A mean condition on forcing term for multiplicity of periodic solutions for nonlinear dissipative hyperbolic equations

A MEAN CONDITION ON FORCING TERM FOR MULTIPLICITY OF PERIODIC SOLUTIONS FOR NONLINEAR DISSIPATIVE HYPERBOLIC EQUATIONS

Wan Se Kim

Abstract. A condition on forcing term insuring the multiplicity of Dirichlet-periodic solutions of nonlinear dissipative hyperbolic equations is discussed. The nonlinear term is assumed to have co- ercive growth.

1. Introuction

Let R be the set of all reals and ⊆ R

, n ≥ 1, be a bounded domain with smooth boundary ∂ which is assumed to be of class C

.

Let Q = (0, 2π) × and L

(Q) be the space of measurable and Lebesgue square integrable real-valued functions on Q with usual inner product < · , · > and corresponding norm ∥ · ∥

.

By H

() we mean the completion of C

() with respect to the norm

∥ · ∥

dened by

∥φ∥

= ∫ ∑

|α|≤1

|D

φ(x) |

dx.

H

() stands for the usual Sovolev space ; i.e., the completion of C

() with respect to the norm ∥ · ∥

dened by

∥φ∥

= ∫ ∑

|α|≤2

|D

φ(x) |

dx.

Received June 9, 2003.

2000 Mathematics Subject Classication: 35L20, 35L70.

Key words and phrases: multiplicity, nonlinear, dissipative hyperbolic equations, coercive growth.

This work was supported by by grant R01-2000-000-00002-0 from the Basic Re- search Program of the Korea Science and Engineering Foundation.

We denote by λ

< λ

≤ · · · the eigenvalues of the problem

− △

u = λu, u ∈ H

()

and ϕ

the positive normalized eigenfunction corresponding to λ

. The purpose of this work is to investigate a condition on forcing term insuring multiplicity of periodic solutions of the nonlinear dissipative hyperbolic equations of the form

(E) ∂

u

∂t

− △

u + β ∂u

∂t − λ

u + g(u) = h(t, x) in Q, (B

) u(t, x) = 0 on (0, 2π) × ∂ ,

(B

) u(0, x) = u(2π, x) on

where β( ̸= 0) ∈ R, g : R → R is a continuous nonlinear function with coercive growth, and with a restriction on the left-hand side growth by the dierence of λ

and λ

and h ∈ L

(Q) is 2π-periodic with respect to t.

Moreover, we assume that there exist constants a

> 0 and b

≥ 0 such that

(H

) |g(u)| ≤ a

|u| + b

for all u ∈ R.

Dissipative hyperbolic equations of the form (E) are related to the mathematical interpretation of undistorted plane waves. A more general interpretation is given by the telegraph equations of the form in one space dimension

u

− c

u + (α + β)u

+ αβu = 0,

where c is the speed of light and α the capacity and β the inductive damping factor, satised by the voltage or the current u as a function of time t and the position x along a cable, here x measures the length of the cable from an initial point. For the existence of a generalized solution, Mawhin[18] give a solvabality of the doubly-periodic problem to the equation of the form

(1.1) u

− u

+ βu

+ g(u) = h(t, x), (t, x) ∈ R

,

where β ̸= 0, u = u(t, x) and g is a continuous function on R of at

most linear growth. He obtain results for resonance and nonresonance

situations related to the eigenvalues of the linear dierential operator

u

− u

+ βu

for the doubly-periodic problem. The results do not

depend upon the size of |β| and non-vanishing of this number bing only

assumed to insure the compactness properties required by a treatment

using topological degree. The results are more in sprit of work on ordi- nary and elliptic partial dierential equations. Mawhin[18], and Fucik and Mawhin[4] deal with the existence of a periodic solution in both variables to the equations of the form

(1.2) u

− u

+ βu

− µu

+ νu

+ g(u) = h(t, x), (t, x) ∈ R

, where β, µ, ν ∈ R, β ̸= 0, u = u(t, x) and g is a continuous bounded function. The authors decompose the set R

of all pairs (µ, ν) (see [21]) and obtain existence results of dierent types on dierent component.

