Fredholm solvability of a periodic Neumann problem for a linear telegraph equation
We investigate a periodic problem for the linear telegraph equation $$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$ with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions. This resul...
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2013
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2425 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508310423207936 |
|---|---|
| author | Kmit, I. Ya. Кміть, І. Я. |
| author_facet | Kmit, I. Ya. Кміть, І. Я. |
| author_sort | Kmit, I. Ya. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:15:16Z |
| description | We investigate a periodic problem for the linear telegraph equation
$$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$
with Neumann boundary conditions. We prove that the operator of the problem is modeled by a
Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions.
This result is stable under small perturbations of the equation where p becomes variable and discontinuous or an additional zero-order term appears.
We also show that the solutions of this problem possess smoothing properties. |
| first_indexed | 2026-03-24T02:23:10Z |
| format | Article |
| fulltext |
UDC 517.9
I. Kmit (Inst. Appl. Problems Mech. and Math. Nat. Acad. Sci. Ukraine, Lviv and Humboldt Univ. Berlin, Germany)
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM
FOR A LINEAR TELEGRAPH EQUATION*
ФРЕДГОЛЬМОВIСТЬ ПЕРIОДИЧНОЇ ЗАДАЧI НЕЙМАНА
ДЛЯ ЛIНIЙНОГО ТЕЛЕГРАФНОГО РIВНЯННЯ
We investigate a periodic problem for the linear telegraph equation
utt − uxx + 2µut = f(x, t)
with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index
zero in the scale of Sobolev spaces of periodic functions. This result is stable under small perturbations of the equation
where µ becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions of
this problem possess smoothing properties.
Дослiджується перiодична задача для лiнiйного телеграфного рiвняння
utt − uxx + 2µut = f(x, t)
з крайовими умовами Неймана. Доведено, що оператор задачi моделюється фредгольмовим оператором нульового
iндексу у шкалi просторiв Соболєва перiодичних функцiй. Цей результат є стiйким щодо малих збурень рiвняння,
де µ стає змiнною i розривною або з’являється додатковий член нульового порядку. Також показано, що розв’язки
задачi мають властивiсть пiдвищення гладкостi.
1. Introduction. The telegraph equation
utt − uxx + 2µut + F (x, t, u) = 0, (x, t) ∈ (0, 1)× R, (1.1)
combines features of diffusion and wave equations. It describes dissipative wave processes, e. g.,
in transmission and propagation of electrical signals, dynamical processes in biological populations,
economical and ecological systems etc. (see [1, 6, 9, 10, 21] and references therein). Hillen [8]
discusses the appearance of Hopf bifurcations for (1.1) with Neumann boundary conditions in the
case when µ is a negative constant. An important step towards a rigorous bifurcation analysis (via
the Implicit Function Theorem and the Lyapunov – Schmidt reduction [4, 12]) is to establish the
Fredholm solvability of a linearized problem. Note in this respect that the Fredholm property for
hyperbolic PDEs is much less studied than for ODEs and parabolic PDEs.
We will investigate a linearization of (1.1) known as the damped wave equation
utt − uxx + 2µut = f(x, t), (x, t) ∈ (0, 1)× R, (1.2)
where µ is a constant. This equation describes a correlated random walk under the assumption that
particles with density u have a constant speed and a constant turning rate µ. Furthermore, we impose
time-periodicity conditions
*Supported by the Alexander von Humboldt Foundation and the DFG Research Center MATHEON mathematics for
key technologies (project D8).
c© I. KMIT, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3 381
382 I. KMIT
u
(
x, t+
2π
ω
)
= u(x, t), (x, t) ∈ [0, 1]× R,
(1.3)
ut
(
x, t+
2π
ω
)
= ut(x, t), (x, t) ∈ [0, 1]× R,
where ω > 0 is a particle speed, and Neumann boundary conditions
ux(0, t) = ux(1, t) = 0, t ∈ R. (1.4)
Solvability of periodic problems for the equation (1.2) and its nonlinear perturbation (1.1) is inves-
tigated, in particular, in [2, 3, 5, 8, 13 – 15, 19, 22]. Our main result (Theorem 4.2) is the Fredholm
alternative for the problem (1.2) – (1.4). Previously, the Fredholm zero index property was known
only in the double-periodic case [13, 15]. Our result is interesting in view of a general question:
Which function spaces can be used to capture Fredholm solvability for second-order hyperbolic
PDEs (for first-order hyperbolic PDEs, this question is investigated in [17, 18]).
