Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 81–86.

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

A SEMILINEAR WAVE EQUATION WITH NON-MONOTONE

NONLINEARITY

JOSÉ F. CAICEDO, ALFONSO CASTRO, RODRIGO DUQUE, ARTURO SANJUAN

Abstract. We prove the existence of weak solutions to a semilinear wave with
non-monotone asymptotically linear nonlinearity when the forcing is domi-

nated by a trigonometric polynomial.

1. Introduction

Let Ω = (0, π)× [0, 2π] and Ψ a trigonometric polynomial of the form

Ψ(x, t) =
∑

i,j=1,N, i6=j

akj sin(kx) cos(jt) + bkj sin(kx) cos(jt), (1.1)

where N is a positive integer. We study the existence of weak solutions to the
Dirichlet-periodic problem

�u+ τu+ h(u) = f(x, t) := CΨ(x, t) + g(x, t),

u(0, t) = u(π, t) = 0,

u(x, t) = u(x, t+ 2π), (x, t) ∈ [0, π]× R,
(1.2)

where � denotes the D’Alembert operator ∂tt−∂xx, τ > 0 and τ 6∈ σ(�) = {k2−j2 :
k = 1, 2, . . . , , j = 0, 1, 2, . . .}, C ∈ R, g ∈ L2(Ω), and∫∫

Ω

Ψ(x, t)g(x, t) dx dt = 0, (1.3)

We assume that h is bounded and differentiable, that h′(u) < −τ for some u ∈ R,
and that

lim
|u|→∞

h′(u) = 0. (1.4)

That is, H(u) := τu+ h(u) is non-monotone and asymptotically linear. We denote
by ‖ · ‖2 the norm in L2(Ω). Our main result is the following theorem.

Theorem 1.1. For each g ∈ L2(Ω) satisfying (1.3), there exists C0(‖g‖2) such that
if |C| > C0(‖g‖2) then (1.2) has a solution.

2020 Mathematics Subject Classification. 35J25, 58J05.
Key words and phrases. Semilinear wave equation; weak solution; Nazarov-Turan lemma;

non-monotone nonlinearity; characteristic lines.
©2023 This work is licensed under a CC BY 4.0 license.

Published March 27, 2023.

81



82 J. F. CAICEDO, A. CASTRO, R. DUQUE, A. SANJUAN EJDE/SI/02

This result is in the spirit of determining the range of semilinear wave operators
with non-monotone nonlinearities which goes back to the results in [9, 12] where the
range of such operators was proven to be dense in L2(Ω). The reader is referred to
[6] for a review in the subject and to [4] for a recent result on (1.2) with Ψ replaced
by functions that may be flat on characteristics. For earlier results on semilinear
wave equations with monotone nonlinearities see [1, 10, 11].

A key piece in our arguments is the Nazarov-Turan lemma which we state next
for the sake of completeness in the presentation. For a role of the Nazarov-Turan
lemma in bifurcation at infinity, the reader is referred to [8].

Lemma 1.2 (Nazarov-Turan lemma). If Φ is a trigonometric polynomial with∫
Ω

Φ(x, t) sin(kx) sin(kt) dx dt =

∫
Ω

Φ(x, t) sin(kx) sin(kt) dx dt = 0, (1.5)

for any positive integer k then there exists α > 0 such that for any δ ∈ (0, 1),

m({(x, t) ∈ Ω; |Φ(x, t)| < δ}) < δα. (1.6)

Moreover,

m(Ar,δ) := m({x ∈ [0, π] : |Φ(x, r + x)| < δ}) < δα, (1.7)

uniformly for r ∈ [0, 2π].

In the above lemma and in what follows m denotes the Lebesgue measure in one
or two dimensions as given by the context.

2. Preliminaries

Let N denote the closure of the linear subspace of L2(Ω) spanned by

{sin(kx) cos(kt), sin(kx) sin(kt), k = 1, 2, . . .}. (2.1)

That is, N is the kernel of the wave operator � in (1.2). If v ∈ N , then there exists

a unique 2π-periodic function p : R → R such that p ∈ L2([0, 2π]),
∫ 2π

0
p(t)dt = 0,

and

v(x, t) = p(t+ x)− p(t− x). (2.2)

We let H1 denote the Sobolev space of functions u : [0, π] × R → R that are
2π-periodic in their second variable, with u and its first order partial derivatives in
L2(Ω), and vanishing on {0, π} × R. The norm in in H1 by ‖ · ‖1,2. We also let
Y = N⊥ ∩H1. We say that u = y + v ∈ Y ⊕N is a weak solution to (1.2) if∫∫

Ω

{(ytŷt − yxŷx)− (H(u)− f)(ŷ + v̂)} dx dt = 0, (2.3)

for all ŷ + v̂ ∈ Y⊕N . We let ΠN : L2(Ω)→ N and ΠY : L2(Ω)→ N⊥ denote the
corresponding orthogonal projections.

