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On the continuous dependence of solutions to nonlinear problems in Hilbert space
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
80,
305-3
11 (198 1)
On the Continuous Dependence of Solutions Nonlinear Problems in Hilbert Space
to
M. ARON* Department of Mathematics, University of Strathclyde, Glasgow, Scotland Submitted by W. F. Ames
1. INTRODUCTION Various physical problems lead to the study of equations of the form Ati = qt>u,
t > 0,
(l-1)
with 40) = u,,
(1.2)
where A and Y(t) are operators on some suitable specified subspace of a Hilbert space R and ti is the derivative with respect to the time t of the vector-valued function u(t) E GP’. It is known that the problem (l.l), (1.2) is not, in general, well posed in the senseof Hadamard [ 11. For example, solutions to (l.l), (1.2) might not depend continuously on the initial data u,,. It is then necessary to know under which conditions continuous dependence can be recovered. An investigation of John (21 has shown that in certain situations it is sufficient to impose a uniform bound on solutions to achieve this. In some of these cases continuity must be replaced by the corresponding concept of Holder continuity. In the last years a number of workers 13-101 have given numerous examples of this type of continuous dependenceand physical justification for it. The following simple example, considered in [9], may provide an intuitive idea of Holder stability and the situations in which it might be applicable. Let the solutions of a linear system be such that their time dependenceis of the type e”‘. Then, the solutions which are bounded at t = t1 would be of the form emnt’-rl) and thus remain arbitrarily near the null solution for t E ]O, t,) but not for t E [0, r,]. It results that there exists continuous dependenceon the initial data in all compact sub-intervals of [0, tt) but not on the interval [0, t, J itself. -* Present England.
address:
Department
of Mathematics,
Plymouth
Polytechnic,
Plymouth,
Devon.
305 409 ‘X012 2
0022-247X/8
l/040305-07$02.00/O
Copynght ‘C 1981 by Academic Press, Inc. A,, riehts of rrnrd~wtinn in ~lnsr C,T.r- ..,-..-..-A
M.ARON
306
In the present paper we are concerned with the Holder continuous dependence on the initial data of solutions to the problem (l.l), (1.2). More precisely, we show that under certain conditions imposed on A, Y, and u,,, there exists, for each suffkiently regular solution u of (l.l), (1.2), a time interval I, = (0, T,) such that in every compact sub-interval of I, the solution ZJ depends Holder continuously on u, provided it belongs to the class of solutions which admit of a uniform bound (in a sense which will be made precise later). The technique used in proving this result is that of logarithmic convexity (see, e.g., [9, 11, 121).
2. CONTINUOUS
DEPENDENCE
Let SF be a real Hilbert space provided with the norm /I . (1induced by the scalar product (., .I. Let g CR be a dense domain and let A: G -+Z be a linear, symmetric, time-independent operator. Assume that A satisfies b&j
> 0,
for all
24E G?!,24# 0.
Assume also that the operator Y(t): GJ -+R
(2.1)
satisfies (2.2) (2.3)
and g(t) exists in the strong sense for t E [0, T]. Let u E g, u f 0 be a strong solution of the problem (l.l), (1.2) (i.e., a solution for which ti exists in the strong sense, is continuous, and is 9 valued). Define F(t) = [u, Au],
2.4 E g,
t E [O, z-1,
(2.4)
Using (1.1) and the symmetry of A we see that
Our hypotheses imply that f is continuously differentiable with respect to t
SOLUTIONS TO NONLINEAR
307
PROBLEMS
in IO, T) and also that f(t) > 0, t E (0, r],f(O) = 0. We shall prove that, in this case, there exists an interval I = [O, r,l, 7’, < T, throughout which
If f(O)# 0 the assertion follows by continuity. If j(O) = 0 we see (by appplying the mean value theorem to the function f) that in every interval (0, ?) with i< T either f z 0 or there exists at least one point t* at which f(r*) > 0. The continuity reveals the existence of an interval (t,, f2) c (0, 6, t” E (tl , t2) throughout which f(t) > 0 and either t, = 0 or f(t,) = 0. If f(r) > 0 for t E [0, t,] we may take T,, = t, and the assertion is proved. Assume then that there exists a point to E (0, tl) such that f(t”) < 0. It then follows that for every interval (0, i) there exists a sub-interval (t,, t:!), where f > 0 and also a sub-interval (tj, t4), to E (f3, f4), where f < 0. Therefore the point I = 0 cannot be a point of (strict) extremum for $ Thus, we have reached a contradiction which proves our claim. By meansof expansions in finite Taylor series,or alternatively by Jensen’s inequality [ 131, we obtain from (2.7) the inequality
F(t) < F(0)‘f”-f”‘0F(To)u70,
f E (0,
(2.8)
To).
