Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 91–99.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

A SEMILINEAR WAVE EQUATION WITH NON-MONOTONE

NONLINEARITY AND FORCING FLAT ON CHARACTERISTICS

JOSÉ F. CAICEDO, ALFONSO CASTRO, RODRIGO DUQUE

Abstract. We provide sufficient conditions on the forcing term for a semi-

linear wave equation with non-monotone asymptotically linear nonlinearity to
have a weak solution. Earlier results required the forcing term not to be flat

on characteristics, now we remove those requirements. Also we provide esti-

mates on the measure of the level sets of the forcing term, that suffice for the
equation to have a weak solution.

1. Introduction

We consider the existence of weak solutions to the Dirichlet-periodic problem

∂ttu− ∂xxu+H(u) := �u+H(u) = G(x, t), x ∈ (0, π), t ∈ R,
u(0, t) = u(π, t) = 0,

u(x, t) = u(x, t+ 2π),

(1.1)

with H not monotone and asymptotically linear. More precisely we assume that
H(u) = τu+ h(u) with τ ∈ R− {0} and

lim
|u|→+∞

h′(u) = 0. (1.2)

For the sake of simplicity in the estimates, we assume that h is bounded. We
also assume that −τ 6∈ {k2 − j2; k = 1, 2, . . . , j = 0, 1, 2, . . .} := σ(�). The set
σ(�) is the spectrum of the wave operator � subject to the boundary conditions
in (1.1). The main difficulty in studying the solvability of (1.1) is the fact that 0 is
an eigenvalue of infinite multiplicity. This renders useless compactness techniques
extensively used in the study of related semilinear elliptic boundary value problems.
If H is a monotonic function, for each G ∈ L2(Ω) := L2((0, π)× (0, 2π)), equation
(1.1) has a solution (see [2]). For H non-monotone it has been known from [13] and
[9] that (1.1) has a solution for G in a dense subset of L2(Ω). However the proofs
in [13] and [9] do not shed light on the nature of the functions G for which (1.1)
has a solution. In [8], [3] and [6] it is shown that when the forcing term G is large
and not flat in characteristics then (1.1) has a weak solution. Here we extend such
results to cases where G may be flat in characteristics and provide an estimate on
the size of the subsets of characteristics on which G may be flat (constant). To date

2010 Mathematics Subject Classification. 35L70, 35D30.
Key words and phrases. Wave equation; flat on characteristic; non-monotone nonlinearity;

infinite multiplicity eigenvalue.
c©2021 This work is licensed under a CC BY 4.0 license.

Published October 6, 2021.
91



92 J. F. CAICEDO, A. CASTRO, R. DUQUE EJDE/SI/01

we do not know of Gs for which (1.1) has no solution under our hypothesis on H.
However, in [4] a class of continuous G’s for which the wave equation in (1.1) has
no continuous solution 2π-periodic in both x and t is provided. For related results
on wave equations with non-monotone nonlinearities the reader is referred to [1]
and [5].

For the sake of simplicity in the notations we assume that τ > 0. We denote
by ‖ · ‖2 the norm in L2, and by N the closure of the linear subspace of L2(Ω)
generated by

{sin(kx) cos(kt), sin(kx) sin(kt); k = 1, 2, . . .}. (1.3)

That is, N is the kernel of the wave operator � subject to the boundary conditions
in (1.1). We denote by N⊥ the orthogonal complement of N in L2(Ω), and by
PN : L2(Ω) → N , PN⊥ : L2(Ω) → N⊥ the corresponding orthogonal projections.
If v ∈ N , then there exists a 2π-periodic function p : R→ R such that

v(x, t) = p(t+ x)− p(t− x), p ∈ L2([0, 2π]). (1.4)

We denote by H1 the Sobolev space of the functions u : (0, π)× R → R such that
u, ux, ut ∈ L2(Ω), and satisfy the boundary conditions in (1.1). The norm in H1 is
denoted by ‖ · ‖1,2 and Y denotes the subspace of functions y in H1, such that∫∫