Our argument is based on a reduction to a related periodic Neumann problem for a hyperbolic
system. When splitting u = v + w into the density v of particles moving right and the density w of
particles moving left, (1.2) – (1.4) is brought into the form:
vt + vx = g(x, t) + µ(w − v), (x, t) ∈ (0, 1)× R,
(1.5)
wt − wx = g(x, t) + µ(v − w), (x, t) ∈ (0, 1)× R,
v
(
x, t+
2π
ω
)
= v(x, t), (x, t) ∈ [0, 1]× R,
(1.6)
w
(
x, t+
2π
ω
)
= w(x, t), (x, t) ∈ [0, 1]× R,
v(0, t) = w(0, t), t ∈ R,
(1.7)
v(1, t) = w(1, t), t ∈ R.
System (1.5) describes a random walk process introduced by Taylor [23], where a particle moving
right dies with rate µ and is reborn as a particle moving left with the same rate. Homogeneous
Neumann boundary conditions (1.7) describe reflection of particles from the boundary.
For (1.5) – (1.7) we proved a Fredholm alternative in [17]. Now, we show the equivalence between
the models (1.2) – (1.4) and (1.5) – (1.7) in a certain functional analytic sense, which allows us to
derive the Fredholm property for (1.2) – (1.4) from the result about (1.5) – (1.7) obtained in [17].
By a perturbation argument, our results extend to the equation
utt − uxx + ν(x, t)ut + α(x, t)u = f(x, t), (x, t) ∈ (0, 1)× R, (1.8)
where ν(x, t)−2µ and α(x, t) are sufficiently small in appropriate function spaces (see Remark 4.3).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM FOR A LINEAR . . . 383
In Section 2 we introduce function spaces of solutions and give account of their useful properties.
Equivalence of the models (1.2) – (1.4) and (1.5) – (1.7) is proved in Section 3. Finally, in Section 4
we prove our main result (Theorem 4.2).
2. Function spaces and operators. We will use yet another representation of the problem (1.5) –
(1.7), namely,
ut + zx = g(x, t), (x, t) ∈ (0, 1)× R,
(2.1)
zt + ux = −2µz, (x, t) ∈ (0, 1)× R,
u
(
x, t+
2π
ω
)
= u(x, t), (x, t) ∈ [0, 1]× R,
(2.2)
z
(
x, t+
2π
ω
)
= z(x, t), (x, t) ∈ [0, 1]× R,
(2.3)
z(0, t) = z(1, t) = 0, t ∈ R,
where
u =
v + w
2
, z =
v − w
2
. (2.4)
Here 2u stands for the total particle density, while 2z for particle flux. The relationship between the
models (1.2) – (1.4), (1.5) – (1.7), and (2.1) – (2.3) is discussed in Section 3. We will show that they
are equivalent in a certain sense.
For solutions and right-hand sides of the problems (1.2) – (1.4), (1.5) – (1.7), and (2.1) – (2.3) we
now introduce pairs of spaces
(
Uγb , H
0,γ−1), (V γ
b ,W
γ
d
)
, and
(
Zγb ,W
γ
0
)
, respectively. Here γ ≥ 1
denotes a real scaling parameter. The subscript b indicates that we construct spaces constrained by
boundary conditions, d stands for the diagonal subspace of pairs (u, u), and 0 for the subspace of
pairs (u, 0).