DEFINITION. If, at time To, the solution u to the problem (1.1) (1.2) is uniformly bounded in the sensethat
(2.9)
F(To) < M,
for some positive finite constant M, then we say that u belongs to the class M and denote u E M. For u E M, we deduce from (2.8) that
F(t) GM’-“[F(O)]“,
6= 1 -t/T,,
0<6<
1.
(2.10)
The last relation implies the Holder continuous dependenceof the solution u on the initial data u,, in the measure F(t), on every compact sub-interval of I, = IO, To). Thus, we have the following. THEOREM. Let the operators A, P(t), and the function u. in (l.l), (1.2) satisfy (2.1)--(2.3). Assume that g(t) exists (in the strong sense) in some time interval [0, T]. Then for each strong solution u E &j there exists a time interval I, = [0, To) c [O, T] such that u depends Hiilder continuous& on the initial data uO, in the measure F(t) on every compact sub-interval of I,.
308
M.
ARON
Remark 1. The inequality (2.8) may be written in the form
which shows that there exists an interval I, = [0, T,) throughout which F(t) cannot be greater than some time-increasing exponential function. Remark 2. satisfied:
Suppose that the following supplementary conditions are
(a) g is a connected set included in some ball B,. = (u(t) EE Ilull c VI. (b) P(t) has a linear Gateaux differential (dP(f)~)h at every point of C?Jand for each t > 0. (c)
The functional [(diP(t)u) h, , h2] 1s . continuous in u at every point
of g. Then, every operator i”(t)
which satisfies (2.2) can be written in the form
1141
(2.13) where p,(u) belongs to the class of all functionals on Z continuous first Gateaux differential satisfying
P,(U) > 0,
for
u # 0,
P,(O) = 0,
that have a
(2.14)
and where U(t) ranges over all continuous operators such that
I wu, u1= 0,
UER,
(2.15)
[ WP,
hEZ.
(2.16)
h 1= 0,
3. EXAMPLES Here we give two examples demonstrating the applicability of the above theorem. EXAMPLE 1. Let 0 c R” be a bounded region with the boundary r. Assume that R is sufftciently regular and r is smooth enough to make
309
SOLUTIONSTONONLINEARPROBLEMS
integration by parts and application of the divergence theorem permissible. Let R = L’(Q) be equipped with the scalar product (3.1)
We consider the initial-boundary
value problem
ti + tAu = u3(e’ - l), u/r=%
in Q,
t>o,
(3.2)
u(x, 0) = u&), and choose 69 = {u(t): u(t) E C’(.n),
u Ir = 0, t > 0).
(3.3)
It is known that 63 is a linear set, dense in L’(Q) 115, Chap. IX]. It follows then that in this case all the requirements of the above Theorem are satisfied.
EXAMPLE
2.
Let fin, I-, and GS be defined as before and let i”(t)
be given
i, k, I = 1, 2,. .., n,
(3.4)
by
where @(., .) is a function which wk.,;
*) E qo,
satisfies Tl,
@(Uk,, ; t) > 0, t > 0,
wuk.,; 0) = 0 Bu,,j ’ a = const, a > 0,
(3.5) (3.6) (3.7)
and where we have denoted
uk-?
auk
k, I = 1, 2 ,..., n.
(3.8)
310
M. ARON
Consider the following
initial-boundary
uII-= 0,
value problem.
t z 0,
(3.9)
u(x, 0) = u()(x). Integrating
by parts and applying the divergence theorem we obtain [tt,Y(t>u]
u E ti.
= I_ 2 . I> i.j:
(3.10)
I
Now, it is easily seen,from (3.5k(3.7), that the Theorem in the previous section can be applied.