Ω

y(t, x)v(t, x) dx dt = 0, for all v ∈ N . (1.5)

A function u = y + v ∈ Y ⊕N is called a weak solution of (1.1) if∫∫
Ω

{(ytŷt − yxŷx)− (H(u)−G)(ŷ + v̂)} dx dt = 0, (1.6)

for all ŷ + v̂ ∈ Y ⊕N .
If τ > 0, −τ /∈ σ(�), and z ∈ L2(Ω), the equation �u + τu = z subject to

the boundary condition in (1.1) has only one weak solution v+ y, which we denote
(�+τI)−1(z). An elementary Fourier series argument shows that there exists κ > 0
such that

‖(� + τI)−1(PN⊥(z))‖1,2 + ‖(� + τI)−1(PN⊥(z))‖C1/2 ≤ κ‖z‖2 ,
‖(� + τI)−1(PN (z))‖2 ≤ κ‖z‖2 ,

(1.7)

where C1/2 denote the space of Hölder continuous functions with exponent 1/2.
Throughout this paper we denote by µ the Lebesgue measure in R. Our main

result is the following theorem.

Theorem 1.1. Let f̂ ∈ N with f̂(x, t) = q̂(x + t) − q̂(t − x) and ‖q̂‖2 = 1. Let
g ∈ N⊥, and G(x, t) = Cf(x, t) + g(x, t) with f ∈ N and C ∈ R. If

µ({x ∈ [0, 2π] : q̂(x) = y}) <
π(2τ + |h′|∞ −

√
4τ |h′|∞ + |h′|2∞)

|h′|∞
(1.8)

for all y ∈ R, then there exists η > 0 and C0 > 0 such that, if ‖f − f̂‖2 < η and
|C| > C0 then problem (1.1) has a weak solution.

Since the smallest root of the quadratic polynomial Q(s) = (2πτ − s|h′|∞)2 −
2π|h′|2∞s is the right-hand side in (1.8), if

0 ≤ α1 <
π(2τ + |h′|∞ −

√
4τ |h′|∞ + |h′|2∞)

|h′|∞
(1.9)



EJDE-2021/SI/01 A SEMILINEAR WAVE EQUATION 93

then Q(α1) > 0. That is,

2π|h′|2∞α1 < (πτ − α1|h′|∞)2. (1.10)

Also, since we are assuming H to be non-monotone, |h′|∞ > τ . Because ϕ(s) =

s −
√

4τs− s2 defines a decreasing function in [0,∞) we have ϕ(|h′|∞) < ϕ(τ) =

(1−
√

3)τ . Hence, if (1.9) holds then

α1|h′|∞ < π(3−
√

5)πτ < 2πτ. (1.11)

Preliminary lemmas

In this section we state and prove some properties of the measure of level sets
that play important roles in the proof of Theorem 1.1.

Lemma 1.2. Let (X,B,m) be a measure space. If q ∈ L1(X) and m(X) < +∞
then there exists y ∈ R such that

m({x ∈ X : q(x) = y}) = max
{
m({x ∈ X : q(x) = z}); z ∈ R

}
:= α(q). (1.12)

Proof. If m({x ∈ X : q(x) = z}) = 0 for all z ∈ R then α(q) = 0 and we can take y
to be any real number.

If there exists ẑ ∈ R such that m({x ∈ X : q(x) = ẑ}) > 0 then {z;m({x ∈ X :
q(x) = z}) ≥ m({x ∈ X : q(x) = ẑ})} is finite, say {z1, . . . , zn}. Therefore, there
exists j ∈ {1, . . . , n} such that m({x ∈ X : q(x) = zj}) ≥ m({x ∈ X : q(x) = zi})
for i = 1, . . . , n. Taking y = zj the lemma is proven. �

Lemma 1.3. Let q̂ ∈ L2([0, 2π]) with ‖q̂‖2 = 1, and α(q̂) as in Lemma 1.2. If
α(q̂) < α1 then there exists δ > 0 such that if ‖q − q̂‖2 < δ, then α(q) < α1 for all
y ∈ R.

