 21 has given a comprehensive survey of the work done prior to 1963. The theory described there applies for the most part to the case of righthand sides F which grow no faster than y’2 although she does give several results for the Robin problem (that is, uz + bz > 0) in the case that F is superquadratic. Since then most writers in singular perturbation theory (see [5] and [6] for extensive lists of references) have concentrated on the case of F = @(Iy’ 1’). R ecent results of this author [7,8] show however that a general theory for the perturbed problem (P,) is possible although there are many details which have to be worked out. The unperturbed problem (P,) for the case of Dirichlet data (that is, a2 = b, = 0) has been the object of much study since the turn of the century. We mention only the fundamental papers of Bernshtein [9] and Nagumo [lo] and refer the reader to the paper of Hartman and Wintner [ 1 I], the survey article of Jackson [ 121 and the book of Hartman [ 13 ; Chap. 121 for modern treatments of the Bernshtein-Nagumo theory and many additional references. Indeed, to study the perturbed problems (P,) and (P,) in this paper we will use some natural extensions of Nagumo’s results which are given in a paper of Heidel [14]. Much less is known about the perturbed problem (P,) independently of the size ofthe righthand side F as a function of y’. The most complete treatment of this and related problems is contained in the doctoral dissertation of Lee [ 151 (cf. also [ 16, 171) for the case that F = O(lyf2 1). There is however a substantial literature on the unperturbed problem. We mention only the papers of Hartman and Wintner [1 I], Shcherbina, [18, 191 and Myshkis and Shcherbina [3] which the reader can consult for iurther references. The last two papers also contain a discussion of several physical phenomena which are described by solutions of (P,). Finally we note that a complete study of the one-dimensional capillary tube problem has been given by Johnson and Perko [2] and Perko [2Q] for both the finite and the semi-infinite interval. We take this opportunity to thank Professor L. Perko of Northern Arizona University for his extensive and most helpful correspondence with us on some of the problems discussed below.
2. THE
Consider
the Dirichlet E2y”
(D)
DIRICHLET
PROBLEM
problem =
h(t,
~‘1 fk
Y, $1
L’(O,E)= A,L’(l,&) = B,
=
F(t,
Y, Y’),
O
Singularly perturbed superquadratic boundary value problems
and the corresponding
reduced
problem
h(t, u) “m
4
4 = 0,
U(0) = A, The function the domain
h is assumed
177
to be continuous
o
U(l) = B. with respect to t and of class C(“) with respect to y in
Q(u(t,) = ((t, L‘): 0 d t < 1, IL’ - u(t)( < d] where 0 = 2q + 1 or y1for integral values of q 2 0 and IZ B 2, u = u(t) is a solution of (R,) and d is a small positive constant. Similarly the function f is assumed to be continuous in the domain _QJu(t)) = 9@(t)) x R1 and strictly positive there, that is, f(t, Y, Y’) 2 1 > 0
in Bl(Mt))
for a positive In order to describe how of the reduced in 9(u(t))
constant 1. deduce the existence of solutions of (D) for small positive values of c: and then to these solutions behave in the limit F, + 0’ we isolate the stable solutions u = u(t) problem (R,). Namely if the function h is of class Cczq+ ” (q 2 0) with respect to y we will say that u is &)-stable provided that
(i) d;,h(t,u(f))=O
I andO
forO
(ii) thtre exists a positive
constant
m such that
a y2qf1 h(t, y) 2 l-‘m2 Similarly provided
if h is of class P(n that
(i) a;,h(t, u(t)) > 0
> 0
for (t, y) in ~(u(L)),
> 2) with respect to Y in 9(u(t)) then we will say that u is (II”)-stable
for 0 < f d 1 and I G j 6 n constant m such that
1;
(ii) there exists a positive
ij;h(t, y) 2 l- ‘m2 > 0 Finally if h is of class C(“)(n 2 2) with respect provided that (i)
i3; h(t, u(t)) 3 0
for (t, y) in g(~(t)).
to y in 9(~(t)) we will say that u is (III,,)-stable
and 8; h(t, u(t)) < 0
for 0 d t 61
and 1 6 j,,, je d n - 1 where j,,o,) denotes an odd (even) integer; (ii) there exists a positive constant m such that a;h(t,y)
2 I-‘m2
> 0 or a;h(t,y)
d -l-‘m2
< 0
for (t,~)
in 9(~(t)) if n is odd or even, respectively. These definitions (Iu), (II,) and (III,) of “stability with respect to $’ have been used by the author in [7], [S] and [21] and they are based on earlier work of Boglaev [22]. For ease of presentation, in what follows we will simply say that h is a-smooth in 9(~(t)) if h is continuous with respect to t and of class C@) with respect to y in 9(ucr)). The results of this section extend the work of the author [21] to the case in which the function f is superquadratic in 9,@(t)), that is, f = O(l J’ I’) as 1y’ / --+ co for (t, y) in 9(u(t)) and v > 2.
178
F.
THEOREM 2.1. Assume (1) the reduced that U” 2 0; (2) the function superquadratic
A. HOWES
that
problem
(R,) has an (IQ)- or (II&stable
solution
u = u(t) of class P’[O,
l] such
h is a-smooth in g(~(t)) for c = 2q + I or n and the function f is continuous and in g,(~(t)); moreover, there is a positive constant I such that f > 1 > 0 in
9#(t)). Then there exists a positive constant s0 such that the problem whenever 0 < i: 6 F~. In addition, for t in [0,11 we have that
(D) has a solution
y = y(t, 6)
u(t) 6 y(t,c) < u(t) + (E$F-2)l’~
Proof of Theorem 2.1. To prove this theorem we will apply a generalization of the classical Nagumo theorem (cf. [IO, 12 Sec. 71) on second-order differential inequalities due to Heidel [ 141. Specifically if there exist C ‘2’-bounding functions a and /3 such that for 0 < I: < F~ a G P, andfor
a@, 8) d A < pco, e),
U(l, E) d B $ /3(1, E)
< t < 1, E2U” 2 F(L, M,a’),
E2pI < F(l, B, p’),
then Heidel has shown that the boundary value problem (D) has a solution y = y(t,c) satisfying Nagumo condition with ccO a(t, E) = u(t),
P(t, E) = u(t) + (2]m-2)1’o.
Here g = 2q + 1 or n and 1~is a positive constant such that 1~2 a!M for )Iu” IIm ,< hf. For ease of presentation let us set r = r(c) = (E~~vY~)~‘~. Clearly TV< b, $0, e) < A < p(O, 8), cc(1,E) d B < p(1, E)and EMU”2 F(t, a, a’) (since U” 3 0). The only verification remaining is the one showing that F’/?’ G F(t, p, ,Y): F(t, p, /I’) - E~P” = h(t, p) f(t, j3, p’) - FLU” o-l = (h(t, u) + c
l/o’!) a@,
u) I-J + l/b!)
j=l >
{l/(0!) I-‘m%+Y-2}
2 0
since 3’ > a!fM.
1 - s2M
dyl(t,
4) I-“} f(t,
p, p’) - E2zi’
Singularly
perturbed
superquadratic
boundary
179
value problems
Here (and throughout the paper) t = ((t, F) = (t, u + 0(/3 - u)) (0 < 0 < 1) belongs to Q(W) if E is sufficiently small. Consequently o! and p satisfy all of the required inequalities. The proof will be complete if we can show that whenever
E2L.l’= ~(t, y, y’)
and
Y < y < ,G on
J c [0, 11
follows because u’(O) Q y ‘(t, E) < u’(l) for any such on [0, I]. To see this note that ~(0, e) = u(0) = 40, E) and a(4 E) G ~(4 E) imply that ~‘(0, E) 2 u’(O,E) = u’(O), while y(1,~) = u(1) = cr(l,~) and c~(t,e) < y(t,e) imply that ~‘(1~6)G ~‘(1,E) = u’(1). Therefore u’(0) < y’(t, E) d u’(1) since y” 2 0 for CI< y G P. We conclude from Heidel’s theorem that the problem (D) has a solution with the stated properties. then
]y’(t, E)I < N on J x (0, ~~1. This
solution
Suppose next that the solution u = u(t) of the reduced problem (R,)is concave, that is, u” G 0. Then a result analogous to Theorem 2.1 holds if u is either (IJ- or (IIIJ-stable. The precise statement is contained in Theorem 2.2. THEOREM 2.2. Make the same assumptions (1) is replaced by (1’) :
(1’) the reduced problem that u” < 0. Then the conclusion
as in Theorem
(R,) has an (14)- or (III,,)-stable
of Theorem
2.1 with the exception
solution
that assumption
u = u(t) of class C”‘[O, l] such
2.1 is valid with the inequality
replaced
by
u(t) - (E2;~m-2)‘io < y(t, .7)6 u(t).
Proof of Theorem 2.2. This result is proved by making the change of dependent variable y -+ - y and then applying Theorem 2.1 to the transformed problem. Note that under this change of variable (Ia)-stability is left unchanged while &)-stability is transformed into (III&.tability. We remark that the above results are valid if the function u has only an absolutely continuous derivative on [0,11. In this case we assume that u” 2 0 where it exists (in Theorem 2.1) or that u” < 0 where it exists (in Theorem 2.2). Heidel’s result can be extended to AC”‘-bounding functions by arguing as in [12] which treats only the classical Nagumo condition. However it frequently happens that the function u is only piecewise differentiable on (0,l) as would be the case for example when two distinct solutions of h(t, U) = 0 intersect in (0,l) with unequal slopes (cf. [21]). Heidel’s theorem can also be extended to cover this case provided we impose the following additional conditions on the bounding functions c( and p. Namely if
u’(t;) # u’(t,‘)
for some t, in (0, 1)
then we require that a’(t;) d cr’(t,‘) and p(t;) 2 P’(t,‘). These inequalities follow essentially from the fact that the supremum of a family of lower solutions (that is, functions satisfying the c(inequalities) is again a lower solution while the infimum of a family of upper solutions (that is, functions satisfying the B-inequalities) is again an upper solution. As regards the boundary value problem (D) we will see in the next two theorems that the occurrence of a finite jump in u’(t) has the effect of introducing an interior layer into the derivative of any solution. For simplicity we consider only the case of a single jump at the point f, in (0, 1).