For the existence of a solution for the doubly-periodic problem for the equations of the form (1.1) whose nonlinearity grow exponentially, we re- fer Kim[11]. The author impose a condition on the mean of source term to apply continuation theorem(see [5]). In [8] and [17], the existence of a solution for the Dirichlet-periodic resonance problem for (1.1) are treated. The authors imposed on the source term the orthogonal con- dition and the Landesman-Lazer condition, respectively, with the rst eigenfunction of the linear dierential operator to prove the resonance problem. In [10], the existence of a solution for the Dirichlet-periodic problem for the equation of the form (1.1) is discussed when the rate of growth allowed in the nonlinear term g is any polynomial growth. The only condition on the nonlinear term ia a weak type of sign condition which is a strong tool in the proof of the result. Any restriction on the forcing term is not needed except that term is in L

(cf. [9]). The author use Leray-Schauder theory to prove the result. Brezis and Nirenberg[3]

apply their general results on the ranges of nonlinear operators to the Dirichlet-periodic problem for the equations of the form

(1.3) u

+ Eu + βu

+ g(t, x, u) = h(t, x), (t, x) ∈ R × , where β ̸= 0, u = u(t, x) and

Ev(x) = ∑

|α|,|β|≤m

(−1)

D

(a

(x)D

v(x)).

The function g is supposed to be continuous with sublinear growth in

u, satises a sign condition and Landesman-Lazer type conditions. The

authors prove the existence of a solution with some additional condi-

tions related to the kernel of the operator E and the regularity results

are also stated. In [20], Nkashama and Willem treat the existence of a

solution for the Dirichlet-periodic problem for (1.3) whose nonlinearity

g is jumping over two conesecutive eigenvalues. They prove results by

using coincidence degree. The readers can refer to the book[22] for a

survey of boundary value problems for dissipative hyperbolic equations.

But few seems to be known about the multiple existence of gener- alized solutions of Dirichlet-periodic solutions of the form (E) with a mean condition on the forcing term from which is motivated the coer- civity of the nonlinear term(cf. [1], [6], [7], [13], [15], [19]). In [12] and [14]. The multiple existence of doubly-periodic solutions of the equa- tions of the form (1.1) and Dirichlet-periodic solutions of the equations of the form (E) are treated, respectively(cf. [16]). These results are re- lated to Ambrosetti-prodi[2] who initiate so call Ambrosetti-Prodi type multiplicity in the study of the Dirichlet problem to elliptic equations.

In our result, we discuss a mean condition on the forcing term insuring the multiple existnce of solutions. We impose the coercive growth con- dition on nonlinear term. We take advantage of the properties of the rst eigenfunction ϕ

in the proofs of our results. Our proof of the main theorem is based on Mawhin’s continuation theorem(see [5]).

2. Preliminary results Let’s dene the linear operator

L : DomL ⊆ L

(Q) → L

(Q) by

DomL = {u ∈ L

((0, 2π), H

() ∩ H

()) | ∂u

∂t ∈ L

(Q), ∂

u

∂t

∈ L

(Q), u(0, x) = u(2π, x), x ∈ } and

Lu = ∂

u

∂t

− △

u + β ∂u

∂t − λ

u.

By Fourier series and Parseval inequality, we have easily

< Lu, ∂u

∂t >= β ∥ ∂u

∂t ∥

for all u ∈ DomL.

Hence kerL = ker( △

+ λ

I) = kerL

since △

+ λ

I is self-adjoint and ker( △

+ λ

I) is one space dimension generated by the eigenfunction ϕ

. Therefore L is a closed, densely dened linear operator and Im( L) = [kerL]

; i.e., L

(Q) = kerL ⊕

ImL.

Let us consider a continous projection P

: L

(Q) → L

(Q) such that ImP

= KerL. Then L

(Q) = kerL ⊕

kerP

. We consider another continuous projection P

: L

(Q) → L

(Q) dened by

(P

h)(t, x) = 1 2π

∫∫

Q

h(t, x)ϕ

(x)dtdxϕ

(x).

Then we have L

(Q) = ImP

⊕ ImL, kerP

= ImL, and L

(Q)/ImL is isomorphic to ImP

.

Since dim[L

(Q)/ImL] = dim[ImP

] = dim[kerL] = 1, we have an isomorphism J : ImP

→ kerL.

By the closed graph theorm, the generalized right inverse of L dened by

K = [L|

]

: ImL → ImL is continuous.