Proof. Suppose there are sequences {qj}j in L2(0, 2π) such that limj→∞ ‖qj− q̂‖2 =
0, {δj}j in (0,∞) such that limj→+∞ δj = 0, and {yj}j in R such that

µ({x ∈ [0, 2π] : |qj(x)− yj | < δj}) ≥ α1. (1.13)

Since {qj}j is bounded in L1(0, 2π), α1 > 0, and limj→+∞ δj = 0, {yj}j is
bounded. By passing to a subsequence we may assume that {yj}j converges. Let
ŷ = limj→+∞ yj . By Egoroff’s theorem, see [10], there exists E ⊂ [0, 2π] such that
µ(E) < (α1−α(q̂))/4 such that {qj}j converges uniformly to q̂ in [0, 2π]−E. Since

∩∞n=1 {x ∈ [0, 2π] : |q̂(x)− ŷ| < 1/n} = {x ∈ [0, 2π] : q̂(x) = ŷ}, (1.14)

there exists η > 0 such that

µ({x ∈ [0, 2π] : |q̂(x)− ŷ| < η}) < α+ α1

2
. (1.15)

Let J be such that, for j ≥ J , |qj(x)− q̂(x)| < η/4 for x ∈ [0, 2π]−E and δj < η/4.
Hence, for j ≥ J ,

µ({x ∈ [0, 2π] : |qj(x)− ŷ| < δj})

≤ µ({x ∈ [0, 2π] : |qj(x)− ŷ| < η

2
})

≤ µ({x ∈ [0, 2π] : x ∈ E and |qj(x)− ŷ| < η})
+ µ({x ∈ [0, 2π] : x ∈ [0, 2π]− E and |q̂(x)− ŷ| < η})

<
α1 − α

8
+
α+ α1

2
< α1,

(1.16)



94 J. F. CAICEDO, A. CASTRO, R. DUQUE EJDE/SI/01

which contradicts (1.13). The proof is complete. �

Proof of Theorem 1.1

Let f(x, t) = q(x + t) − q(t − x). An elementary Fourier series argument and

Parseval’s identity prove that
√

2π‖q− q̂‖2 = ‖f− f̂‖2. Hence taking δ as in Lemma

1.3, for ‖f − f̂‖2 <
√

2πδ := η we have

α(q) < α1. (1.17)

From [13] and [9], there exist sequences {φn}, {un} ⊂ L2(Ω) with un = zn + wn ∈
N ⊕Y, and limn→+∞ ‖φn‖2 = 0 such that

�wn + τ(zn + wn) + h(zn + wn) = Cf(x, t) + g(x, t) + φn(x, t) (1.18)

in the weak sense. Projecting onto N⊥ and N one sees that (1.18) is equivalent to

(� + τI)wn = g + PN⊥(φn − h(zn + wn)), (1.19)

τzn + PN (h(zn + wn)) = Cf + PN (φn). (1.20)

In turn, (1.19) and (1.20) are equivalent to

wn = (� + τI)−1(g) + (� + τI)−1PN⊥(φn − h(zn + wn)), (1.21)

zn =
C

τ
f +

1

τ
PN (φn − h(zn + wn)) ≡ C

τ
f +

1

τ
vn. (1.22)

Since limn→∞ ‖φn‖2 = 0 in L2 and h is bounded, there exist k1 such that ‖φn −
h(zn + wn)‖2 ≤ k1. Hence, (1.21) and (1.22) imply

‖wn‖1,2 ≤ κk1 and ‖vn‖2 ≤ k1 + ‖h(un)‖2. (1.23)