F. A.HOWES
180 THEOREM
2.3. Assume
that
(1) the reduced problem (R,) has an (I )- or (II”)-stable solution u = u(t) of class 15~’ on [0, I]\{ toI such that u’(t;) < u’(tl), U” 2 0 on 10, l]\{t,> and u”(ti) 2 0; (2) the function h is o-smooth in g@(t)) for 0 = 2q + 1 or n and the function fis continuous and superquadratic in 5@(t)); moreover, there is a positive constant 1 such that f 3 1 > 0 in -%(4t)). Then there exists a positive constant e0 such that the problem whenever 0 < E < co. In addition, for t in [0, l] we have that
(D) has a solution
4’ = y(t, e)
u(t) < y(t, E) < u(t) + w(t, E) -f (+r-y where CJ= 2q + 1 or n, 1~2 a! I/ u” Ilooand w(t, E) = +I-~ ~(u’(f+) - u’(t;)) exp[ - rnE- ’ 1t - t, I] w(t,e) = ;?-I
-t ,c-l’(q+l)lt
qE1!(q+l)(u’(tO+)- u’(t;))(l
(for T(q) = [(+(u’(t,‘) - u’(t;)))2q q2q+2m2/((q w(t,c) = +T-‘((n -
We remark satisfies
that
w(t, e) is a solution
(n -
q = 0,
- t,I)-l/(q+l)
if
+ 1)(2q -t 1)!)]1’(2q+2’)
1)/2)~~~‘“+‘)(u’(t0+) - u’(t;))(l
(for 7(n) = [(+(u’(t,‘) - u’(t;)))“-’
if
q 2 I and
+ ~.~-~~@+l)lt - t,l)-2i(n-1)
if ~7= n
I)n+1m2/(2n(n + l)!)]‘“““‘)
of the differential
w’(t;, E) = $n’(t,‘)
equation
- u’tt;))
w’(t,+, 8) = -+(u’(t,‘)
s2w” = (m2/o!) w0 which
and
- u’(tJ).
2.3. Define for E > 0 and 0 d t < 1
Proof of Theorem
Gl(t,E) = u(t), P(t, F) = u(t) + w(t, e) -f (E2ym-2)“” where 1’ is a positive constant such that 1~2 o!M for 11 u” Ilrn G hf. Since u’(t;) < u’(t,‘) by assumption (1) we have that cc’(t;) < cr’(t,‘). As regards /I it follows by our choice of w that p’(t;) It only remains have that
to verify
that
= P’(to’) = $(u’(t,,
+
u’(t,‘,).
s2p” < F(t, p, /I’). Let us set I = I%) = (s21m2)1’o,
then
we
F(t, p, B’) - ~~8” = h(t, p) f(t, j3, p’) - t2P” o-l h(t, u) -f C (l/j!) a;h(t, u)(w + r)j
= i
j=l
2 (I- ‘m2/a!)(w”
+
ry 1 -
+ (l/a!)
a;h(t, t)(w
c2A4 -
E’W”
+ I-‘bll f(t, p, p’) - ~~11”- dew”
181
Singularly perturbed superquadratic boundary value problems
= (m2/a!) w” - E2W’I+ E2J’jO!- E2M
> 0
since
&v” = (m’/cr!) wO‘ and
‘)J2 a!M.
Thus c1and p satisfy the required inequalities. Since it follows as in the proof of Theorem 2.1 that U’(0) ,< $(t, E) < U’(1) for any solution y of s2$’ = F(t, JJ,y’) with d < JJ < p we conclude from the extended version of Heidel’s theorem that the problem (D) has a solution with the stated properties. The analogous result in the case that a”< 0 holds provided we replace (I$)-stability with (IIIJ-stability. For completeness we state this remark in the form of a theorem which follows from Theorem 2.3 via the change of variable 4: --t -y. THEOREM 2.4. Make the same assumptions as in Theorem 2.3 with the exception that assumption (1) is replaced by (1’):
(1’) the reduced problem (R,) has an (I )-or {III~)-stable solution u = u(t) of class C’*‘on [0, l]\(t,~ such that u’(t;) > u’(t,f), u” < 0 on 10, l]\(r 0J and d’(td) < 0. Then the conclusion of Theorem 2.3 is valid with the inequality replaced by u(t) + 1%(t,E) - (&m-2)“”
< L’(t,E) < u(t).
We close this section with four remarks on the theory developed here. Remurk 2.1. It is sometimes necessary to extend the definitions of (Iy)-, (II,,)- and (III,,)-stability to the case when Z;h is strictly bounded away from zero only along the junctionu (cf. [21]). The extended definitions are as follows.
Let u = U(C)be a solution of h(t, U) = 0. The function u is called (I;)-stable if (i) J;,h(t, u(t)) 5 0 for 0 G t < 1 and 1 <
> 0
for
0 < t < 1 and
8fq+* h(t, y) > 0
(t, y)
in
5+4(t)).
for
Similarly the function u is called (Uhf-stable if (i) d:,h(t,u(t))>O
for
O
1
and
1
I;
(ii) there exists a positive constant m such that a;h(t, u(t)) 3 /-‘mz > 0 for d;+lh(t, y) 2 0 Finally the function (i) drh(t, u(t)) & 0
for
(t, y)
0 G t G 1 and in L+(t)).
II is called (III~~-stable if and
i;h(t,
u(t)) d 0
for 0 < t G 1 and 1 < jO,je < n - 1 where j&J denotes an odd (even) integer;
182
F. A. How=
(ii) there exists a positive constant d;h(t, u(t)) > l-‘m2 > 0 and
m such that d;“h(t, y) < 0
g@(t)) or a;h(C, u(t)) d -l- ‘m2 < 0 and f;+‘h(t, .5@(t)) if n is odd or even, respectively.
y) 2 0
for for
(t, Y)
in
(t, 4’) in
Theorems 2.1 and 2.3 are clearly valid with (Iq)- or (II”)-stability replaced by (Ii)- or (II;)-stability, respectively. Similarly Theorems 2.2 and 2.4 hold with (Ii)- or (III:)-stability, respectively. Since the extended definitions are somewhat technical they are not included in the main body of the paper. We note here that the results of the following sections are also valid with (I&, (II”)- and (IIIJstability replaced by their primed counterparts as appropriate. Remark 2.2. The restriction that the solution u of the reduced equation be convex or concave can be removed in the case of (14)-stability but not in the case of (II,,)-stability. For example, in Theorems 2.1 and 2.3 if u is an (14)-stable solution of (R,) which is not necessarily convex then a(t, 9) = u(t) - (,52ym-2)‘1(2q+ ‘) (for 1’ 2 (2q + 1) ! I(u” /I,) is clearly a lower solution 2.1 to show that a solution y of E2y” = F(t, y, y’) has a uniformly
bounded
derivative
of(D). We can argue as in the proof ofTheorem
with
c( B 4’ < p
and so the results follow once more from Heidel’s theorem.
Remark 2.3. Since the function f is assumed to be strictly positive in the region of interest the solutions u of the reduced problem (R,) are roots of the equation h(t, u) = 0. As such it would be less restrictive to impose inequality constraints on u(0) - A and u(l) - B rather than to require that u(0) = A and u(l) = B. However in the presence of superquadratic nonlinearities it does not appear possible to relax this requirement. This follows on the one hand from the fact that the method of proof used above breaks down unless u(0) = A and u(1) = B and on the other from the observations of Vishik and Liusternik (cf. [4; Chap. 21). In short these authors showed that solutions of the Dirichlet problem cannot exhibit boundary layer behavior in the presence of superquadratic nonlinearities. If the zeros u(t) of h(t, u) do not satisfy the boundary conditions exactly or are unstable then it is likely that solutions of the full problem (D) either do not exist for small values of c: or (if they do) are not of bounded t-variation as E -+O+ (cf. [7]). Remark 2.4. Our treatment of the interior layer problem (Theorems 2.3 and 2.4) deserves a brief comment. In Theorem 2.3 for example the solution u of the reduced problem satisfies u’(t;) < u’(t,‘) and so the interior layer correction term w must be chosen so that p(t;) 2 p’(t,‘). Actually for the function w constructed in the proof of Theorem 2.3 we have that @‘(t;) = p(ti). This function w is uniformly small on [0, 11; however, w'(tt , E)is of order U( 1). Thus w serves to smooth out the discontinuity in the derivative of u in the sense that u’(t) + w’(t, E) is smooth at t = t,. 3. ONE
Consider
TYPE
OF
ROBIN
PROBLEM
next the problem E2JY = h(t, L’)f(t, y, y’) = F(t, y, y’),
0 < t < 1,
183
Singularly perturbed superquadratic boundary value problems
(E)
Jl(l,E) = B,
q(0, t;‘) - y’i0, E) = A,
where a is a nonnegative constant. Since the derivative of y is prescribed in the boundary condition at f = 0 the appropriate reduced problem corresponding to (E) is (cf. [S]) (R,)
U(l) = B.
o
h(r, u) f(r, u, u’) = 0,
We require of the functions h and j’ the same properties as in the previous section and we will again restrict attention to stable solutions of (Rr). The definitions of the various types of y-stability are exactly the same as above. We note that a solution u = u(t) of (Rr) is only required to satisfy the right-hand boundary condition and as will become clear momentarily we must require that au(O) - u’(O) < A
if
a” > 0
au(O) - U’(O)2 A
if
U” d 0.
or that The first two theorems are the analogs of Theorems 2.1 and 2.2. THEOREM3.1.
Assume that
(1) the reduced that U” 2 0 and (2) the function superquadratic
(R,) has an (IJ- or (II,)-stable solution u = u(t) of class C2’[0, l] such au(O) - u’(0) ,< A; h is e-smooth in &J(u(t))for (r = 2q -+ 1 or n and the function f is continuous and in .Q:‘,(u(t)):moreover, there is a positive constant 1 such that f b I > 0 in
problem
I&). Then there exists a positive constant so such that the problem (E) has a solution Y = Y(F,R) whenever 0 < E < R,,.In addition, for t in [0, 11 we have that u(t) < y(t, E) < U(t) .+ U(t,E) where d =29 u(t, R) 5 0
(E2yR2-2)1’”
+lorn,y>,a!//u”//,and
if
au(O) - u’(0) = A,
u(t, E) = m- 1E(A - au(O) + u’(0)) exp[ - me- 1t] ~(t, F) = T- &l!(q+
(for
+
if
q = 0
“(A - ff~(O) 4- ~‘~0))~~ + zc- l!fq+lJt)- “*
r(q) = [(A - uu(O)-+ u’(0))24q2q+2m2/((q
+ 1)(2q
if 9 3 1 and
au(O) - u’(O)< A
+ J)!)]1”2q+2))
and c(t, s) = r-‘((~3 - 1)/Z) E~~(*+~)(A - au(O) + ~‘(O)~(l+ TE-~‘(~+~)t)-l’(tt+l)
if
u = n
and au(O) - u’(0) < A (for We note
z(n) = [(A - au(O) + u’(O)),-* (n - l)“+’
that ~(t, e) is a solution of the differential equation I
F*U”
=
which satisfies ~‘(0,a) = -(A - au(O) + u’(0)).