For each f ∈ L2(Ω), the equation �u+ τu = f has a unique weak solution v+ y
which we denote as (� + τI)−1(f). Moreover, there exists a real number κ such
that

‖(� + τI)−1(ΠY (f))‖1,2 + ‖(� + τI)−1(ΠY (f))‖C1/2 ≤ κ‖f‖2,
‖(� + τI)−1(ΠN (f))‖2 ≤ κ‖f‖2

(2.4)

where C1/2 denotes the Hölder space of continuous functions with exponent 1/2.



EJDE-2023/SI/02 A SEMILINEAR WAVE EQUATION 83

3. Proof of Theorem 1.1

By (1.4), there exists K1 ≥ 128|h′|∞
τπ such that if |s| ≥ K1 then |h′(s)| < τ/128.

Let φ = (� + τI)−1(Ψ), α > 0 be as in Lemma 1.2 applied to Φ = φ, and

δ =
( τ2π2

(128(1 + τ + |h′|∞))2

)1/α

, (3.1)

C0 =
(K1 + 2K2

1 + κ(
√

2π[|h|∞ + |h′|∞] + ‖g‖2 + 1))(τ + 1)

sin(δ)
. (3.2)

Since δα < 1, we have

δα <
τπ

128(1 + τ + |h′|∞)
.

From [9] and [12], there exist sequences {φn}, {un} ⊂ L2 with un = zn+wn ∈ N⊕Y
such that

�wn+τ(zn+wn)+h(zn+wn) = CΨ(x, t)+g(x, t)+φn(x, t), ‖φn‖2 → 0. (3.3)

Without loss of generality we may assume that ‖φn‖2 ≤ 1 for n = 1, 2, . . .. Let
vn ∈ N and yn ∈ Y be such that

� yn + τ(vn + yn) = g(x, t) + φn(x, t), ‖φn‖2 → 0. (3.4)

Subtracting (3.3) from (3.4), we obtain

�(wn−yn)+τ(wn−yn+zn−vn)+h(zn+wn) = CΨ(x, t) = C(�+τI)(ψ). (3.5)

Letting Wn = wn − yn − Cψ and Vn = zn − vn,

�Wn + τ(Wn + Vn) + h
(
Vn + vn +Wn + yn + Cψ(x, t)

)
= 0. (3.6)

Equation (3.6), in turn, is equivalent to the equations

Wn = −(�+τI)−1ΠY (h (Vn + vn +Wn + yn + Cψ)) , (3.7)

τVn = −ΠN

(
h
(
Vn + vn +Wn + yn +

C

τ + 1
Ψ(x, t)

))
. (3.8)

Since h is assumed to be bounded, by the continuity of (� + τI)−1: L2 → Y and
Arzela-Ascoli’s theorem we may assume that {Wn} converges uniformly in Ω.

By (2.2), there exists g1, pn, Pn ∈ L2(0, 2π) such that

ΠN (g)(x, t) = g1(t+ x)− g1(t− x), vn(x, t) = pn(x, t)− pn(t− x),

Vn(x, t) = PN (x, t).
(3.9)

By (3.4), the sequence {τpn} converges to g1 in L2([0, 2π]). From (3.8) and [2],

2πτPn(r) = −I1n(r) + I2n(r), a.e. in [0, 2π], (3.10)

where

I1n(r) =

∫ π

0

h ((Wn + yn)(x, r − x) + qn(r)− qn(r − 2x) + Cψ(x, t)) dx

I2n(r) =

∫ π

0

h ((Wn + yn)(x, r + x) + qn(r + 2x)− qn(r) + Cψ(x, t)) dx,

(3.11)

and

qn(s) = pn(s) + Pn(s) for all s ∈ R. (3.12)

Since h is bounded, from (3.10) and (3.11) we see that the sequence {Pn} is bounded
in L∞.