Since vn, PN (φn) ∈ N , there exist 2π-periodic functions pn, γn : R → R, with
pn, γn ∈ L2([0, 2π]) such that

vn(x, t) = pn(t+ x)− pn(t− x), PN (φn)(x, t) = γn(t+ x)− γn(t− x), (1.24)

for all n ∈ Z+, x ∈ [0, π], t ∈ R.
By (1.20) and [3, Lemma 5.2], we have

2πpn(r) = 2πγn(r)− In(r) + Γn(r), a.e. in [0, 2π], (1.25)

with

In(r) =

∫ π

0

h
(
wn(x, r − x) +

1

τ
(pn(r)− pn(r − 2x) + Cf(x, r − x))

)
dx

=
1

2

∫ r

r−2π

h
(
w̃n(r, y) +

1

τ
(qn(r)− qn(y))

)
dy,

Γn(r) =

∫ π

0

h(wn(x, r + x) +
1

τ
(pn(r + 2x)− pn(r) + Cf(x, r + x)))dx

=
1

2

∫ r+2π

r

h(ŵn(r, y) +
1

τ
(qn(y)− qn(r)))dy,

where qn(s) = pn(s)+Cq(s), w̃n(r, y) = wn( r−y2 , r+y2 ), and ŵn(r, y) = wn(y−r2 , r+y2 ).

Next we prove that the sequence {pn}, defined in (1.25), converges in L2(0, 2π).
By (1.10) and (1.11), there exists ε1 > 0 such that

2π|h′|2∞α+ 2π2ε21
τ2

< (π − |h
′|∞α+ 2πε1

τ
)2 and |h′|∞α+ πε1 < 2πτ. (1.26)



EJDE-2021/SI/01 A SEMILINEAR WAVE EQUATION 95

Let ε ∈ (0, ε1). By (1.7), {wn} is bounded in the Holder space C1/2. Hence it is
bounded and equicontinuous. This and the Arzela-Ascoli theorem imply that {wn}
has a uniformly convergent subsequence. Thus, without loss of generality, we may
assume that {wn} converges uniformly on Ω. Hence there exists N1 such that if
n,m ≥ N1 then

|wn(x, t)− wm(x, t)| < ε for all (x, t) ∈ Ω. (1.27)

Since {γn}n converges to zero in L2(0, 2π), by Egoroff’s theorem there exists D ⊂
[0, 2π] be such that µ(D) < ε and {γn}n converges uniformly to zero in [0, 2π]−D.
Hence, there exists N2 ≥ N1 such that if n ≥ N2 then

‖γn‖2 < ε and |γn(s)| < ε for almost all s ∈ [0, 2π]−D. (1.28)

Hence, ∫
D

(|In(r)− Im(r)|+ |Γn(r)− Γm(r)|)dr ≤ 4π|h|∞ε (1.29)

for all n,m ≥ N2. For r ∈ [0, 2π]−D, we have

|In(r)− Im(r)| ≤ 1

2

∫
Dr

∣∣∣h(w̃n(r, y) +
1

τ
(qn(r)− qn(y))

)
− h
(
w̃m(r, y) +

1

τ
(qm(r)− qm(y))

)∣∣∣ dy
+

1

2

∫
Er

∣∣∣h(w̃n(r, y) +
1

τ
(qn(r)− qn(y))

)
− h
(
w̃m(r, y) +

1

τ
(qm(r)− qm(y))

)∣∣∣ dy,
(1.30)

where Dr = {s ∈ [r−2π, r] : s ∈ D or s+2π ∈ D} and Er = [r−2π, r]−Dr. Since
µ(Dr) = µ(D) < ε, applying the mean value theorem we have

|In(r)− Im(r)|

≤ |h|∞ε+
1

2

∫
Er

∣∣∣h(w̃n(r, y) +
1

τ
(qn(r)− qn(y))

)
− h
(
w̃m(r, y) +

1

τ
(qm(r)− qm(y))

)∣∣∣ dy
≤ |h|∞ε+

1

2

∫
Er

∣∣∣ (qn − qm)(r)− (qn − qm)(y)

τ
h′(w̃m(r, y) + ζnm(r, y))