(m2/o!)
flu
m2/(2”(n
.-i- l)!)]licn+l)),
184
F. A. HOWES
Proof of’ Theorem 3.1. The proof is only slightly different from the proof of Theorem 2.3. This follows because Heidel’s theorem [ 141 as stated in the proof of Theorem 2.1 also applies to boundary conditions of the form
a,y(O, e) - a,y’(O, E) = A, b,y(l, for u2, b, 3 0 and UT + uz > 0, bf -t b: > 0. Namely the inequalities G!G P,
a,r(O,e)
b,a(l,E)
-t b,a’(l,c)
E) -+ b,y’(l, E) = B, if the bounding
functions
CIand p satisfy
- u2cQo, E) < A < U$(O,E) - a,P’(O,E), d /I < b,P(l, E) .f b,B’(l,c)
and 8%” > F(t, a, cr’), E’P,’ d F(t, fl, p’) then the problem ,2y’l = F(t, L’,y’), a,y(O, e) - a&(0,
o
b,y(l,e)
+ b,y’(l,s)
has a solution y = y(t, E) with CI < y < ,6’provided that F satisfies a generalized with respect to c1and ,0. Define for F: > 0 and 0 d t d 1
= B Nagumo
condition
cl(t, e) = u(t), P(t, E) = u(t) -t v(t, e) -t (c2pm-2)“u. Here c = 24 + 1 or n and 1~is a positive I- = L-(E) = (~2ym-2)1i0. Clearly
such that 1’ 3 o!M for I/u” II?: < M. Let us set
constant
acc(0, E) - r*‘(O,R) d A d @(O, 8) - P’(O, e)
!x < B,
and a( I, E) < B < ,B(l, E) since c’ > 0 satisfies ~‘(0, F) = -(A - au(O) + u’(O)). Finally ~(r, a, cx’)since U” > 0 and so it remains to verify that a2p” G F(t, fi, p’):
E~CZ”2
F(t, ,!I, p’) - E~/I” = h(t, /?) f(t, p, B’) - c2p” 0-l
h(t, u) t
=
i
C (l/j!) a;,h(t, U)(L) + l-)‘+ (l/o!) d;h(t, 5)(u -f rYj 1 f(t, p, P’) ,=l - ,&’ _ &’ t
3 (l- ‘m2/a!)(F =
(m2/g!)
2 0 Therefore that if
u”
since
-
c2v”
I -
e2fM
=
-
c2v“
.t (.~~]))/a! - E2A4
E~V” = (m2/a!)ti0
E and p satisfy the required
E2y”
ry
inequalities.
F(t. y, y’)
and
1~2 o!M.
To conclude
and CI < y G B
the proof it is sufficient
to show
then
W(0) - A < y’(t, 8) d U’(1) since this inequality establishes that F satisfies a generalized c( and fl (cf. the proof of Theorem 2.1).
Nagumo
condition
with respect to
Singularly
perturbed
superquadratic
boundary
problems
value
18.5
Clearly ~‘(0, t) 2 au(O) - A since 4”(0,E) = aJ(o, F) - A 2 ~~~0) - A
(a 2 O),
and since 4”’ > 0 for c1d 4‘ d j? it follows that au(O) - A d y’(t, E) G y’(l, ST).
However ~(1, e) = u(l) = ~(1, a) and y >, x imply that ~‘(1, E) d ~‘(1, E) = u’(1) and this gives the desired inequality. We conclude from Heidel’s theorem that the problem (E) has a solution with the stated properties. If the solution u = u(r) of (R,) is concave, that is, ifu” d 0, then a similar result holds provided we replace (I&J-stability with (III,,)-stability and provided we require that au(O) - u’(0) 2 A. The precise result is stated in the next theorem which follows from the previous one via the change of variable y + - y. THEOREM 3.2. Make the same assumptions as in Theorem 3.1 with the exception that assumption (1) is replaced by (1’) :
(1’) the reduced problem (Rn) has an (I,)- or ~IIr~)-stable solution u = u(t) of class P’[O, I] such that u” $ 0 and au(O) - u’(0) 2 A. Then the conclusion of Theorem 3.1 is valid with the inequality replaced by
As in the case of the Dirichlet problem (D) it often happens that solutions u of the reduced problem (RE) have jump discontinuities in their first derivatives, that is, for some t, in (0, 1) u’(t;) + u’(r,‘). By arguing as in the proof of Theorems 2.3 and 3.1 we can establish the following analog of Theorem 2.3. (The formulation of the analog of Theorem 2.4 is straightforward and is left to the reader.) THEOREM 3.3.Assumethat
(1) the reduced such that u’(t;) (2) the function superquadratic ~~(~~t)).
problem (R,) has an (I f- or (II,)-stable solution u = u(t) of class C”’ on [0, I]\(t,). < u’(t,i), 11” < 0 on 10, l]\(t,>, u”(tz) G 0 and au(O) - u’(0) L4; h is o-smooth in S(u(t)) for (T= 2q + 1 or rr and the function f is continuous and in gl(u(t)); moreover, there is a positive constant 1 such that f 2 1 > 0 in
Then there exists a positive constant e0 such that the problem (E) has a solution Y = y(t, E) whenever 0 < F < ee. In addition, for t in [0, I] we have that u(t) < yft, F) < uit) + oft, F) + wtt, 4 +
fe2Ym2Pu
where (T = 2q + 1 orn,y > a! /Iu”/)~ and the functions t:and w are as in Theorem 3.1 andTheorem 2.3, respectively.
F. A HOWES
186
The results of this section
are easily seen to apply to the “reflected” E2fl
=
44
Y)
.I-(&
L’,
problem
O
Y’),
(E’) Y(O,
&)
=
by(1,
4
E)
+
y’(1, F) = B, f + 1 - t in (E’) and applies
for nonnegative values of b. One simply makes the change of variable the previous theory to the transformed problem. 4. ANOTHER
It is now a straightforward
matter
TYPE
OF ROBIN
to consider
the problem o
&*Y” = 44 Y) f(4 L’,Y’), (G)
PROBLEM
ay(0, E) - y’(0, F) = A,
by(1, E) + y’(l,s)
= B,
for nonnegative values of the constants a and b. Since the derivative of ): is prescribed at both endpoints it is reasonable to consider stable solutions u = u(t) of the corresponding reduced equation o
44 u) f(4 u, u’) = 0,
(R,)
which may satisfy neither of the boundary conditions in (G). However we find it necessary to restrict such functions u so that
as in the previous
au(O) - u’(0) d A
and
bu(1) + u’(1) < B
if
U” > 0
au(O) - u’(0) > A
and
bu(1) + u’(l) > B
if
u” < 0.
section
or
The expected results for the problem (G) are valid if u is (14)- or (II,,)-stable and U” > 0 or if u is (I )- or (III”)-stable and U” < 0. In the first case (u” 2 0) we note that if y = y(t, F) is a solution of $y” = h(t, y) f(t, y, y’) which satisfies TV< y < p for bounding functions c( and /I then au(O) - A < y’(t, E) < bu(1) + B, while in the second case (u” < 0) the inequality
is of the form
bu(1) + B < y’(t, E) < au(O) - A. As an illustration of the analog of Theorems is left to the reader. THEOREM 4.1.
Assume
2.1 and 3.1 we state the following
result whose proof
that
(1) the reduced equation (R,) has an (Iq)- or (II,,)-stable solution u = u(t) of class C(*)[O, l] such that au(O) - u’(O) < A, bu(1) + u’(1) 9 B and U” 3 0; (2) the function h is o-smooth in Q(u(t)) for 0 = 2q + 1 or n and the function f is continuous and superquadratic in Ql(u(t)); moreover, there is a positive constant 1 such that f 2 1 > 0 in 2,
(44).
Then there exists a positive constant E,, such that the problem whenever 0 < B < E,,. In addition, for t in [0, l] we have that U(t)
d
y(t,
E)
d
U(t)
+
fJ(t,
8)
+
fi(t,
E)
+
(&2y62)“0.
(G) has a solution
y = y(t, F)
Singularly
perturbed
superquadratic
boundary
value problems
187
Herea=2q+ lorn, y 2 a! I)~“lllo’ r(r, E)is defined as in Theorem 3.1 and ii(t, E) = ~(1 - t, E) with (A - au(O) + u’(0)) replaced by (B - bu(1) - u’(1)). We remark that u and ii both satisfy the differential equation u’(O,&)= -(A - au(O) + d(O)) and E’(l,&) = B - k(l) - u’(1). 5. THE
SEMI-INFINITE
INTERVAL
E’Z” = (m2/a!)zb and that
PROBLEM
We consider next the following problem on the half-line [0, CD) o
E2Y”= h(t, Y) f(4 y, Y’),
(H)
U,Y(O,&)
-
U,L”(O,&)
= A,
for nonnegative values of the constants a, and a,. If a2 = 0 we assume without loss of generality that oI = 1 and it is natural to associate with (H) the reduced problem h(4 4 f(L 4 u’) = 0,
o
u(0) = A.
On the other hand if cl2 > 0 then since ~‘(0, F)is prescribed (cf. Section 3) the appropriate reduced problem is simply the equation h(t, u) f(t, u, u’) = 0.