ACKNOWLEDGMENTS I express my thanks to the referee, whose improving the presentation of the material.
constructuve
criticism
has been very
helpful
in
REFERENCES I. J. HADAMARD, “Lectures on Cauchy’s Problem in Linear Partial Differential Equations,” Yale Univ. Press, New Haven, 1923; reprint, Dover, New York, 1952. 2. F. JOHN. Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math. 13 (1960), 551-585. 3. R. J. KNOPS AND L. E. PAYNE, Stability in linear elasticity, Internat. J. Solid. Structure 4 (1968) 1233-1242. 4. R. J. KNOPS AND L. E. PAYNE, On the stability of solutions to the Navier-Stokes equations backward in time, Arch. Rationnl Mech. Anal. 29 (1968), 33 l-335. 5. L. E. PAYNE, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960) 35-45. 6. L. E. PAYNE. On some non well-posed problems for partial differential equations, in “Proceedings, Adv. Sympos. Numerical Solutions of Nonlinear Differential Equations (Madison. Wisconsin, 1966),” pp. 2399263. Wiley, New York, 1966. 7. L. E. PAYNE AND D. SATHER, On some improperly posed problems for the Chaplygin equation, J. Math. Anal. Appl. 19 (1967), 67-17. 8. L. E. PAYNE AND D. SATHER, On some improperly posed problems for quasilinear equations of mixed type, Trans. Amer. Math. Sot. 128 (1967), 135-141. 9. R. J. KNOPS AND E. W. WILKES, Theory of elastic stability, in “Encyclopedia of Physics,” Vol. Via/3, Springer-Verlag, New York/Berlin, 1973. IO. N. S. WILKES, Continuous dependence and instability in linear viscoelasticity, J. Mkanique 17 (1978), 7 17-726.
SOLUTIONS
II.
12.
13. 14. 15.
TO
NONLINEAR
PROBLEMS
311
R. J. KNOPS, Logarithmic convexity and other techniques applied to problems in continuum mechanics “Proceedings, Symposium on Non Well-Posed Problems and Logarithmic Convexity (Edinburgh, Scotland1972),” Springer-Verlag, New York/Berlin, 1973. L. E. PAYNE, “Improperly Posed Problems in Partial Differential Equations,” SIAM Regional Conference Series in Applied Mathematics Vol. 22, Sot. Ind. Appl. Math., Philadelphia, Pa., 1975. C. B. MONEY, “Multiple Integrals in the Calculus of Variations,” Springer-Verlag. Berlin/New York, 1966. D. G. B. EDELEN, On decomposition of operators and solutions of functional inequalities, Arch. Rational Mech. Anal. 54 (1974) 212-222. M. M. VAINBERG. “Variational Methods for the Study of Nonlinear Operators.” HoldenDay. New York, 1964.
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
80,
305-3
11 (198 1)
On the Continuous Dependence of Solutions Nonlinear Problems in Hilbert Space
to
M. ARON* Department of Mathematics, University of Strathclyde, Glasgow, Scotland Submitted by W. F. Ames
1. INTRODUCTION Various physical problems lead to the study of equations of the form Ati = qt>u,
t > 0,
(l-1)
with 40) = u,,
(1.2)
where A and Y(t) are operators on some suitable specified subspace of a Hilbert space R and ti is the derivative with respect to the time t of the vector-valued function u(t) E GP’. It is known that the problem (l.l), (1.2) is not, in general, well posed in the senseof Hadamard [ 11. For example, solutions to (l.l), (1.2) might not depend continuously on the initial data u,,. It is then necessary to know under which conditions continuous dependence can be recovered. An investigation of John (21 has shown that in certain situations it is sufficient to impose a uniform bound on solutions to achieve this. In some of these cases continuity must be replaced by the corresponding concept of Holder continuity. In the last years a number of workers 13-101 have given numerous examples of this type of continuous dependenceand physical justification for it. The following simple example, considered in [9], may provide an intuitive idea of Holder stability and the situations in which it might be applicable. Let the solutions of a linear system be such that their time dependenceis of the type e”‘. Then, the solutions which are bounded at t = t1 would be of the form emnt’-rl) and thus remain arbitrarily near the null solution for t E ]O, t,) but not for t E [0, r,]. It results that there exists continuous dependenceon the initial data in all compact sub-intervals of [0, tt) but not on the interval [0, t, J itself. -* Present England.
address:
Department
of Mathematics,
Plymouth
Polytechnic,
Plymouth,
Devon.
305 409 ‘X012 2
0022-247X/8
l/040305-07$02.00/O
Copynght ‘C 1981 by Academic Press, Inc. A,, riehts of rrnrd~wtinn in ~lnsr C,T.r- ..,-..-..-A
M.ARON
306
In the present paper we are concerned with the Holder continuous dependence on the initial data of solutions to the problem (l.l), (1.2). More precisely, we show that under certain conditions imposed on A, Y, and u,,, there exists, for each suffkiently regular solution u of (l.l), (1.2), a time interval I, = (0, T,) such that in every compact sub-interval of I, the solution ZJ depends Holder continuously on u, provided it belongs to the class of solutions which admit of a uniform bound (in a sense which will be made precise later). The technique used in proving this result is that of logarithmic convexity (see, e.g., [9, 11, 121).