∣∣∣ dy
+

1

2

∫
Er

∣∣∣(w̃n − w̃m)(r, y)h′
(1

τ
(qn(r)− qn(y) + ζ̄nm(r, y))

)∣∣∣dy,

(1.31)

where ζnm(r, y) is in the bounded interval defined by (qn(r)−qn(y))/τ and (qm(r)−
qm(y))/τ , and ζ̄nm(r, y) is in the bounded interval defined by wn((r−y)/2, (r+y)/2)
and wm((r − y)/2, (r + y)/2). From (1.27), for n,m ≥ N2, we have∫ r

r−2π

∣∣∣(w̃n − w̃m)(r, y)h′
(1

τ
(qn(r)− qn(y)) + ζ̄nm(r, y)

)∣∣∣dy < 2πε|h′|∞. (1.32)



96 J. F. CAICEDO, A. CASTRO, R. DUQUE EJDE/SI/01

From the definition of qn we have∫
Er

∣∣∣((qn − qm)(r)− (qn − qm)(y))h′(w̃m(r, y) + ζnm(r, y))
∣∣∣ dy

≤ |(pn − pm)(r)|
∫
Er

∣∣∣h′(w̃m(r, y) + ζnm(r, y))
∣∣∣ dy

+

∫
Er

∣∣∣(pn − pm)(y)h′(w̃m(r, y) + ζnm(r, y))
∣∣∣ dy.

(1.33)

By (1.2), there exists b > 0 such that |h′(s)| < ε for any |s| ≥ b. For C > 0 and
r ∈ [0, 2π]−D, we define the set

A(r, C) =
{
y ∈ Er :

∣∣C
τ

(q(r)− q(y))
∣∣ < b

}
=
{
y ∈ Er : |q(r)− q(y)| < τb

|C|
}
.

Let δ be as in Lemma 1.3. By Lemma 1.3 and (1.17) for |C| > τb/δ := C0 we have

µ(A(r, C)) < α1 for all r ∈ [0, 2π]. (1.34)

If y /∈ A(r, C) and C ≥ C0, then C
τ |q(r)− q(y)| ≥ b. Hence

|h′(w̃m(r, y) + ζnm(r, y))| < ε. (1.35)

Also, for all r ∈ [0, 2π]−D and C ≥ C0,∣∣∣ ∫ r

r−2π

h′(w̃m(r, y) + ζnm(r, y)) dy
∣∣∣

=
∣∣∣ ∫
A(r,C)

h′(w̃m(r, y) + ζnm(r, y)) dy

+

∫
Ac(r,C)

h′(w̃m(r, y) + ζnm(r, y))dy
∣∣∣

< (|h′|∞α1 + 2πε).

(1.36)

By the Cauchy-Schwartz inequality∫ r

r−2π

|h′(w̃m(r, y) + ζnm(r, y)) · (pn(y)− pm(y))| dy

≤
(∫ r

r−2π

[h′(w̃m(r, y) + ζnm(r, y))]
2
dy
)1/2(∫ 2π

0

[pn(y)− pm(y)]2dy
)1/2

≤ ‖pn − pm‖2
(∫ r

r−2π

[h′(wm(r, y) + ζnm(r, y))]2dy
)1/2

.

Reasoning as in (1.36) we have∫ r

r−2π

[h′(w̃m(r, y) + ζnm(r, y))]2dy

=

∫
A(r,C)

[h′(w̃m(r, y) + ζnm(r, y)]2dy

+

∫
Ac(r,C)

[h′(w̃m(r, y) + ζnm(r, y))]2dy

< |h′|2∞α1 + πε2 := K + π2ε2.