(R,)
The theory developed above carries over with little change to the problem (H) because Heidel’s theorem is easily extended to problems on the half-line (cf. [12]). In particular the bounding functions a and /? are required to satisfy the same inequalities on [0, co) and the function F(t, y, y’) = h(t, y) f(t, y, y’) is required to satisfy a generalized Nagumo condition on each compact subinterval of [0, co). Under the additional assumptions stated above the problem
Y” = F(4 Y,Y’), u,Y(~)
- a,~‘@)
o
has a solution J = y((t) such that a(t) d y(t) < j?(t) for 0 < t < co. This inequality can be used to estimate the behavior of a solution near t = 03 if p(t) - a(t) is close to zero for such values. For the problem (H) it turns out (not surprisingly) that up to terms of order O(E’!“) the behavior of solutions for large values oft is determined by the behavior of the reduced solutions U. As in the previous section it would be tedious to write down the analogs of all the theory developed above and so we content ourselves with stating only the analog of Theorem 3.1 for the case that a, = 1 (and a, = 0). For convenience we define the region &(u(t)) to be the region g(u(t)) defined above with t in [0, co). Then @,(u(t)) = 9(u(t)) x RI. The definitions of y-stability are similarly extended to the half-line [0, co). THEOREM
5.1. Assume that
(1) the reduced equation (R,) has an (IB)-or (I&)-stable solution u = u(t) of class C’*‘[O, co) such that U” 2 0, au(O) - u’(0) 6 A a_nd Ij u” 11 m < co ; (2)‘the function h $a-smooth in 9(u(t)) for B = 2q + 1 or n and the function f’ is continuo_us and superquadratic in gd,(u(t)); moreover, there is a positive constant 1such that ,f 2 I > 0 in gI(u(t)). Then there exists a positive
constant
&0 such that the problem
(H) has a solution
y = y(t, E)
188
whenever
F.
A. HOWB
for t in [0, co) we have that
0 < F < Ed. In addition,
U(t) < L’(t,E) < U(t) + U(t, E) + where CJ= 2q + 1 or n, If the function has this property
(E+-
y 2 a! 1)u” 11 m and u(t, 8) is defined
2)1’u
in Theorem
u has a iimit at t = 00 then it is clear that the solution also. We isolate this remark as a Corollary.
COROLLARY 5.1. In addition
to the assumptions
of Theorem
u0 = lim u(t) t+cO Then the problem (H) has a solution lim y(t, E) exists and satisfies f’rn
y = y(t,
3.1. J of(H) just constructed
5.1 suppose
exists.
for each F > 0 sufficiently
E)
that
small such that L.~(E)=
u0 d L.&E) d U* + (&2ym-2)“0.
6. EXAMPLES In this section
Example
we discuss several examples
6.1. Consider
first the differential
which illustrate
values of q > 0. The function
13~~+lh = (2q + l)! P
for
Since u(O) = 0, u(l) = I and u” = 2 the Dirichlet
Y(0,4 = 0, 2.1 a solution
= F([, y, y’) u(t) = t2 is the only real zero of F and u
h(t,y) = (y - t2)24+‘. problem o
E2Y” = F(t, Y, Y’),
has by Theorem
above.
equation
E2Y” = (y - t2)24+ 1 (1 + y’y for 0 < t < 1 and integral is clearly (Q-stable with
the theory developed
L’(l,&) = I
y = y(t, E) for each E > 0 which satisfies for 0 < t 6 1 t2 < y(t, F) < t2 + (2&2)l’(2q+Q.
On the other hand since u’(0) = 0 the Robin
problem o
E2Y” = F(4 Y, Y’), -l:‘(O, 8) = A, for any value of A 2 0 has by Theorem O,
Y(l,&) = 1
3.1 a solution
Y = y(t,
E)
for each E > 0 which satisfies for
t2 < y(t, E) < t2 + u(t, E) + (2&2)“‘24+”
Singularly
perturbed
superquadratic
v(t, F) = 0
if
boundary
value problems
189
A = 0,
u(t, 8) = AE eerie
if
q = 0
and
A > 0,
and u(t, 8) = Az-‘q& l/(q+l) ( 1 + TE-l/(q+l)t)-l/qa
if
q > 1
and
A>0
for 7 = z(q) = [A2qq2q+2/(q + I)]1ic2q+2). Example
6.2. Consider
now the differential
equation
a2Y’ = (y2 - 9) (1 + y’4) = F(t, y, y’) where for convenience we allow t to be in the interval ( has only the straight lines ui(t) = t and u2(t) = --t as that ul is only stable on [0, l] while u2 is only stable on path which is stable on [ - 1, l] is the angular one u(t)
1,l). The reduced equation F(t, U, a’) = 0 differentiable real solutions and we note [ - 1, 01. Consequently the only reduced = ItI ; indeed, u is (II,)-stable there. Since
u’(O-) = - 1 < U’(o+) = 1 and u”-
0 on [ - 1, l]\(O)
the Dirichlet
problem -1
a2Y = F(r, Y, Y’), L’(--I,&) = 1, has by Theorem
2.3 a solution
L’(l,&) = 1
y = y(v(t,E) for each E > 0 which satisfies for - 1 < t < 1 ItI d L‘(t,E) < ItI + w(t, E)
where w(t, E) = (12)“3 &z’3(1 + (12)_ 1’3 &-2’31t/)-? On the other hand u’( - 1) = - 1 and u’(1) = 1 and so the Neumann e2L.” = F(4 Y, Y’), -y’(-
1, E) = A,
for values of A 2 1 and B 2 1 has by the analog E > 0 which satisfies for - 1 < t < 1
-1
problem
L”(l,&) = B of Theorem
2.3 a solution
Y = J@, E) for each
1t I < y(t, E) < I t I + u(t, &) + qt, E) + w(t, E). Here w is the function
just defined,
u(t,s)=G(t,~)=O
if
A=B=
1,
u(t, E) = (12/(A -
1))‘j3 (A -
1) ~~‘~(1 + ((A - 1)/12)‘13~-213(1 + t))-1’3
if
A > 1
iY(t,s) = (12/(B -
l))“3(B
1).s213(1 + ((B -
if
B > 1.
and -
1)/12)‘13~-213(1
- t))-1’3
190
F. A. HOWFS
Example 6.3. Consider finally the problem described in the Introduction liquid confined by a rigid wall at t = 0; namely
for the free surface of a
o
,2y” = y(1 + y’2)3’2, - ~‘(0, E) = tan 8
where -rc/2 < 8 < 7r/2 is the contact angle (measured from the horizontal axis). Suppose first that the liquid is water and therefore that 0 < 0 < 7r/2. Since u E 0 is an ($,)-stable solution of the reduced equation ~(1 + u’2)3!2 = 0 and -u’(O) = 0 < tan 13Theorem 5.1 allows us to conclude that this problem has a (unique) solution y = y(t, E) for each E > 0 such that for 0 < t -c cc 0 d y(t, c) d (tan 0) se-“”
(Uniqueness follows from the maximum principle; cf. [l l] or [23, Chap 11.) In particular, for each E > 0 lim
t+m
y(t, E) = 0,
a result which is usually assumed in finding the exact solution by quadratures (cf. [l; Chap. 11). If however the liquid is mercury then the contact angle is negative, that is, -n/2 < 0 < 0. Since -u’(O) = 0 > tan 0 we can apply the “concave” version of Theorem 5.1 to see that the problem has a unique solution y = y(t, E)for each E > 0 in this case also. In addition, for 0 < t < cc we have that (tan
e) Ee - ve <
y(t, E) < o-
and so for each E > 0 lim y(t, c) = 0
f-02
as before. 7. CONCLUDING
REMARKS
We consider finally (and very briefly) an example suggested to us by Perko [24] which shows one type of behavior that solutions of the problem E’y”
03
=
Y(O,4
w, Y) f(t, = 4
o
Y, Y'),
Y(L
4 = B
can exhibit when we remove the restriction that f z 1 > 0. The example is
PI)
E2Y" = Y(l
Ye2 E)=
0,
-
o
IY’l”)1
Y(L4
=t
where n is a positive integer greater than or equal to 3. It is not difficult to see that any solution y = y(t, E) of (D,) satisfies the a priori estimates (cf. [25]): O
and
0 < y’(t, E) < 1 for
0 < t < 1.
Singularly
perturbed
superquadratic
boundary
value problems
191
Moreover, 0,
O
t - $,
;
cr(t, E) = and &t, E) I=: it are bounding functions of (D,) and so by Heidel’s theorem [14] this problem has a (unique) solution y = y(t, E)for all E > 0 such that a@,E) d y(t, E) sCj?(t, E) on [0, I]. (Uniqueness follows from the maximum principIe; cf. [ll J or [23, Chap. 11.) With a little additional work one can in fact show that
lim y(t, 8) = u(t) = E-o+
0, i t - $
o
+rt1,
that is, y converges uniformly to the angular function
192
F. A. HOWES
23. PROTTJBM. H. & WEINBERCEXH. F., Muximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, New Jersey (1967). 24. PERKO L. M., Private communication. 25. DORR F. W., PARTER S. V. & SHAMPINEL. F., Applications of the maximum principle to singular perturbation problems, SIAM Rw. IS, 43-88 (1973). 26. HOWES F. A., Singularly perturbed boundary value problems with angular limiting solutions, Trans. Am. math. Sm., 241,155-182(1978).
SINGULARLY
&Applications. Vol. 3. No in Great Britain.
2. pp
PERTURBED VALUE
0362.546x/79/0301-Ol7OSO2.00/0
175-192.
SUPERQUADRATIC PROBLEMS*
BOUNDARY
F. A. HOWEST School of Mathematics,
University
of Minnesota,
Minneapolis,
MN 55455, U.S.A
f Received in revised form 7 March 1978) Key Words: Singular perturbation, greater than quadratic.
reduced problem,
differential
inequality,
rate of growth in the derivative
1. INTRODUCTION IN THIS
paper we study the existence
of solutions
of the two singularly
perturbed
boundary
value
problems o
s2Y’ = Mt, L’)j-(4 L‘,Y’) = F(4 L’,Y’), (P,)
b,~‘(l, F) + b&(1, E) = B,
a,L’(O, E) - Q&(0,&) = A,
and s2Y’ = F(4 L’,v’), (Pz)
o
U,L‘(O,&)- U,L”(O,&) = A,
for small positive values of the parameter s and for nonnegative values of the constants a,, a,, b, and b,. Our principal assumptions are that the function h has certain zeros which are stable in a sense to be described below and that the function f is strictly positive and grows faster than JJ” as /JJ’/ + co (that is, f is superquadratic). In the course of proving the existence of solutions of (Pi) and (P,) we will also derive rather sharp estimates on the behavior of these solutions which improve as F decreases to zero. As will become apparent below the assumption that f is strictly positive implies that the existence and the asymptotic behavior of solutions of (P,) and (P,) are governed by the nature of the solutions of the equation h(t, U) = 0. However the assumption that f is superquadratic has an equally significant effect on the types of solutions which these problems possess. Our interest in such problems was generated by the lack of any coherent theory for singularly perturbed superquadratic boundary value problems and also by the occurrence ofproblems ofthis type in several applied areas. For example (cf. [ 1; Chap. 11) the elevation y of the free surface of a liquid meeting a plane, vertical, rigid wall (t = 0) is described by a problem of the form (P,) with h(t, L‘) = L‘,f (t, y,y')= (1 + y’2)3!2, s2 = T/(pg), a, = O,a, = 1 andA = -tan @for0 < 8 < 42. Here p is the density of the liquid, g is the gravitational constant, T is the coefficient of surface
* Dedicated + Supported
to my father on the occasion of his seventieth birthday. in part by the National Science Foundation under grant 175
No. MCS 76-05979.