Given l ∈ N0 and γ ≥ 0, we first introduce the space H l,γ of all measurable functions u : (0, 1)×
× R→ R such that
u(x, t) = u
(
x, t+
2π
ω
)
for a. a. (x, t) ∈ [0, 1]× R
and
‖u‖2Hl,γ :=
∑
k∈Z
(1 + k2)γ
l∑
m=0
1∫
0
∣∣∣∣∣∣∣
2π/ω∫
0
∂mx u(x, t)e
−ikωt dt
∣∣∣∣∣∣∣
2
dx <∞. (2.5)
It is well-known (see, e. g., [24], Chapter 2.4) that H l,γ is a Banach space. In fact, this is the space
of all
2π
ω
-periodic maps u : R→ H l(0, 1) that are locally L2-Bochner integrable together with their
generalized derivatives up to the (possibly noninteger) order γ. Furthermore, we define
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
384 I. KMIT
W γ = H0,γ ×H0,γ ,
W γ
d =
{
(g, f) ∈W γ : g =
}
,
W γ
0 =
{
(g, f) ∈W γ : f = 0
}
,
V γ =
{
(v, w) ∈W γ : (vt + vx, wt − wx) ∈W γ
}
,
Zγ =
{
(u, z) ∈W γ : (ut + zx, zt + ux) ∈W γ
}
,
Uγ =
{
u ∈ H0,γ : ux ∈ H0,γ−1, utt − uxx ∈ H0,γ−1},
where ut, ux, utt, uxx, vt, vx, zt, and zx are understood in the sense of generalized derivatives. The
function spaces W γ , V γ , Zγ , and Uγ will be endowed with the norms∥∥(g, f)∥∥2
W γ = ‖g‖2H0,γ + ‖f‖2H0,γ ,∥∥(v, w)∥∥2
V γ
=
∥∥(v, w)∥∥2
W γ +
∥∥(vt + vx, wt − wx)
∥∥2
W γ ,∥∥(u, z)∥∥2
Zγ
=
∥∥(u, z)∥∥2
W γ +
∥∥(ut + zx, zt + ux)
∥∥2
W γ ,∥∥u∥∥2
Uγ
=
∥∥u∥∥2
H0,γ +
∥∥ux∥∥2H0,γ−1 +
∥∥utt − uxx∥∥2H0,γ−1 .
In the following two lemmas we collect some useful properties of the function spaces V γ and
Uγ , respectively.
Lemma 2.1 ( [17], Section 2). (i) The space V γ is complete.
(ii) If γ ≥ 1, then V γ is continuously embedded into
(
H1,γ−1)2 .
(iii) For any x ∈ [0, 1] there exists a continuous trace map
(v, w) ∈ V γ 7→
(
v(x, ·), w(x, ·)
)
∈
(
L2
(
0,
2π
ω
))2
.
Similar properties are encountered in the function spaces Uγ .
Lemma 2.2. (i) The space Uγ is complete.
(ii) If γ ≥ 2, then Uγ is continuously embedded into H2,γ−2.
(iii) If γ ≥ 1, then for any x ∈ [0, 1] there exists a continuous trace map
u ∈ Uγ 7→ ux(x, ·) ∈ L2
(
0,
2π
ω
)
.
Proof. (i) Let (uj) be a fundamental sequence in Uγ . Then (uj) is fundamental in H0,γ and
(∂xuj) and
(
∂2t uj − ∂2xuj
)
are fundamental in H0,γ−1. Since H0,γ is complete, there exist u ∈ H0,γ
and v, w ∈ H0,γ−1 such that
uj → u in H0,γ , ∂xuj → v in H0,γ−1, and ∂2t uj − ∂2xuj → w in H0,γ−1
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM FOR A LINEAR . . . 385
as j → ∞. It remains to show that ∂xu = v and ∂2t u − ∂2xu = w in the sense of generalized
derivatives. For this purpose take a smooth function ϕ : (0, 1)×
(
0,
2π
ω
)
→ R with compact support
and note that
−
2π/ω∫
0
1∫
0
u∂xϕdx dt = − lim
j→∞
2π/ω∫
0
1∫
0
uj∂xϕdx dt =
= lim
j→∞
2π/ω∫
0
1∫
0
∂xujϕdx dt =
2π/ω∫
0
1∫
0
vϕ dx dt.
Similarly,
2π/ω∫
0
1∫
0
u(∂2t − ∂2x)ϕdx dt = lim
j→∞
2π/ω∫
0
1∫
0
uj(∂
2
t − ∂2x)ϕdx dt =
= lim
j→∞
2π/ω∫
0
1∫
0
(∂2t − ∂2x)ujϕdx dt =
2π/ω∫
0
1∫
0
wϕdxdt.