84 J. F. CAICEDO, A. CASTRO, R. DUQUE, A. SANJUAN EJDE/SI/02

Let us show that the sequence {Pn} converges en L2([0, 2π]). Indeed, let us show
that {Pn} is a Cauchy sequence in L2([0, 2π]). Let Yj(r, x) = (Wj + yj)(x, r − x),
Qj(r, x) = qj(r)− qj(r−2x), Fmn(s, r, x) = Cψ(x, r−x) + (Qn+s(Yn−Ym))(r, x),
and Gmn(s, r, x) = Cψ(x, r − x) + (Ym + s(Qn −Qm))(r, x). Hence

∣∣I1n(r)− I1m(r)
∣∣ ≤ ∫ π

0

∫ 1

0

|h′ (Fmn(s, r, x)) | ds · |(Yn − Ym)(r, x)| dx

+

∫ π

0

∫ 1

0

|h′ (Gmn(s, r, x)) | ds · |(Qn −Qm)(r, x)|dx.
(3.13)

Let |C| ≥ C0 with C0 given by (3.2). If x 6∈ Ar,δ and |(Qn − Qm)(r, x)| ≥ 2K2
1 ,

then m[{s ∈ [0, 1]; |Ym(r, x) + Cφ(x, r − x) + s(Qn − Qm)(r, x)| ≤ K1)}] ≤ 1/K1.
Hence

∣∣ ∫ 1

0

h′ (Ym(r, x) + Cφ(x, r − x) + s(Qn −Qm)(r, x)) ds
∣∣ ≤ |h′|∞

K1
+
πτ

128

≤ πτ

64
.

(3.14)

On the other hand, if |(Qn − Qm)(r, x)| ≤ 2K2
1 , then |Ym(r, x) + Cψ(x, r − x) +

s(Qn−Qm)(r, x))| > K1. Thus |h′(Ym(r, x) +Cψ(x, t) + s(Qn−Qm)(r, x))| < τ
128 .

Hence

∣∣ ∫ 1

0

h′ (Ym(r, x) + Cψ(x, r − x) + s(Qn −Qm)(r, x)) ds
∣∣ ≤ τ

128
<
πτ

64
. (3.15)

From (3.13), (3.14), and (3.15),

|I1n(r)− I1m(r)|

≤
∫ π

0

|h′|∞|(Yn − Ym)(r, x)|dx

+
τπ

64

∫
[0,π]\Ar,δ

|(Qn −Qm)(r, x)| dx

+

∫
Ar,δ

|h′|∞|Qn(r, x)−Qm(r, x)|dx

≤
∫ π

0

|h′|∞|(Yn − Ym)(r, x)| dx+
τπ

64

(
π(|(Pn − Pm + pn − pm)(r)|

+

∫
[0,π]\Ar,δ

|(pn + Pn − pm − Pm)(r − 2x)| dx
)

+m(Ar,δ)|h′|∞|(Pn − Pm + pn − pm)(r)|

+

∫
Ar,δ

|h′|∞|(Pn − Pm + pn − pm)(r − 2x)|dx.

(3.16)



EJDE-2023/SI/02 A SEMILINEAR WAVE EQUATION 85

Similarly,

|I2n(r)− I2m(r)| ≤
∫ π

0

|h′|∞|(Yn − Ym)(r, x)| dx

+
τπ

64

(
π(|(Pn − Pm + pn − pm)(r)|

+

∫
[0,π]\Ar,δ

|(pn + Pn − pm − Pm)(r + 2x)| dx
)

+m(Ar,δ)|h′|∞ |(Pn − Pm + pn − pm)(r)|

+

∫
Ar,δ

|h′|∞|(Pn − Pm + pn − pm)(r + 2x)| dx.

(3.17)

Since ∫
Ar,δ

|Pn(r − 2x)− Pm(r − 2x)|dx ≤ δα/2‖Pn − Pm‖2,

by (3.10), (3.16) y (3.17) we have

2πτ |Pn(r)− Pm(r)| ≤ 2

∫ π

0

|h′|∞|(Yn − Ym)(r, x)| dx

+
τπ

32

(∫
[0,π]\Ar,δ

|(pn + Pn − pm − Pm)(r − 2x)| dx

+ π(|Pn(r)− Pm(r)|+ |(pn − pm)(r)|)
)

+ 2δα|h′|∞(|Pn − Pm + pn − pm)(r)|)

+ 2|h′|∞δα/2 (‖Pn − Pm‖2 + ‖pn − pm‖2) .

(3.18)

By the definition of δ,

64τπ − τπ2 − 128δα|h′|∞ > 64τπ − τπ2 − τπ > 59τπ.