If we equip the space DomL with the norm

∥u∥

=

∫∫

Q

[u

+ ( ∂u

∂t )

+ ( ∂

u

∂t

)

+ ∑

|β|≤2

(D

u)

]dtdx.

Then there exist a constant c > 0 independent of h ∈ ImL, u = Kh such that

∥Kh∥

≤ c∥h∥

.

Therefore K : ImL → ImL is continuous and by the compact imbedding of DomL in L

(Q), we have that K : ImL → ImL is compact

Lemma 2.1. L is a closed, densely dened linear operator such that kerL = [ImL]

and such that the right inverse K : ImL → ImL is completely continuous.

Proof. See [3] and [20].

3. Multiplicity results

Let us consider the following (E

) ∂

u

∂t

− △

u + β ∂u

∂t − λ

u + µg(u) = µh(t, x) in Q, (B

) u(t, x) = 0 on (0, 2π) × ∂ ,

(B

) u(0, x) = u(2π, x) on , where µ ∈ [0, 1].

Let L : DomL ⊆ L

(Q) → L

(Q) be dened as before. If we dene a sustitution operator N : L

(Q) → L

(Q) by

(N u)(t, x) = g(u(t, x)) − h(t, x)

for u ∈ L

(Q) and (t, x) ∈ Q, then N maps L

(Q) continuously into

itself and transforms bounded sets into bounded sets. Let G be any

open bounded subset of L

(Q), then P

N : G → L

(Q) is bounded

and K(I − P

) : G → L

(Q) is compact and continuous. Thus N is L-compact on G (see [5]).

The coincidence degree D

(L + µN, G) is well dened and constant in µ if Lu + µN u ̸= 0 for µ ∈ [0, 1] and u ∈ DomL ∩ ∂G. It is easy to check that (u, µ) is a weak solution of (E

)(B

)(B

) if and only if u ∈ DomL and

(3.1

) Lu + µN u = 0.

Here, we assume the following

(H

) lim inf

|u|→∞

g(u) = + ∞,

(H

) m = inf

u∈R

g(u) > 0.

Moreover, we assume that

(H

) lim sup

u→−∞

| g(u)

u | < λ

− λ

.

Then it follows from (H

) that there exists a ∈ (0, λ

− λ

) and b ≥ 0 satisfying

|g(u)| ≤ a|u| + b for all u ≤ 0.

For h ∈ L

(Q), we write ∧h = ∫∫

Q

h(t, x)ϕ

(x)dtdx.

Lemma 3.1. Let c > 0 be given. Then for every δ > 0 there exists a constant K(δ) > 0 such that, for every ˜ u ∈ ImL, α ∈ R, the following implication holds.

(3.2)

∫∫

Q

g(αϕ

+ ˜ u)ϕ

dtdx ≤ c ⇒ |α| ≤ δ||˜u||

+ K(δ).

Proof. Let δ > 0 be arbitrarily given, and put

(3.3) r :=

√

2π + 1 δ

− 1

δ . We have chosen ϕ

to satisfy the condition (3.4)

∫

ϕ

(x)dx = 1,

and nd η > 0 such that for every subset M ⊆ Q we have the following

(3.5) |M| < η ⇒

∫∫

M

ϕ

(x)dtdx < r

,

where | · | denotes the Lebesgue measure.

Let M

be the set {x ∈ | ϕ

(x) < ξ }. We x ξ > 0 such that

|M

| < η/(4π). Next, from (H

) we can choose suciently large ν > 0 such that

|u| > ν ⇒ g(u) > 2c ξη .

We dene the sets A = {(t, x) ∈ Q \ ((0, 2π) × M

) | |αϕ

+ ˜ u | ≥ ν}, B = A ∪ ((0, 2π) × M

). Then we have

c ≥

∫∫

A

g(αϕ

+ ˜ u)ϕ

dtdx ≥ 2c η |A|, and hence

(3.6) |A| ≤ η/2, |B| ≤ η.

On the other hand, we have

(3.7),

0 = 2

∫∫

Q

αϕ

udtdx ˜

≤

∫∫

Q\B

( |αϕ

+ ˜ u |

− |αϕ

|

)dtdx + 2

∫∫

B

|αϕ

||˜u|dtdx and, from (3.4), it follows that

2πα

=

∫∫

Q\B

|αϕ

|

dtdx +

∫∫

B

|αϕ

|

dtdx.