(1.37)



EJDE-2021/SI/01 A SEMILINEAR WAVE EQUATION 97

From (1.31), (1.36), and (1.37), for r ∈ [0, 2π]−D and n,m ≥ N2, we have

|In(r)− Im(r)|

< |h′|∞ε+
(K + 2πε2)1/2

2τ
‖pn − pm‖2 +

|h′|∞α1 + 2πε

2τ
|pn(r)− pm(r)|,

(1.38)

Repeating the arguments leading to (1.38) for |Γn(r) − Γm(r)| it is seen that the
estimate on the right of (1.38) also holds for |Γn − Γm|. This and (1.25) give

π|pn(r)− pm(r)| < π|γn(r)− γm(r)|+ (K + 2πε2)1/2

2τ
‖pn − pm‖2

+
|h′|∞α1 + 2πε

τ
|pn(r)− pm(r)|+ |h′|∞ε,

(1.39)

for r ∈ [0, 2π]−D. Letting C1 = π − |h
′|∞α1

2τ > 0 we have

(
C1 −

επ

τ

)
|pn(r)− pm(r)| < (π + |h′|∞)ε+

(K + 2πε2)1/2

2τ
‖pn − pm‖2 (1.40)

for all r ∈ [0, 2π]−D and all n,m ≥ N2. Let M > 0 be such that

2π(π + |h′|∞)2ε1 + 4π(π + |h′|∞)
(K + 2πε21)1/2

τ
‖pn − pm‖2

+ C2
1 (8|h|2∞π2 + 2ε1) ≤M,

(1.41)

for all n,m ≥ N2. Squaring in (1.40) and integrating with respect to r we obtain

(C1 −
επ

τ
)2

∫ 2π

0

|pn(r)− pm(r)|2dr

< 2π(π + |h′|∞)2ε2 + 2π(π + |h′|∞)ε
(K + πε2)1/2

τ
‖pn − pm‖2

+
π(K + 2πε2)

2τ2
‖pn − pm‖22 + C2

1

∫
D

|pn − pm|2dr.

(1.42)

Also, by (1.25), (1.28), and (1.29), we have∫
D

|pn − pm|2dr ≤ 2

∫
D

|(pn − γn)− (pm − γm)|2dx+ 2

∫
D

|γm − γn|2dr

= 2

∫
D

|(Im − Γm − In + Γn)/(2π)|2dr + 2ε2

≤ 8|h|2∞ε+ 2ε2.

(1.43)

By (1.26), taking

ρ =
(
C1 −

ε1π

τ

)2 − πK + π2ε21
2τ2

> 0, (1.44)

we have

‖pn − pm‖22 ≤
M

ρ
ε. (1.45)

Since ε > 0 may be chosen arbitrarily small, {pn}n is a Cauchy sequence in
L2([0, 2π]). Hence {vn}n and {zn}n are Cauchy sequences in L2(Ω).



98 J. F. CAICEDO, A. CASTRO, R. DUQUE EJDE/SI/01

Let w ∈ Y and z ∈ N be such that limn→+∞ wn = w and limn→zn zn = z in
L2(Ω). Because of (1.21), (1.22), and limn→+∞ φn = 0 in L2(Ω), we have

w = (� + τI)−1(g) + (� + τI)−1PN⊥(h(z + w)),

z =
C

τ
f +

1

τ
PN (h(z + w)),

(1.46)

which proves that z + w is a weak solution to (1.1). The proof of the theorem is
complete.

Acknowledgements. The authors are grateful to the anonymous referees for their
careful reading of the original manuscript and for their helpful suggestions. The
authors are also grateful to editor Julio G. Dix for obtaining referee reports and
accepting this article.

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José F. Caicedo
Departamento de Matemáticas, Universidad Nacional de Colombia Sede Bogotá, Bo-

gotá, Colombia
Email address: jfcaicedoc@unal.edu.co

Alfonso Castro
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA

Email address: castro@g.hmc.edu



EJDE-2021/SI/01 A SEMILINEAR WAVE EQUATION 99

Rodrigo Duque

Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Palmira, Palmira,

Colombia
Email address: rduqueba@unal.edu.co


	1. Introduction
	Preliminary lemmas
	Proof of Theorem ??
	Acknowledgements

	References