Therefore,

59τπ

32
|Pn(r)− Pm(r)|

≤ 2

∫ π

0

|h′|∞|(Yn − Ym)(r, x)| dx+
5τπ

32
|(pn − pm)(r)|

+
(πτ√2π

64
+ 2δα/2|h′|∞

)
(‖pn − pm‖2 + ‖Pn − Pm‖2)

+
5τπ

64
‖(pn − pm‖2.

(3.19)

Hence,

59τπ

32
‖Pn(r)− Pm(r)‖2

≤ 2
(∫ 2π

0

∫ π

0

|h′|∞|(Yn − Ym)(r, x)| dx dr
)1/2

+
(πτ√2π

64
+ |h′|∞2δα/2

)√
2π(‖pn − pm‖2 + ‖Pn − Pm‖2)

+
5τπ

64
‖(pn − pm‖2.

(3.20)



86 J. F. CAICEDO, A. CASTRO, R. DUQUE, A. SANJUAN EJDE/SI/02

Since {pn} converges in L2([0, 2π]), {Yn} converges uniformly and(τπ√2π

64
+ |h′|∞2δα/2

)√
2π <

τπ

4
<

59τπ

32
. (3.21)

Thus {Pn} is a Cauchy sequence in L2([0, 2π]), which proves the theorem.

Obituary. With great sadness, the last three authors report the passing of Pro-
fessor José Francisco Caicedo on February 23, 2022. He was our friend, teacher,
and mentor. The results in this paper were proven prior to his death. He was a
leading force in the understanding the solvability of semilinear wave equations.

References

[1] H. Brézis, L. Nirenberg; Forced vibrations for a nonlinear wave equation, Comm. on Pure

and Appl. Math., Vol. XXXI No. 1 (1978), pp. 1-30.
[2] A. Castro, B. Preskill; Existence of solutions for a semilinear wave equation with non-

monotone nonlinearity, Continuous and Discrete Dynamical Systems, Series A, Vol. 28, No.

2, (2010), pp. 649-658.
[3] J. Caicedo, A. Castro; A semilinear wave equation with smooth data and no resonance having

no continuous solution, Discrete and Continuous Dynamical Systems, Vol. 24, No. 3 (2009),

pp. 653-658.
[4] J. Caicedo, A. Castro, R. Duque; A semilinear wave equation with non-monotone nonlinear-

ity and forcing flat on characteristics, Electronic Journal of Differential Equations, Special

Issue 01 (2021), pp. 91-99.
[5] J. Caicedo, A. Castro, R. Duque; Existence of solutions for a wave equation with non-

monotone nonlinearity and a small parameter, Milan J. Math., Vol. 79, (2011), 207-220.
[6] J. Caicedo, A. Castro, R. Duque, A. Sanjuan; Existence of Lp-solutions for a semilinear wave

equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, Se-

ries S, Vol. 7, No. 6, (2014), 1193-1202.
[7] J. Caicedo, A. Castro, R. Duque, A. Sanjuan; The semilinear wave equation with non-

monotone nonlinearity: a review, Rend. Istit. Mat. Univ. Trieste Volume 49 (2017), 207-214

[8] J. Caicedo, A. Castro, A. Sanjuan; Bifurcation at infinity for a semilinear wave equation
with non-monotone nonlinearity, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 1857-1865.

[9] H. Hofer; On the range of a wave operator with nonmonotone nonlinearity, Math. Nachr.

106 (1982), pp. 327-340.
[10] P. Rabinowitz; Periodic solutions of nonlinear hyperbolic partial differential equations,

Comm. Pure Appl. Math. Vol. 20, (1967), pp. 145-205.

[11] P. H. Rabinowitz; Large amplitude time periodic solutions of a semilinear wave equation,
Comm. Pure Appl. Math. 37 (1984), no. 2, pp. 189-206.

[12] M. Willem; Density of the range of potential operator, Proc. Amer. Math. Soc. 83 (1981),
No. 2, pp. 341-344.

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

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

Email address: castro@hmc.edu

Rodrigo Duque

Departamento de Matemáticas, Universidad Nacional de Colombia, Palmira, Colombia
Email address: rduqueba@unal.edu.co

Arturo Sanjuan
Departamento de Matemáticas, Universidad Distrital Francisco José de Caldas, Bo-
gotá, Colombia

Email address: aasanjuanc@udistrital.edu.co


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem ??
	Obituary

	References