176
F. A. HOWES
tension and 8 is the contact angle, that is, the angle which the surface makes with the wall. If T is small then E = [T/(pg)] “’ is small and the problem for y is singularly perturbed. As a second example consider a liquid confined between two plane vertical walls at t = 0 and t = 1 or equivalently, a liquid contained in an open capillary tube. Then (cf. [2,3]) the elevation y of the free surface of the liquid satisfies a problem of the form (P,) with E and 0 as before and with a, = b, = 0, a2 = b, = 1 and -A = tan 8 = B for 0 < 0 < 71/2. Here however the right-hand side F is equal to F(t, y, y’, F) = (L. - &y’/[t(l + L.“)~‘*]) (1 + L.“)~‘*. Once again if the coefficient of surface tension T is small then E is small and the problem is singularly perturbed. (We will not consider functions such as F” in this paper per se although the results we obtain for the simpler problem (P,) can be extended to this more general setting with little difficulty.) Before commencing our study of the problems (P,) and (P,) we review briefly some of the literature in both the perturbed (0 < E 6 1) and the unperturbed (e = 1) cases. For the perturbed problem (P,) (and for its system analogs) Vasil’eva [4; Chap. 21 has given a comprehensive survey of the work done prior to 1963. The theory described there applies for the most part to the case of righthand sides F which grow no faster than y’2 although she does give several results for the Robin problem (that is, uz + bz > 0) in the case that F is superquadratic. Since then most writers in singular perturbation theory (see [5] and [6] for extensive lists of references) have concentrated on the case of F = @(Iy’ 1’). R ecent results of this author [7,8] show however that a general theory for the perturbed problem (P,) is possible although there are many details which have to be worked out. The unperturbed problem (P,) for the case of Dirichlet data (that is, a2 = b, = 0) has been the object of much study since the turn of the century. We mention only the fundamental papers of Bernshtein [9] and Nagumo [lo] and refer the reader to the paper of Hartman and Wintner [ 1 I], the survey article of Jackson [ 121 and the book of Hartman [ 13 ; Chap. 121 for modern treatments of the Bernshtein-Nagumo theory and many additional references. Indeed, to study the perturbed problems (P,) and (P,) in this paper we will use some natural extensions of Nagumo’s results which are given in a paper of Heidel [14]. Much less is known about the perturbed problem (P,) independently of the size ofthe righthand side F as a function of y’. The most complete treatment of this and related problems is contained in the doctoral dissertation of Lee [ 151 (cf. also [ 16, 171) for the case that F = O(lyf2 1). There is however a substantial literature on the unperturbed problem. We mention only the papers of Hartman and Wintner [1 I], Shcherbina, [18, 191 and Myshkis and Shcherbina [3] which the reader can consult for iurther references. The last two papers also contain a discussion of several physical phenomena which are described by solutions of (P,). Finally we note that a complete study of the one-dimensional capillary tube problem has been given by Johnson and Perko [2] and Perko [2Q] for both the finite and the semi-infinite interval. We take this opportunity to thank Professor L. Perko of Northern Arizona University for his extensive and most helpful correspondence with us on some of the problems discussed below.
2. THE
Consider
the Dirichlet E2y”
(D)
DIRICHLET
PROBLEM
problem =
h(t,
~‘1 fk
Y, $1
L’(O,E)= A,L’(l,&) = B,
=
F(t,
Y, Y’),
O
Singularly perturbed superquadratic boundary value problems
and the corresponding
reduced
problem
h(t, u) “m
4
4 = 0,
U(0) = A, The function the domain
h is assumed
177
to be continuous
o
U(l) = B. with respect to t and of class C(“) with respect to y in
Q(u(t,) = ((t, L‘): 0 d t < 1, IL’ - u(t)( < d] where 0 = 2q + 1 or y1for integral values of q 2 0 and IZ B 2, u = u(t) is a solution of (R,) and d is a small positive constant. Similarly the function f is assumed to be continuous in the domain _QJu(t)) = 9@(t)) x R1 and strictly positive there, that is, f(t, Y, Y’) 2 1 > 0
in Bl(Mt))
for a positive In order to describe how of the reduced in 9(u(t))
constant 1. deduce the existence of solutions of (D) for small positive values of c: and then to these solutions behave in the limit F, + 0’ we isolate the stable solutions u = u(t) problem (R,). Namely if the function h is of class Cczq+ ” (q 2 0) with respect to y we will say that u is &)-stable provided that
(i) d;,h(t,u(f))=O
I andO
forO
(ii) thtre exists a positive
constant
m such that
a y2qf1 h(t, y) 2 l-‘m2 Similarly provided
if h is of class P(n that
(i) a;,h(t, u(t)) > 0
> 0
for (t, y) in ~(u(L)),
> 2) with respect to Y in 9(u(t)) then we will say that u is (II”)-stable
for 0 < f d 1 and I G j 6 n constant m such that
1;
(ii) there exists a positive
ij;h(t, y) 2 l- ‘m2 > 0 Finally if h is of class C(“)(n 2 2) with respect provided that (i)
i3; h(t, u(t)) 3 0
for (t, y) in g(~(t)).
to y in 9(~(t)) we will say that u is (III,,)-stable
and 8; h(t, u(t)) < 0
for 0 d t 61
and 1 6 j,,, je d n - 1 where j,,o,) denotes an odd (even) integer; (ii) there exists a positive constant m such that a;h(t,y)
2 I-‘m2
> 0 or a;h(t,y)
d -l-‘m2
< 0
for (t,~)
in 9(~(t)) if n is odd or even, respectively. These definitions (Iu), (II,) and (III,) of “stability with respect to $’ have been used by the author in [7], [S] and [21] and they are based on earlier work of Boglaev [22]. For ease of presentation, in what follows we will simply say that h is a-smooth in 9(~(t)) if h is continuous with respect to t and of class C@) with respect to y in 9(ucr)). The results of this section extend the work of the author [21] to the case in which the function f is superquadratic in 9,@(t)), that is, f = O(l J’ I’) as 1y’ / --+ co for (t, y) in 9(u(t)) and v > 2.
178
F.
THEOREM 2.1. Assume (1) the reduced that U” 2 0; (2) the function superquadratic
A. HOWES
that
problem
(R,) has an (IQ)- or (II&stable
solution
u = u(t) of class P’[O,
l] such
h is a-smooth in g(~(t)) for c = 2q + I or n and the function f is continuous and in g,(~(t)); moreover, there is a positive constant I such that f > 1 > 0 in
9#(t)). Then there exists a positive constant s0 such that the problem whenever 0 < i: 6 F~. In addition, for t in [0,11 we have that
(D) has a solution
y = y(t, 6)
u(t) 6 y(t,c) < u(t) + (E$F-2)l’~
Proof of Theorem 2.1. To prove this theorem we will apply a generalization of the classical Nagumo theorem (cf. [IO, 12 Sec. 71) on second-order differential inequalities due to Heidel [ 141. Specifically if there exist C ‘2’-bounding functions a and /3 such that for 0 < I: < F~ a G P, andfor
a@, 8) d A < pco, e),
U(l, E) d B $ /3(1, E)
< t < 1, E2U” 2 F(L, M,a’),
E2pI < F(l, B, p’),
then Heidel has shown that the boundary value problem (D) has a solution y = y(t,c) satisfying Nagumo condition with cc
P(t, E) = u(t) + (2]m-2)1’o.
Here g = 2q + 1 or n and 1~is a positive constant such that 1~2 a!M for )Iu” IIm ,< hf. For ease of presentation let us set r = r(c) = (E~~vY~)~‘~. Clearly TV< b, $0, e) < A < p(O, 8), cc(1,E) d B < p(1, E)and EMU”2 F(t, a, a’) (since U” 3 0). The only verification remaining is the one showing that F’/?’ G F(t, p, ,Y): F(t, p, /I’) - E~P” = h(t, p) f(t, j3, p’) - FLU” o-l = (h(t, u) + c
l/o’!) a@,
u) I-J + l/b!)
j=l >
{l/(0!) I-‘m%+Y-2}
2 0
since 3’ > a!fM.
1 - s2M
dyl(t,
4) I-“} f(t,
p, p’) - E2zi’
Singularly
perturbed
superquadratic
boundary
179
value problems
Here (and throughout the paper) t = ((t, F) = (t, u + 0(/3 - u)) (0 < 0 < 1) belongs to Q(W) if E is sufficiently small. Consequently o! and p satisfy all of the required inequalities. The proof will be complete if we can show that whenever
E2L.l’= ~(t, y, y’)
and
Y < y < ,G on
J c [0, 11
follows because u’(O) Q y ‘(t, E) < u’(l) for any such on [0, I]. To see this note that ~(0, e) = u(0) = 40, E) and a(4 E) G ~(4 E) imply that ~‘(0, E) 2 u’(O,E) = u’(O), while y(1,~) = u(1) = cr(l,~) and c~(t,e) < y(t,e) imply that ~‘(1~6)G ~‘(1,E) = u’(1). Therefore u’(0) < y’(t, E) d u’(1) since y” 2 0 for CI< y G P. We conclude from Heidel’s theorem that the problem (D) has a solution with the stated properties. then
]y’(t, E)I < N on J x (0, ~~1. This
solution
Suppose next that the solution u = u(t) of the reduced problem (R,)is concave, that is, u” G 0. Then a result analogous to Theorem 2.1 holds if u is either (IJ- or (IIIJ-stable. The precise statement is contained in Theorem 2.2. THEOREM 2.2. Make the same assumptions (1) is replaced by (1’) :
(1’) the reduced problem that u” < 0. Then the conclusion
as in Theorem
(R,) has an (14)- or (III,,)-stable
of Theorem
2.1 with the exception
solution
that assumption
u = u(t) of class C”‘[O, l] such
2.1 is valid with the inequality
replaced
by
u(t) - (E2;~m-2)‘io < y(t, .7)6 u(t).