Using (3.5), (3.6), (3.7) and the denition of the set B, we obtain that 2πα

≤

∫∫

Q\B

|αϕ

+ ˜ u |

dtdx + 2

∫∫

B

|αϕ

||˜u|dtdx +

∫∫

B

|αϕ

|

dtdx

≤ |Q|ν

+ 2|α|r||˜u||

+ α

r

. This yields that

α

≤ 2|α|r

2π − r

||˜u||

+ |Q|ν

2π − r

. We conclude that

(3.8) |α| ≤ 2r

2π − r

||˜u||

+ ν

√ |Q|

2π − r

.

Hence, (3.2) holds as a consequence of (3.3), (3.8), and the proof is complete.

From Lemma 3.1, we have the following.

Lemma 3.2. Let h ∈ L

(Q) be given such that ∧h > 0 and suppose

hypotheses (H

) − (H

) hold.

Then there exist constsnts γ = γ(h) > 0, M = ˜ ˜ M (h) > 0 such that, for every µ ∈ [0, 1], every solution of problem (3.1

) of the form u = αϕ + ˜ u with ˜ u ∈ ImL and α ∈ R, we have

|α| ≤ γ, ||˜u||

≤ ˜ M .

Proof. Let us take inner product of (3.1

) with

, then from the periodicity of u in t, we have

|| ∂u

∂t ||

≤ 1

|β| ||h||

.

Again, take inner product of (3.1

) with u. This yields that (λ

− λ

) ||˜u||

− || ∂u

∂t ||

+ µ

∫∫

Q

g(u)udtdx ≤ µ

∫∫

Q

|h||u|dtdx.

By hypotheses (H

) − (H

), we have that there exists a constsnt C

> 0 such that g(u)u ≥ −a|u|

− C

for every u ∈ R. There exists therefore a constant C

> 0 such that

(λ

− λ

− a)||˜u||

≤ C

(1 + α

) + ||h||

||u||

. Hence we have

(3.9) ||˜u||

≤ C

(1 + α

) for some constant C

> 0 independent of µ .

Taking inner product of (3.1

) with ϕ

and using the fact that µ > 0, we obtain that ∫∫

Q

g(αϕ

+ ˜ u)ϕ

dtdx = ∧h.

By Lemma 3,1 with c = ∧h and δ = 1/(2 √

C

), we nd a constant C

> 0 such that

(3.10) |α| ≤ 1

2 √ C

||˜u||

+ C

.

Combining (3.10) with (3.9), we obtain that |α| ≤ (1+|α|)/2+C

, hence

|α| ≤ γ, ||˜u||

≤ ˜ M for some constant γ > 0, ˜ M > 0, and the proof is complete.

Remark. Let h ∈ L

(Q), then, by hypothesis (H

) and Lemma 3.2, we have a number ˜ γ(h) > γ such that

(3.11)

∫∫

Q

g(αϕ

(x) + ˜ u)ϕ(x)dtdx >

∫∫

Q

hϕ

(x)dtdx

for |α| ≥ ˜γ.

Lemma 3.3. Let h ∈ L

(Q) be given such that ∧h > 0 and let hypotheses (H

) − (H

) hold. Suppose G(h) ⊆ L

(Q) be a bounded open set such that the implication

|α| < ˜γ

, ||˜u||

< ˜ M

⇒ u = αϕ + ˜u ∈ G(h)

holds where ˜ γ

and ˜ M

are numbers such that ˜ γ

> ˜ γ and ˜ M

> ˜ M . Then, for any bounded open set G in L

(Q) such that G(h) ( G, we have D

(L + N, G) = 0.

Proof. By Lemma 3.2, the equation

(3.12) Lu + g(u) = νh

has no solution on ∂G for ν ∈ [0, 1]. Hence, we may apply the homotopy argument with respect to ν. Taking inner product of (3.12) with ϕ

, we obtain, from hypothesis (H

), that

ν ∧ h =

∫∫

Q

g(u)ϕ

dtdx ≥ 2πm

∫

ϕ

dtdx > 0.

Hence, (3.12) has no solution for ν = 0. Therefore, we have D

(L + N, G) = D

(L + g, G) = 0, and the proof is complete.

