We can now make use of Theorem 10.2. If Q is a large ball in N, then

sup G ¤ inf G.

(12.43)

M

‚Q

By Example 1 of Section 10.3, ‚ Q links M weakly. Moreover, if {u k } ‚ E is a

sequence such that v k = Pu k ’ v = Pu weakly on N and wk = (I ’ P)u k ’

w = (I ’ P)u strongly in M, where P is the projection of E onto N, then {u k } has

a renamed subsequence that converges strongly in L 2 ( , ρ). The reason is that {v k }

N0 in L 2 ( , ρ) is compact.

has such a subsequence because the embedding of E

Thus, G (u n ) ’ G (u) weakly in E. Hence, all of the hypotheses of Theorem 10.2

are satis¬ed, and we can conclude that there is a sequence {u k } satisfying (10.3). Write

u k = v k + wk + yk , where v k ∈ N, wk ∈ M N0 , yk ∈ N0 . Then

(G (u k ), v k )/2 = ([L ’ μ]u k , v k ) ’ ( p(u k ), v k )

(12.44)

and, consequently,

θ

vk ¤ G (u k ) v k E /2 + C vk ( u k + 1)

2

(12.45) E

in view of (12.31) and (12.37). Similarly,

θ

wk ¤ G (u k ) wk E /2 + C wk ( u k + 1).

2

(12.46) E

If N0 = {0}, then it follows from (12.45) and (12.46) that u k E is bounded and

consequently there is a renamed subsequence that converges weakly in E and strongly

in L 2 ( , ρ) to a function u. Thus, G (u k ) ’ G (u) weakly. But G (u k ) ’ 0.

Consequently G (u) = 0 and the proof for this case is complete. If N0 = {0}, we note

that

θ

¤ G (u k ) yk ) E /2 + C yk ||( u k + 1)

2

(12.47) yk E

as well. Again, this together with (12.45) and (12.46) implies that u k || E is bounded

and has a renamed subsequence that converges weakly in E and such that u k = v k +wk

converges strongly in L 2 ( , ρ). Now

(G (u k ), yk ’ y)/2 = ([L ’ μ](yk ’ y), yk ’ y)

(12.48)

’( p(u k ) ’ p(u k + y), yk ’ y)

+( p(u k + y) ’ p(u), yk ’ y)

+([L ’ μ]y, yk ’ y),

where yk ’ y weakly in E and L 2 ( , ρ) and u k ’ u weakly in E and strongly in

L 2 ( , ρ). By hypothesis

( p(u k ) ’ p(u k + y), yk ’ y) ≥ 0

(12.49)

12.4. Notes and remarks 147

if μ > »0 . Moreover,

(G (u k ), yk ’ y) ’ 0,

( p(u k + y) ’ p(u), yk ’ y) ’ 0,

and

([L ’ μ]y, yk ’ y) ’ 0.

(12.50)

Hence

2

yk ’ y ¤ o(1), k ’ ∞.

(12.51) E

This shows that yk ’ y in E, and the proof proceeds as before. If »0 > μ, we apply

Theorem 12.1 to ’G(u) and come to the same conclusion. In this case the inequality

in (12.49) is reversed. This completes the proof.

12.4 Notes and remarks

Many authors have studied the one-dimensional periodic-Dirichlet problem for the

semilinear wave equation

u t t ’ u x x = p(t, x, u), t ∈ R, x ∈ (0, π),

u(t, x) = 0, t ∈ R, x = 0, x = π,

u(t + 2π, x) = u(t, x), t ∈ R, x ∈ (0, π)

A basic problem in this one dimensional case is that the null space N of the linear part

u = u t t ’ u x x is in¬nite dimensional. On the other hand, has a compact inverse on

the orthogonal complement of N. In contrast to this, the higher dimensional periodic-

Dirichlet problem for the semilinear wave equation (13.1)“(13.2) has the additional

dif¬culty that does not have a compact inverse on N ⊥ . In fact, it has a sequence

of eigenvalues of in¬nite multiplicities stretching from ’∞ to ∞. This is a serious

complication that causes all of the methods used to solve the one-dimensional case to

fail.

Recently some authors have examined the radially symmetric counterpart of

(13.1)“(13.3) that was considered in this chapter (cf. [146], [15],[14],[23],and [121]).

It is assumed that the function f (t, x, u) is radially symmetric in x. This allows one to

reduce the problem to

u t t ’ u rr ’ r ’1 (n ’ 1)u r = f (t, r, u),

u(2π, r ) = u(0, r ), u t (2π, r ) = u t (0, r ), 0 ¤ r ¤ R,

u(t, R) = u R (t, R) = 0, t ∈ R.

This is much more dif¬cult than the one-dimensional problem for the wave equation,

but the techniques used in solving it cannot be used to solve the n-dimensional problem

for the wave operator when the region and functions are not radially symmetric. This

will be addressed in Chapter 13.

Chapter 13

Semilinear Wave Equations

13.1 Introduction

In this chapter we shall consider the higher-dimensional periodic-Dirichlet problem for

the semilinear wave equation

u ≡ ut t ’ u = p(x, t, u), (x, t) ∈ ,

(13.1)

t ∈ R, x ∈ ‚(0, π)n ,

u(x, t) = 0,

(13.2)

u(x, t + 2π) = u(x, t), (x, t) ∈ ,

(13.3)

= (0, π)n — (0, 2π). Here, x = (x 1 , . . . , x n ) ∈ Rn and

where

(0, π)n = {x ∈ Rn : 0 < x k < π, 1 ¤ k ¤ n}.

In studying this problem, we shall make use of the theory of saddle points.

13.2 Convexity and lower semi-continuity

A set M is called convex if (1’t)w0 +tw1 ∈ M whenever w0 , w1 ∈ M and 0 ¤ t ¤ 1.

Let M be a convex subset of a Hilbert space E, and let G be a functional (real-

valued function) de¬ned on M. We call G convex on M if

G((1 ’ t)w0 + tw1 ) ¤ (1 ’ t)G(w0 ) + t G(w1 ), w0 , w1 ∈ M, 0 ¤ t ¤ 1.

We call it strictly convex if the inequality is strict when 0 < t < 1, w0 = w1 .

G(v) is called upper semi-continuous (u.s.c.) at w0 ∈ M if wk ’ w0 ∈ M

implies

G(w0 ) ≥ lim sup G(wk ).

It is called lower semi-continuous (l.s.c.) if the inequality is reversed and lim sup is

replaced by lim inf. We have

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_13,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

150 13. Semilinear Wave Equations

Lemma 13.1. If M is closed, convex, and bounded in E and G is convex and l.s.c. on

M, then there is a point w0 ∈ M such that

G(w0 ) = min G.

(13.4)

M