Proof of Theorem 2.2. This result is proved by making the change of dependent variable y -+ - y and then applying Theorem 2.1 to the transformed problem. Note that under this change of variable (Ia)-stability is left unchanged while &)-stability is transformed into (III&.tability. We remark that the above results are valid if the function u has only an absolutely continuous derivative on [0,11. In this case we assume that u” 2 0 where it exists (in Theorem 2.1) or that u” < 0 where it exists (in Theorem 2.2). Heidel’s result can be extended to AC”‘-bounding functions by arguing as in [12] which treats only the classical Nagumo condition. However it frequently happens that the function u is only piecewise differentiable on (0,l) as would be the case for example when two distinct solutions of h(t, U) = 0 intersect in (0,l) with unequal slopes (cf. [21]). Heidel’s theorem can also be extended to cover this case provided we impose the following additional conditions on the bounding functions c( and p. Namely if
u’(t;) # u’(t,‘)
for some t, in (0, 1)
then we require that a’(t;) d cr’(t,‘) and p(t;) 2 P’(t,‘). These inequalities follow essentially from the fact that the supremum of a family of lower solutions (that is, functions satisfying the c(inequalities) is again a lower solution while the infimum of a family of upper solutions (that is, functions satisfying the B-inequalities) is again an upper solution. As regards the boundary value problem (D) we will see in the next two theorems that the occurrence of a finite jump in u’(t) has the effect of introducing an interior layer into the derivative of any solution. For simplicity we consider only the case of a single jump at the point f, in (0, 1).
F. A.HOWES
180 THEOREM
2.3. Assume
that
(1) the reduced problem (R,) has an (I )- or (II”)-stable solution u = u(t) of class 15~’ on [0, I]\{ toI such that u’(t;) < u’(tl), U” 2 0 on 10, l]\{t,> and u”(ti) 2 0; (2) the function h is o-smooth in g@(t)) for 0 = 2q + 1 or n and the function fis continuous and superquadratic in 5@(t)); moreover, there is a positive constant 1 such that f 3 1 > 0 in -%(4t)). Then there exists a positive constant e0 such that the problem whenever 0 < E < co. In addition, for t in [0, l] we have that
(D) has a solution
4’ = y(t, e)
u(t) < y(t, E) < u(t) + w(t, E) -f (+r-y where CJ= 2q + 1 or n, 1~2 a! I/ u” Ilooand w(t, E) = +I-~ ~(u’(f+) - u’(t;)) exp[ - rnE- ’ 1t - t, I] w(t,e) = ;?-I
-t ,c-l’(q+l)lt
qE1!(q+l)(u’(tO+)- u’(t;))(l
(for T(q) = [(+(u’(t,‘) - u’(t;)))2q q2q+2m2/((q w(t,c) = +T-‘((n -
We remark satisfies
that
w(t, e) is a solution
(n -
q = 0,
- t,I)-l/(q+l)
if
+ 1)(2q -t 1)!)]1’(2q+2’)
1)/2)~~~‘“+‘)(u’(t0+) - u’(t;))(l
(for 7(n) = [(+(u’(t,‘) - u’(t;)))“-’
if
q 2 I and
+ ~.~-~~@+l)lt - t,l)-2i(n-1)
if ~7= n
I)n+1m2/(2n(n + l)!)]‘“““‘)
of the differential
w’(t;, E) = $n’(t,‘)
equation
- u’tt;))
w’(t,+, 8) = -+(u’(t,‘)
s2w” = (m2/o!) w0 which
and
- u’(tJ).
2.3. Define for E > 0 and 0 d t < 1
Proof of Theorem
Gl(t,E) = u(t), P(t, F) = u(t) + w(t, e) -f (E2ym-2)“” where 1’ is a positive constant such that 1~2 o!M for 11 u” Ilrn G hf. Since u’(t;) < u’(t,‘) by assumption (1) we have that cc’(t;) < cr’(t,‘). As regards /I it follows by our choice of w that p’(t;) It only remains have that
to verify
that
= P’(to’) = $(u’(t,,
+
u’(t,‘,).
s2p” < F(t, p, /I’). Let us set I = I%) = (s21m2)1’o,
then
we
F(t, p, B’) - ~~8” = h(t, p) f(t, j3, p’) - t2P” o-l h(t, u) -f C (l/j!) a;h(t, u)(w + r)j
= i
j=l
2 (I- ‘m2/a!)(w”
+
ry 1 -
+ (l/a!)
a;h(t, t)(w
c2A4 -
E’W”
+ I-‘bll f(t, p, p’) - ~~11”- dew”
181
Singularly perturbed superquadratic boundary value problems
= (m2/a!) w” - E2W’I+ E2J’jO!- E2M
> 0
since
&v” = (m’/cr!) wO‘ and
‘)J2 a!M.
Thus c1and p satisfy the required inequalities. Since it follows as in the proof of Theorem 2.1 that U’(0) ,< $(t, E) < U’(1) for any solution y of s2$’ = F(t, JJ,y’) with d < JJ < p we conclude from the extended version of Heidel’s theorem that the problem (D) has a solution with the stated properties. The analogous result in the case that a”< 0 holds provided we replace (I$)-stability with (IIIJ-stability. For completeness we state this remark in the form of a theorem which follows from Theorem 2.3 via the change of variable 4: --t -y. THEOREM 2.4. Make the same assumptions as in Theorem 2.3 with the exception that assumption (1) is replaced by (1’):
(1’) the reduced problem (R,) has an (I )-or {III~)-stable solution u = u(t) of class C’*‘on [0, l]\(t,~ such that u’(t;) > u’(t,f), u” < 0 on 10, l]\(r 0J and d’(td) < 0. Then the conclusion of Theorem 2.3 is valid with the inequality replaced by u(t) + 1%(t,E) - (&m-2)“”
< L’(t,E) < u(t).
We close this section with four remarks on the theory developed here. Remurk 2.1. It is sometimes necessary to extend the definitions of (Iy)-, (II,,)- and (III,,)-stability to the case when Z;h is strictly bounded away from zero only along the junctionu (cf. [21]). The extended definitions are as follows.
Let u = U(C)be a solution of h(t, U) = 0. The function u is called (I;)-stable if (i) J;,h(t, u(t)) 5 0 for 0 G t < 1 and 1 <
> 0
for
0 < t < 1 and
8fq+* h(t, y) > 0
(t, y)
in
5+4(t)).
for
Similarly the function u is called (Uhf-stable if (i) d:,h(t,u(t))>O
for
O
1
and
1
I;
(ii) there exists a positive constant m such that a;h(t, u(t)) 3 /-‘mz > 0 for d;+lh(t, y) 2 0 Finally the function (i) drh(t, u(t)) & 0
for
(t, y)
0 G t G 1 and in L+(t)).
II is called (III~~-stable if and
i;h(t,
u(t)) d 0
for 0 < t G 1 and 1 < jO,je < n - 1 where j&J denotes an odd (even) integer;
182
F. A. How=
(ii) there exists a positive constant d;h(t, u(t)) > l-‘m2 > 0 and
m such that d;“h(t, y) < 0
g@(t)) or a;h(C, u(t)) d -l- ‘m2 < 0 and f;+‘h(t, .5@(t)) if n is odd or even, respectively.
y) 2 0
for for
(t, Y)
in
(t, 4’) in
Theorems 2.1 and 2.3 are clearly valid with (Iq)- or (II”)-stability replaced by (Ii)- or (II;)-stability, respectively. Similarly Theorems 2.2 and 2.4 hold with (Ii)- or (III:)-stability, respectively. Since the extended definitions are somewhat technical they are not included in the main body of the paper. We note here that the results of the following sections are also valid with (I&, (II”)- and (IIIJstability replaced by their primed counterparts as appropriate. Remark 2.2. The restriction that the solution u of the reduced equation be convex or concave can be removed in the case of (14)-stability but not in the case of (II,,)-stability. For example, in Theorems 2.1 and 2.3 if u is an (14)-stable solution of (R,) which is not necessarily convex then a(t, 9) = u(t) - (,52ym-2)‘1(2q+ ‘) (for 1’ 2 (2q + 1) ! I(u” /I,) is clearly a lower solution 2.1 to show that a solution y of E2y” = F(t, y, y’) has a uniformly
bounded
derivative
of(D). We can argue as in the proof ofTheorem
with
c( B 4’ < p
and so the results follow once more from Heidel’s theorem.
Remark 2.3. Since the function f is assumed to be strictly positive in the region of interest the solutions u of the reduced problem (R,) are roots of the equation h(t, u) = 0. As such it would be less restrictive to impose inequality constraints on u(0) - A and u(l) - B rather than to require that u(0) = A and u(l) = B. However in the presence of superquadratic nonlinearities it does not appear possible to relax this requirement. This follows on the one hand from the fact that the method of proof used above breaks down unless u(0) = A and u(1) = B and on the other from the observations of Vishik and Liusternik (cf. [4; Chap. 21). In short these authors showed that solutions of the Dirichlet problem cannot exhibit boundary layer behavior in the presence of superquadratic nonlinearities. If the zeros u(t) of h(t, u) do not satisfy the boundary conditions exactly or are unstable then it is likely that solutions of the full problem (D) either do not exist for small values of c: or (if they do) are not of bounded t-variation as E -+O+ (cf. [7]). Remark 2.4. Our treatment of the interior layer problem (Theorems 2.3 and 2.4) deserves a brief comment. In Theorem 2.3 for example the solution u of the reduced problem satisfies u’(t;) < u’(t,‘) and so the interior layer correction term w must be chosen so that p(t;) 2 p’(t,‘). Actually for the function w constructed in the proof of Theorem 2.3 we have that @‘(t;) = p(ti). This function w is uniformly small on [0, 11; however, w'(tt , E)is of order U( 1). Thus w serves to smooth out the discontinuity in the derivative of u in the sense that u’(t) + w’(t, E) is smooth at t = t,. 3. ONE
Consider
TYPE
OF
ROBIN
PROBLEM
next the problem E2JY = h(t, L’)f(t, y, y’) = F(t, y, y’),
0 < t < 1,
183
Singularly perturbed superquadratic boundary value problems
(E)
Jl(l,E) = B,
q(0, t;‘) - y’i0, E) = A,
where a is a nonnegative constant. Since the derivative of y is prescribed in the boundary condition at f = 0 the appropriate reduced problem corresponding to (E) is (cf. [S]) (R,)
U(l) = B.
o
h(r, u) f(r, u, u’) = 0,
We require of the functions h and j’ the same properties as in the previous section and we will again restrict attention to stable solutions of (Rr). The definitions of the various types of y-stability are exactly the same as above. We note that a solution u = u(t) of (Rr) is only required to satisfy the right-hand boundary condition and as will become clear momentarily we must require that au(O) - u’(O) < A
if
a” > 0
au(O) - U’(O)2 A
if
U” d 0.
or that The first two theorems are the analogs of Theorems 2.1 and 2.2. THEOREM3.1.