2. CONTINUOUS
DEPENDENCE
Let SF be a real Hilbert space provided with the norm /I . (1induced by the scalar product (., .I. Let g CR be a dense domain and let A: G -+Z be a linear, symmetric, time-independent operator. Assume that A satisfies b&j
> 0,
for all
24E G?!,24# 0.
Assume also that the operator Y(t): GJ -+R
(2.1)
satisfies (2.2) (2.3)
and g(t) exists in the strong sense for t E [0, T]. Let u E g, u f 0 be a strong solution of the problem (l.l), (1.2) (i.e., a solution for which ti exists in the strong sense, is continuous, and is 9 valued). Define F(t) = [u, Au],
2.4 E g,
t E [O, z-1,
(2.4)
Using (1.1) and the symmetry of A we see that
Our hypotheses imply that f is continuously differentiable with respect to t
SOLUTIONS TO NONLINEAR
307
PROBLEMS
in IO, T) and also that f(t) > 0, t E (0, r],f(O) = 0. We shall prove that, in this case, there exists an interval I = [O, r,l, 7’, < T, throughout which
If f(O)# 0 the assertion follows by continuity. If j(O) = 0 we see (by appplying the mean value theorem to the function f) that in every interval (0, ?) with i< T either f z 0 or there exists at least one point t* at which f(r*) > 0. The continuity reveals the existence of an interval (t,, f2) c (0, 6, t” E (tl , t2) throughout which f(t) > 0 and either t, = 0 or f(t,) = 0. If f(r) > 0 for t E [0, t,] we may take T,, = t, and the assertion is proved. Assume then that there exists a point to E (0, tl) such that f(t”) < 0. It then follows that for every interval (0, i) there exists a sub-interval (t,, t:!), where f > 0 and also a sub-interval (tj, t4), to E (f3, f4), where f < 0. Therefore the point I = 0 cannot be a point of (strict) extremum for $ Thus, we have reached a contradiction which proves our claim. By meansof expansions in finite Taylor series,or alternatively by Jensen’s inequality [ 131, we obtain from (2.7) the inequality
F(t) < F(0)‘f”-f”‘0F(To)u70,
f E (0,
(2.8)
To).
DEFINITION. If, at time To, the solution u to the problem (1.1) (1.2) is uniformly bounded in the sensethat
(2.9)
F(To) < M,
for some positive finite constant M, then we say that u belongs to the class M and denote u E M. For u E M, we deduce from (2.8) that
F(t) GM’-“[F(O)]“,
6= 1 -t/T,,
0<6<
1.
(2.10)
The last relation implies the Holder continuous dependenceof the solution u on the initial data u,, in the measure F(t), on every compact sub-interval of I, = IO, To). Thus, we have the following. THEOREM. Let the operators A, P(t), and the function u. in (l.l), (1.2) satisfy (2.1)--(2.3). Assume that g(t) exists (in the strong sense) in some time interval [0, T]. Then for each strong solution u E &j there exists a time interval I, = [0, To) c [O, T] such that u depends Hiilder continuous& on the initial data uO, in the measure F(t) on every compact sub-interval of I,.
308
M.
ARON
Remark 1. The inequality (2.8) may be written in the form
which shows that there exists an interval I, = [0, T,) throughout which F(t) cannot be greater than some time-increasing exponential function. Remark 2. satisfied:
Suppose that the following supplementary conditions are
(a) g is a connected set included in some ball B,. = (u(t) EE Ilull c VI. (b) P(t) has a linear Gateaux differential (dP(f)~)h at every point of C?Jand for each t > 0. (c)
The functional [(diP(t)u) h, , h2] 1s . continuous in u at every point
of g. Then, every operator i”(t)
which satisfies (2.2) can be written in the form
1141
(2.13) where p,(u) belongs to the class of all functionals on Z continuous first Gateaux differential satisfying
P,(U) > 0,
for
u # 0,
P,(O) = 0,
that have a
(2.14)
and where U(t) ranges over all continuous operators such that
I wu, u1= 0,
UER,
(2.15)
[ WP,
hEZ.