(ii) Take u ∈ Uγ . Then u ∈ H0,γ and, hence, ∂2t u ∈ H0,γ−2. By the definition of the space Uγ ,
we have ∂2xu ∈ H0,γ−2 as well. Therefore, u ∈ H2,γ−2. Moreover, we have
‖u‖2H2,γ−2 = ‖u‖2H0,γ−2 + ‖∂xu‖2H0,γ−2 +
∥∥∂2xu∥∥2H0,γ−2 ≤
≤ ‖u‖2H0,γ−2 + c‖∂xu‖2H0,γ−1 + c
∥∥∂2xu∥∥2H0,γ−2 ≤
≤ ‖u‖2H0,γ−2 + c‖∂xu‖2H0,γ−1 + c
∥∥∂2xu− ∂2t u∥∥2H0,γ−2 + c
∥∥∂2t u∥∥2H0,γ−2 ≤ C‖u‖2Uγ ,
where the constants c and C do not depend on u.
Claim (iii) follows from the definition of Uγ .
Remark 2.1. By (2.4) and Lemma 2.1 (iii), for any x ∈ [0, 1] there exists a continuous trace
map
(u, z) ∈ Zγ 7→
(
u(x, ·), z(x, ·)
)
∈
(
L2
(
0,
2π
ω
))2
.
Lemmas 2.1 (iii) and 2.2 (iii) and Remark 2.1 motivate consideration of the following closed
subspaces in V γ , Zγ , and Uγ :
V γ
b =
{
(v, w) ∈ V γ : (1.7) is fulfilled for a. a. t ∈ R
}
,
Zγb =
{
(u, z) ∈ Zγ : (2.3) is fulfilled for a. a. t ∈ R
}
,
Uγb =
{
u ∈ Uγ : (1.4) is fulfilled for a. a. t ∈ R
}
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
386 I. KMIT
Finally, we introduce linear operators LWS , L̃WS ∈ L
(
V γ
b ;W
γ
d
)
by
LWS
[
v
w
]
=
[
vt + vx − µ(w − v)
wt − wx − µ(v − w)
]
,
L̃WS
[
v
w
]
=
[
−vt − vx − µ(w − v)
−wt + wx − µ(v − w)
]
,
a linear operator L′WS ∈ L
(
Zγb ;W
γ
0
)
by
L′WS
[
u
z
]
=
[
ut + zx
zt + ux + 2µz
]
,
and linear operators LTE , L̃TE ∈ L
(
Uγb ;H
0,γ−1) by
LTE(u) = utt − uxx + 2µut,
L̃TE(u) = utt − uxx − 2µut.
3. Equivalence of the models. We first describe the reduction of the problem (1.5) – (1.7) to
(1.2) – (1.4) that was suggested in [11], see also [7]. Recall that the simple change of variables (2.4)
transforms the system (1.5) – (1.7) to the form (2.1) – (2.3). Now, assuming two-times differentiability
of z and w, we differentiate the first equation in (2.1) with respect to t and the second equation with
respect to x. After a simple calculation we come to the problem (1.2) – (1.4) with f = gt + 2µg.
Formally, we will show that the two problems (1.2) – (1.4) and (1.5) – (1.7) are equivalent in the
following sense: there exist isomorphisms α1 : V
γ
b → Zγb , α2 : Z
γ
b → Uγb and β1 : W
γ
d → W γ
0 ,
β2 : W
γ
0 → H0,γ−1 between the respective linear spaces such that the diagram
V γ
b
LWS−→ W γ
d
α1
y yβ1
Zγb
L′WS−→ W γ
0
α2
y yβ2
Uγb
LTE−→ H0,γ−1
(3.1)
is commutative, that is,
β1 ◦ LWS = L′WS ◦ α1, β2 ◦ L′WS = LTE ◦ α2. (3.2)
Specifically, we define α1, β1, α2, and β2 by
α1(v, w) =
(
v + w
2
,
v − w
2
)
, β1(g, g) = (g, 0), (3.3)
α2(u, z) = u, β2(g, 0) = gt + 2µg. (3.4)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM FOR A LINEAR . . . 387
Furthermore, let
α = α2 ◦ α1, β = β2 ◦ β1. (3.5)
The following result shows the commutativity of the upper part of the diagram (3.1) or, the same, the
equivalence of the problems (1.5) – (1.7) and (2.1) – (2.3). Its proof is straightforward.
Lemma 3.1. Suppose γ ≥ 2. Then
(i) The maps α1 : V
γ
b → Zγb and β1 : W
γ
d →W γ
0 defined by (3.3) are isomorphisms.