Assume that
(1) the reduced that U” 2 0 and (2) the function superquadratic
(R,) has an (IJ- or (II,)-stable solution u = u(t) of class C2’[0, l] such au(O) - u’(0) ,< A; h is e-smooth in &J(u(t))for (r = 2q -+ 1 or n and the function f is continuous and in .Q:‘,(u(t)):moreover, there is a positive constant 1 such that f b I > 0 in
problem
I&). Then there exists a positive constant so such that the problem (E) has a solution Y = Y(F,R) whenever 0 < E < R,,.In addition, for t in [0, 11 we have that u(t) < y(t, E) < U(t) .+ U(t,E) where d =29 u(t, R) 5 0
(E2yR2-2)1’”
+lorn,y>,a!//u”//,and
if
au(O) - u’(0) = A,
u(t, E) = m- 1E(A - au(O) + u’(0)) exp[ - me- 1t] ~(t, F) = T- &l!(q+
(for
+
if
q = 0
“(A - ff~(O) 4- ~‘~0))~~ + zc- l!fq+lJt)- “*
r(q) = [(A - uu(O)-+ u’(0))24q2q+2m2/((q
+ 1)(2q
if 9 3 1 and
au(O) - u’(O)< A
+ J)!)]1”2q+2))
and c(t, s) = r-‘((~3 - 1)/Z) E~~(*+~)(A - au(O) + ~‘(O)~(l+ TE-~‘(~+~)t)-l’(tt+l)
if
u = n
and au(O) - u’(0) < A (for We note
z(n) = [(A - au(O) + u’(O)),-* (n - l)“+’
that ~(t, e) is a solution of the differential equation I
F*U”
=
which satisfies ~‘(0,a) = -(A - au(O) + u’(0)).
(m2/o!)
flu
m2/(2”(n
.-i- l)!)]licn+l)),
184
F. A. HOWES
Proof of’ Theorem 3.1. The proof is only slightly different from the proof of Theorem 2.3. This follows because Heidel’s theorem [ 141 as stated in the proof of Theorem 2.1 also applies to boundary conditions of the form
a,y(O, e) - a,y’(O, E) = A, b,y(l, for u2, b, 3 0 and UT + uz > 0, bf -t b: > 0. Namely the inequalities G!G P,
a,r(O,e)
b,a(l,E)
-t b,a’(l,c)
E) -+ b,y’(l, E) = B, if the bounding
functions
CIand p satisfy
- u2cQo, E) < A < U$(O,E) - a,P’(O,E), d /I < b,P(l, E) .f b,B’(l,c)
and 8%” > F(t, a, cr’), E’P,’ d F(t, fl, p’) then the problem ,2y’l = F(t, L’,y’), a,y(O, e) - a&(0,
o
b,y(l,e)
+ b,y’(l,s)
has a solution y = y(t, E) with CI < y < ,6’provided that F satisfies a generalized with respect to c1and ,0. Define for F: > 0 and 0 d t d 1
= B Nagumo
condition
cl(t, e) = u(t), P(t, E) = u(t) -t v(t, e) -t (c2pm-2)“u. Here c = 24 + 1 or n and 1~is a positive I- = L-(E) = (~2ym-2)1i0. Clearly
such that 1’ 3 o!M for I/u” II?: < M. Let us set
constant
acc(0, E) - r*‘(O,R) d A d @(O, 8) - P’(O, e)
!x < B,
and a( I, E) < B < ,B(l, E) since c’ > 0 satisfies ~‘(0, F) = -(A - au(O) + u’(O)). Finally ~(r, a, cx’)since U” > 0 and so it remains to verify that a2p” G F(t, fi, p’):
E~CZ”2
F(t, ,!I, p’) - E~/I” = h(t, /?) f(t, p, B’) - c2p” 0-l
h(t, u) t
=
i
C (l/j!) a;,h(t, U)(L) + l-)‘+ (l/o!) d;h(t, 5)(u -f rYj 1 f(t, p, P’) ,=l - ,&’ _ &’ t
3 (l- ‘m2/a!)(F =
(m2/g!)
2 0 Therefore that if
u”
since
-
c2v”
I -
e2fM
=
-
c2v“
.t (.~~]))/a! - E2A4
E~V” = (m2/a!)ti0
E and p satisfy the required
E2y”
ry
inequalities.
F(t. y, y’)
and
1~2 o!M.
To conclude
and CI < y G B
the proof it is sufficient
to show
then
W(0) - A < y’(t, 8) d U’(1) since this inequality establishes that F satisfies a generalized c( and fl (cf. the proof of Theorem 2.1).
Nagumo
condition
with respect to
Singularly
perturbed
superquadratic
boundary
problems
value
18.5
Clearly ~‘(0, t) 2 au(O) - A since 4”(0,E) = aJ(o, F) - A 2 ~~~0) - A
(a 2 O),
and since 4”’ > 0 for c1d 4‘ d j? it follows that au(O) - A d y’(t, E) G y’(l, ST).
However ~(1, e) = u(l) = ~(1, a) and y >, x imply that ~‘(1, E) d ~‘(1, E) = u’(1) and this gives the desired inequality. We conclude from Heidel’s theorem that the problem (E) has a solution with the stated properties. If the solution u = u(r) of (R,) is concave, that is, ifu” d 0, then a similar result holds provided we replace (I&J-stability with (III,,)-stability and provided we require that au(O) - u’(0) 2 A. The precise result is stated in the next theorem which follows from the previous one via the change of variable y + - y. THEOREM 3.2. Make the same assumptions as in Theorem 3.1 with the exception that assumption (1) is replaced by (1’) :
(1’) the reduced problem (Rn) has an (I,)- or ~IIr~)-stable solution u = u(t) of class P’[O, I] such that u” $ 0 and au(O) - u’(0) 2 A. Then the conclusion of Theorem 3.1 is valid with the inequality replaced by
As in the case of the Dirichlet problem (D) it often happens that solutions u of the reduced problem (RE) have jump discontinuities in their first derivatives, that is, for some t, in (0, 1) u’(t;) + u’(r,‘). By arguing as in the proof of Theorems 2.3 and 3.1 we can establish the following analog of Theorem 2.3. (The formulation of the analog of Theorem 2.4 is straightforward and is left to the reader.) THEOREM 3.3.Assumethat
(1) the reduced such that u’(t;) (2) the function superquadratic ~~(~~t)).
problem (R,) has an (I f- or (II,)-stable solution u = u(t) of class C”’ on [0, I]\(t,). < u’(t,i), 11” < 0 on 10, l]\(t,>, u”(tz) G 0 and au(O) - u’(0) L4; h is o-smooth in S(u(t)) for (T= 2q + 1 or rr and the function f is continuous and in gl(u(t)); moreover, there is a positive constant 1 such that f 2 1 > 0 in
Then there exists a positive constant e0 such that the problem (E) has a solution Y = y(t, E) whenever 0 < F < ee. In addition, for t in [0, I] we have that u(t) < yft, F) < uit) + oft, F) + wtt, 4 +
fe2Ym2Pu
where (T = 2q + 1 orn,y > a! /Iu”/)~ and the functions t:and w are as in Theorem 3.1 andTheorem 2.3, respectively.
F. A HOWES
186
The results of this section
are easily seen to apply to the “reflected” E2fl
=
44
Y)
.I-(&
L’,
problem
O
Y’),
(E’) Y(O,
&)
=
by(1,
4
E)
+
y’(1, F) = B, f + 1 - t in (E’) and applies
for nonnegative values of b. One simply makes the change of variable the previous theory to the transformed problem. 4. ANOTHER
It is now a straightforward
matter
TYPE
OF ROBIN
to consider
the problem o
&*Y” = 44 Y) f(4 L’,Y’), (G)
PROBLEM
ay(0, E) - y’(0, F) = A,
by(1, E) + y’(l,s)
= B,
for nonnegative values of the constants a and b. Since the derivative of ): is prescribed at both endpoints it is reasonable to consider stable solutions u = u(t) of the corresponding reduced equation o
44 u) f(4 u, u’) = 0,
(R,)
which may satisfy neither of the boundary conditions in (G). However we find it necessary to restrict such functions u so that
as in the previous
au(O) - u’(0) d A
and
bu(1) + u’(1) < B
if
U” > 0
au(O) - u’(0) > A
and
bu(1) + u’(l) > B
if
u” < 0.
section
or
The expected results for the problem (G) are valid if u is (14)- or (II,,)-stable and U” > 0 or if u is (I )- or (III”)-stable and U” < 0. In the first case (u” 2 0) we note that if y = y(t, F) is a solution of $y” = h(t, y) f(t, y, y’) which satisfies TV< y < p for bounding functions c( and /I then au(O) - A < y’(t, E) < bu(1) + B, while in the second case (u” < 0) the inequality
is of the form
bu(1) + B < y’(t, E) < au(O) - A. As an illustration of the analog of Theorems is left to the reader. THEOREM 4.1.
Assume
2.1 and 3.1 we state the following
result whose proof
that
(1) the reduced equation (R,) has an (Iq)- or (II,,)-stable solution u = u(t) of class C(*)[O, l] such that au(O) - u’(O) < A, bu(1) + u’(1) 9 B and U” 3 0; (2) the function h is o-smooth in Q(u(t)) for 0 = 2q + 1 or n and the function f is continuous and superquadratic in Ql(u(t)); moreover, there is a positive constant 1 such that f 2 1 > 0 in 2,
(44).
Then there exists a positive constant E,, such that the problem whenever 0 < B < E,,. In addition, for t in [0, l] we have that U(t)
d
y(t,
E)
d
U(t)
+
fJ(t,
8)
+
fi(t,
E)
+
(&2y62)“0.