(2.16)
h 1= 0,
3. EXAMPLES Here we give two examples demonstrating the applicability of the above theorem. EXAMPLE 1. Let 0 c R” be a bounded region with the boundary r. Assume that R is sufftciently regular and r is smooth enough to make
309
SOLUTIONSTONONLINEARPROBLEMS
integration by parts and application of the divergence theorem permissible. Let R = L’(Q) be equipped with the scalar product (3.1)
We consider the initial-boundary
value problem
ti + tAu = u3(e’ - l), u/r=%
in Q,
t>o,
(3.2)
u(x, 0) = u&), and choose 69 = {u(t): u(t) E C’(.n),
u Ir = 0, t > 0).
(3.3)
It is known that 63 is a linear set, dense in L’(Q) 115, Chap. IX]. It follows then that in this case all the requirements of the above Theorem are satisfied.
EXAMPLE
2.
Let fin, I-, and GS be defined as before and let i”(t)
be given
i, k, I = 1, 2,. .., n,
(3.4)
by
where @(., .) is a function which wk.,;
*) E qo,
satisfies Tl,
@(Uk,, ; t) > 0, t > 0,
wuk.,; 0) = 0 Bu,,j ’ a = const, a > 0,
(3.5) (3.6) (3.7)
and where we have denoted
uk-?
auk
k, I = 1, 2 ,..., n.
(3.8)
310
M. ARON
Consider the following
initial-boundary
uII-= 0,
value problem.
t z 0,
(3.9)
u(x, 0) = u()(x). Integrating
by parts and applying the divergence theorem we obtain [tt,Y(t>u]
u E ti.
= I_ 2 . I> i.j:
(3.10)
I
Now, it is easily seen,from (3.5k(3.7), that the Theorem in the previous section can be applied.
ACKNOWLEDGMENTS I express my thanks to the referee, whose improving the presentation of the material.
constructuve
criticism
has been very
helpful
in
REFERENCES I. J. HADAMARD, “Lectures on Cauchy’s Problem in Linear Partial Differential Equations,” Yale Univ. Press, New Haven, 1923; reprint, Dover, New York, 1952. 2. F. JOHN. Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math. 13 (1960), 551-585. 3. R. J. KNOPS AND L. E. PAYNE, Stability in linear elasticity, Internat. J. Solid. Structure 4 (1968) 1233-1242. 4. R. J. KNOPS AND L. E. PAYNE, On the stability of solutions to the Navier-Stokes equations backward in time, Arch. Rationnl Mech. Anal. 29 (1968), 33 l-335. 5. L. E. PAYNE, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960) 35-45. 6. L. E. PAYNE. On some non well-posed problems for partial differential equations, in “Proceedings, Adv. Sympos. Numerical Solutions of Nonlinear Differential Equations (Madison. Wisconsin, 1966),” pp. 2399263. Wiley, New York, 1966. 7. L. E. PAYNE AND D. SATHER, On some improperly posed problems for the Chaplygin equation, J. Math. Anal. Appl. 19 (1967), 67-17. 8. L. E. PAYNE AND D. SATHER, On some improperly posed problems for quasilinear equations of mixed type, Trans. Amer. Math. Sot. 128 (1967), 135-141. 9. R. J. KNOPS AND E. W. WILKES, Theory of elastic stability, in “Encyclopedia of Physics,” Vol. Via/3, Springer-Verlag, New York/Berlin, 1973. IO. N. S. WILKES, Continuous dependence and instability in linear viscoelasticity, J. Mkanique 17 (1978), 7 17-726.
SOLUTIONS
II.
12.
13. 14. 15.
TO
NONLINEAR
PROBLEMS
311
R. J. KNOPS, Logarithmic convexity and other techniques applied to problems in continuum mechanics “Proceedings, Symposium on Non Well-Posed Problems and Logarithmic Convexity (Edinburgh, Scotland1972),” Springer-Verlag, New York/Berlin, 1973. L. E. PAYNE, “Improperly Posed Problems in Partial Differential Equations,” SIAM Regional Conference Series in Applied Mathematics Vol. 22, Sot. Ind. Appl. Math., Philadelphia, Pa., 1975. C. B. MONEY, “Multiple Integrals in the Calculus of Variations,” Springer-Verlag. Berlin/New York, 1966. D. G. B. EDELEN, On decomposition of operators and solutions of functional inequalities, Arch. Rational Mech. Anal. 54 (1974) 212-222. M. M. VAINBERG. “Variational Methods for the Study of Nonlinear Operators.” HoldenDay. New York, 1964.