Now, let ˜ γ, ˜ M be dened in Lemma 3.2 and Remark, and h ∈ L

() with ∧h > 0. Let α

∈ [−˜γ, ˜γ], x

∈ , ˜u

∈ [− ˜ M , ˜ M ] be such that

g(α

ϕ

(x

) + ˜ u

) = min

[x∈ ,|α|≤˜γ,|˜u|≤ ˜M ]

g(αϕ

(x) + ˜ u).

Theorem . Let hypotheses (H

) − (H

) hold and suppose that h ∈ L

(Q) and there exists a constant C > 0 be such that

(3.13)

∫∫

Q

g(α

ϕ

(x) + ˜ u(t, x))ϕ

(x)dtdx < C

for all ˜ u ∈ L

with ∧˜u = 0, ∥˜u∥

≤ ˜ M

and satisfying the conditions (B

) and (B

).

Then boundary value problem (E), (B

), (B

) has at least two solu- tions if

(3.14) C <

∫∫

Q

hϕ

(x)dtdx.

Proof. Let G(h) ( G be an open bounded set such that the follow- ing implication

α

< α < ˜ γ

, ||˜u||

< ˜ M

⇒ u = αϕ + ˜u ∈ G(h)

holds and such that if u ∈ ∂ G(h), then necessary u = α

ϕ

(x) + ˜ u or u = ˜ γ

ϕ

(x) + ˜ u with ∥˜u∥

≤ ˜ M

. If u = α

ϕ

(x) + ˜ u with ∥˜u∥

≤ ˜ M

, then, by taking inner product (3.1

) with ϕ

, we have

∫∫

Q

µg(α

ϕ

(x) + ˜ u(t, x))ϕ

(x)dtdx = µ

∫∫

Q

hϕ

(x)dtdx

which, from (3.13) and (3.14), is impossible. If u = ˜ γ

ϕ

(x) + ˜ u with

∥˜u∥

≤ ˜ M

, then, by (3.11), we have

∫∫

Q

g(˜ γ

ϕ

(x) + ˜ u)ϕ

(x)dtdx >

∫∫

Q

hϕ

(x)dtdx

which is also impossible. Thus D

(L + N, G(h)) is well dened and D

(L + N, G(h)) = D

(J P

N, G(h) ∩ kerL, 0),

where D

is Brouwer degree and P

N : L

(Q) → kerL is an operator dened by

(P

N u)(t, x) = 1 2π [

∫∫

Q

g(u(t, x))ϕ

(x)dtdx −

∫∫

Q

hϕ

(x)dtdx]ϕ

(x).

Now, let T : kerL → R be dened by

T (αϕ

(x)) = α.

Then

D

(L + N, G(h) = D

(J P

N, G(h) ∩ kerL, 0)

= D

(T (J P

N )T

, T ( G(h) ∩ kerL), 0).

If we let J : ImP

→ kerL be the identity map, then the operator

= 2 πT (J P

N )T

will be dened by ( α) =

∫∫

Q

g(αϕ

(x))ϕ

(x)dtdx −

∫∫

Q

hϕ

(x)dtdx.

Thus, we have, by (3.13) and (3.14), ( α

) =

∫∫

Q

g(α

ϕ

(x))ϕ

(x)dtdx −

∫∫

Q

hϕ

(x)dtdx < 0, and, by (3.11), we have

(˜ γ

) =

∫∫

Q

g(˜ γ

ϕ

(x))ϕ

(x)dtdx −

∫∫

Q

hϕ

(x)dtdx > 0.

Hence, the coincidence degree exists and the corresponding value

|D

(L + N, G(h)) | = |D

[J P

N, G(h) ∩ KerL, 0]| = 1.

By Mawhin’s continuation theorem, the problem (E), (B

), (B

) has at least one solution in G(h).

By the additivity of the coincidence degree, we have

0 = D

(L + N, G) = D

(L + N, G(h))) + D

(L + N, G − G(h)) and hence

|D

(L + N, G − G(h))| = 1.

Therefore, (E), (B

), (B

) has another solution in G − G(h). This proves our assertion.

Remark. If 1

| |

∫∫

Q

h(t, x)ϕ

(x)dtdx < inf

u∈R

g(u),

then the boundary value problem (E)(B

)(B

) has no solution.

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Department of Mathematics Hanyang University

Seoul 133-791, Korea

E-mail : [email protected]