(G) has a solution
y = y(t, F)
Singularly
perturbed
superquadratic
boundary
value problems
187
Herea=2q+ lorn, y 2 a! I)~“lllo’ r(r, E)is defined as in Theorem 3.1 and ii(t, E) = ~(1 - t, E) with (A - au(O) + u’(0)) replaced by (B - bu(1) - u’(1)). We remark that u and ii both satisfy the differential equation u’(O,&)= -(A - au(O) + d(O)) and E’(l,&) = B - k(l) - u’(1). 5. THE
SEMI-INFINITE
INTERVAL
E’Z” = (m2/a!)zb and that
PROBLEM
We consider next the following problem on the half-line [0, CD) o
E2Y”= h(t, Y) f(4 y, Y’),
(H)
U,Y(O,&)
-
U,L”(O,&)
= A,
for nonnegative values of the constants a, and a,. If a2 = 0 we assume without loss of generality that oI = 1 and it is natural to associate with (H) the reduced problem h(4 4 f(L 4 u’) = 0,
o
u(0) = A.
On the other hand if cl2 > 0 then since ~‘(0, F)is prescribed (cf. Section 3) the appropriate reduced problem is simply the equation h(t, u) f(t, u, u’) = 0.
(R,)
The theory developed above carries over with little change to the problem (H) because Heidel’s theorem is easily extended to problems on the half-line (cf. [12]). In particular the bounding functions a and /? are required to satisfy the same inequalities on [0, co) and the function F(t, y, y’) = h(t, y) f(t, y, y’) is required to satisfy a generalized Nagumo condition on each compact subinterval of [0, co). Under the additional assumptions stated above the problem
Y” = F(4 Y,Y’), u,Y(~)
- a,~‘@)
o
has a solution J = y((t) such that a(t) d y(t) < j?(t) for 0 < t < co. This inequality can be used to estimate the behavior of a solution near t = 03 if p(t) - a(t) is close to zero for such values. For the problem (H) it turns out (not surprisingly) that up to terms of order O(E’!“) the behavior of solutions for large values oft is determined by the behavior of the reduced solutions U. As in the previous section it would be tedious to write down the analogs of all the theory developed above and so we content ourselves with stating only the analog of Theorem 3.1 for the case that a, = 1 (and a, = 0). For convenience we define the region &(u(t)) to be the region g(u(t)) defined above with t in [0, co). Then @,(u(t)) = 9(u(t)) x RI. The definitions of y-stability are similarly extended to the half-line [0, co). THEOREM
5.1. Assume that
(1) the reduced equation (R,) has an (IB)-or (I&)-stable solution u = u(t) of class C’*‘[O, co) such that U” 2 0, au(O) - u’(0) 6 A a_nd Ij u” 11 m < co ; (2)‘the function h $a-smooth in 9(u(t)) for B = 2q + 1 or n and the function f’ is continuo_us and superquadratic in gd,(u(t)); moreover, there is a positive constant 1such that ,f 2 I > 0 in gI(u(t)). Then there exists a positive
constant
&0 such that the problem
(H) has a solution
y = y(t, E)
188
whenever
F.
A. HOWB
for t in [0, co) we have that
0 < F < Ed. In addition,
U(t) < L’(t,E) < U(t) + U(t, E) + where CJ= 2q + 1 or n, If the function has this property
(E+-
y 2 a! 1)u” 11 m and u(t, 8) is defined
2)1’u
in Theorem
u has a iimit at t = 00 then it is clear that the solution also. We isolate this remark as a Corollary.
COROLLARY 5.1. In addition
to the assumptions
of Theorem
u0 = lim u(t) t+cO Then the problem (H) has a solution lim y(t, E) exists and satisfies f’rn
y = y(t,
3.1. J of(H) just constructed
5.1 suppose
exists.
for each F > 0 sufficiently
E)
that
small such that L.~(E)=
u0 d L.&E) d U* + (&2ym-2)“0.
6. EXAMPLES In this section
Example
we discuss several examples
6.1. Consider
first the differential
which illustrate
values of q > 0. The function
13~~+lh = (2q + l)! P
for
Since u(O) = 0, u(l) = I and u” = 2 the Dirichlet
Y(0,4 = 0, 2.1 a solution
= F([, y, y’) u(t) = t2 is the only real zero of F and u
h(t,y) = (y - t2)24+‘. problem o
E2Y” = F(t, Y, Y’),
has by Theorem
above.
equation
E2Y” = (y - t2)24+ 1 (1 + y’y for 0 < t < 1 and integral is clearly (Q-stable with
the theory developed
L’(l,&) = I
y = y(t, E) for each E > 0 which satisfies for 0 < t 6 1 t2 < y(t, F) < t2 + (2&2)l’(2q+Q.
On the other hand since u’(0) = 0 the Robin
problem o
E2Y” = F(4 Y, Y’), -l:‘(O, 8) = A, for any value of A 2 0 has by Theorem O,
Y(l,&) = 1
3.1 a solution
Y = y(t,
E)
for each E > 0 which satisfies for
t2 < y(t, E) < t2 + u(t, E) + (2&2)“‘24+”
Singularly
perturbed
superquadratic
v(t, F) = 0
if
boundary
value problems
189
A = 0,
u(t, 8) = AE eerie
if
q = 0
and
A > 0,
and u(t, 8) = Az-‘q& l/(q+l) ( 1 + TE-l/(q+l)t)-l/qa
if
q > 1
and
A>0
for 7 = z(q) = [A2qq2q+2/(q + I)]1ic2q+2). Example
6.2. Consider
now the differential
equation
a2Y’ = (y2 - 9) (1 + y’4) = F(t, y, y’) where for convenience we allow t to be in the interval ( has only the straight lines ui(t) = t and u2(t) = --t as that ul is only stable on [0, l] while u2 is only stable on path which is stable on [ - 1, l] is the angular one u(t)
1,l). The reduced equation F(t, U, a’) = 0 differentiable real solutions and we note [ - 1, 01. Consequently the only reduced = ItI ; indeed, u is (II,)-stable there. Since
u’(O-) = - 1 < U’(o+) = 1 and u”-
0 on [ - 1, l]\(O)
the Dirichlet
problem -1
a2Y = F(r, Y, Y’), L’(--I,&) = 1, has by Theorem
2.3 a solution
L’(l,&) = 1
y = y(v(t,E) for each E > 0 which satisfies for - 1 < t < 1 ItI d L‘(t,E) < ItI + w(t, E)
where w(t, E) = (12)“3 &z’3(1 + (12)_ 1’3 &-2’31t/)-? On the other hand u’( - 1) = - 1 and u’(1) = 1 and so the Neumann e2L.” = F(4 Y, Y’), -y’(-
1, E) = A,
for values of A 2 1 and B 2 1 has by the analog E > 0 which satisfies for - 1 < t < 1
-1
problem
L”(l,&) = B of Theorem
2.3 a solution
Y = J@, E) for each
1t I < y(t, E) < I t I + u(t, &) + qt, E) + w(t, E). Here w is the function
just defined,
u(t,s)=G(t,~)=O
if
A=B=
1,
u(t, E) = (12/(A -
1))‘j3 (A -
1) ~~‘~(1 + ((A - 1)/12)‘13~-213(1 + t))-1’3
if
A > 1
iY(t,s) = (12/(B -
l))“3(B
1).s213(1 + ((B -
if
B > 1.
and -
1)/12)‘13~-213(1
- t))-1’3
190
F. A. HOWFS
Example 6.3. Consider finally the problem described in the Introduction liquid confined by a rigid wall at t = 0; namely
for the free surface of a
o
,2y” = y(1 + y’2)3’2, - ~‘(0, E) = tan 8
where -rc/2 < 8 < 7r/2 is the contact angle (measured from the horizontal axis). Suppose first that the liquid is water and therefore that 0 < 0 < 7r/2. Since u E 0 is an ($,)-stable solution of the reduced equation ~(1 + u’2)3!2 = 0 and -u’(O) = 0 < tan 13Theorem 5.1 allows us to conclude that this problem has a (unique) solution y = y(t, E) for each E > 0 such that for 0 < t -c cc 0 d y(t, c) d (tan 0) se-“”
(Uniqueness follows from the maximum principle; cf. [l l] or [23, Chap 11.) In particular, for each E > 0 lim
t+m
y(t, E) = 0,
a result which is usually assumed in finding the exact solution by quadratures (cf. [l; Chap. 11). If however the liquid is mercury then the contact angle is negative, that is, -n/2 < 0 < 0. Since -u’(O) = 0 > tan 0 we can apply the “concave” version of Theorem 5.1 to see that the problem has a unique solution y = y(t, E)for each E > 0 in this case also. In addition, for 0 < t < cc we have that (tan
e) Ee - ve <
y(t, E) < o-
and so for each E > 0 lim y(t, c) = 0
f-02
as before. 7. CONCLUDING
REMARKS
We consider finally (and very briefly) an example suggested to us by Perko [24] which shows one type of behavior that solutions of the problem E’y”
03
=
Y(O,4
w, Y) f(t, = 4
o
Y, Y'),
Y(L
4 = B
can exhibit when we remove the restriction that f z 1 > 0. The example is
PI)
E2Y" = Y(l
Ye2 E)=
0,
-
o
IY’l”)1
Y(L4
=t
where n is a positive integer greater than or equal to 3. It is not difficult to see that any solution y = y(t, E) of (D,) satisfies the a priori estimates (cf. [25]): O
and
0 < y’(t, E) < 1 for
0 < t < 1.
Singularly
perturbed
superquadratic
boundary
value problems
191
Moreover, 0,
O
t - $,
;
cr(t, E) = and &t, E) I=: it are bounding functions of (D,) and so by Heidel’s theorem [14] this problem has a (unique) solution y = y(t, E)for all E > 0 such that a@,E) d y(t, E) sCj?(t, E) on [0, I]. (Uniqueness follows from the maximum principIe; cf. [ll J or [23, Chap. 11.) With a little additional work one can in fact show that
lim y(t, 8) = u(t) = E-o+
0, i t - $
o
+rt1,
that is, y converges uniformly to the angular function
192
F. A. HOWES
23. PROTTJBM. H. & WEINBERCEXH. F., Muximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, New Jersey (1967). 24. PERKO L. M., Private communication. 25. DORR F. W., PARTER S. V. & SHAMPINEL. F., Applications of the maximum principle to singular perturbation problems, SIAM Rw. IS, 43-88 (1973). 26. HOWES F. A., Singularly perturbed boundary value problems with angular limiting solutions, Trans. Am. math. Sm., 241,155-182(1978).