(ii) β1 ◦ LWS = L′WS ◦ α1.
To prove the commutativity of the lower part of the diagram (3.1), i. e., the equivalence of the
problems (1.2) – (1.4) and (2.1) – (2.3), we will need the following simple lemma.
Lemma 3.2. If µ 6= 0, then the map f ∈ H0,γ−1 7→ g ∈ H0,γ where
gt + 2µg = f,
(3.6)
g
(
x, t+
2π
ω
)
= g(x, t)
is bijective.
Proof. The problem (3.6) has a unique solution given by the formula
g(x, t) =
e
−2µ
(
t+
2π
ω
)
1− e−2µ
2π
ω
2π/ω∫
0
e2µτf(x, τ) dτ −
t∫
0
e2µ(τ−t)f(x, τ) dτ, (3.7)
which gives us the lemma.
Corollary 3.1. If µ 6= 0, then the map ux ∈ H0,γ−1 7→ z ∈ H0,γ where
zt + 2µz = ux,
(3.8)
z
(
x, t+
2π
ω
)
= z(x, t)
is bijective. Furthermore,
z(x, t) =
e
−2µ
(
t+
2π
ω
)
1− e−2µ
2π
ω
2π/ω∫
0
e2µτux(x, τ) dτ −
t∫
0
e2µ(τ−t)ux(x, τ) dτ. (3.9)
Now we state the desired equivalence result.
Lemma 3.3. Suppose γ ≥ 2. Then
(i) The maps α2 : Z
γ
b → Uγb and β2 : W
γ
0 → H0,γ−1 defined by (3.4) are isomorphisms.
(ii) β2 ◦ L′WS = LTE ◦ α2.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
388 I. KMIT
Proof. (i) We first show that α2 maps Zγb to Uγb and β2 maps W γ
0 to H0,γ−1. The latter is
obvious. To show the former, suppose that (u, z) ∈ Zγb solves (2.1) – (2.3). By the definition of
Zγ , zt ∈ H0,γ−1 and zt + ux ∈ H0,γ−1, hence ux ∈ H0,γ−1. Since z ∈ H1,γ−1, we have zt ∈
∈ H1,γ−2. Now, as zt + ux ∈ H0,γ , we have ux ∈ H1,γ−2 and, hence, uxx ∈ H0,γ−2. Therefore
ztx+uxx ∈ H0,γ−2 and utt+ zxt ∈ H0,γ−2. Consequently, utt−uxx ∈ H0,γ−2 as well. This implies
that u ∈ Uγ . To check the boundary conditions (1.3) and (1.4), we take into account part (iii) of
Lemma 2.2 about the traces of u. Now, conditions (1.4) follow from (2.3) and the second equation
of (2.1). Conditions (1.3) are a straightforward consequence of (2.2).
Further set
α−12 (u) = (u, z) where z is given by (3.9)
and
β−12 (f) = (g, 0) where g is given by (3.7).
We are done if we show that
α−12 maps Uγb into Zγb and β−12 maps H0,γ−1 intoW γ
0 (3.10)
and that
β−12 ◦ β2 = IW γ
0
, β2 ◦ β−12 = IH0,γ−1 , α2 ◦ α−12 = IUγb
, α−12 ◦ α2 = IZγb
. (3.11)
We start with proving (3.10). Since the right-hand side of the representation (3.9) belongs to
H0,γ , we have ∂tz ∈ H0,γ−1. Differentiating (3.9) with respect to t, we easily arrive at the second
equality of (2.1). To meet the first equality, we start from the weak formulation of (1.2) – (1.4): By
Lemma 3.2, any function f ∈ H0,γ−1 admits a unique representation f = gt +2µg where g ∈ H0,γ .
On the account of this fact, for any ϕ ∈ C1
(
[0, 1]×
[
0,
2π
ω
])
with ϕ
(
x, t+
2π
ω
)
= ϕ(x, t) we
have
0 =
2π/ω∫
0
1∫
0
[
− uttϕ− uxϕx − 2µutϕ+ (gt + 2µg)ϕ
]
dxdt =
=
2π/ω∫
0
1∫
0
[
utϕt − gϕt + [zt + 2µz]ϕx − 2µutϕ+ 2µgϕ
]
dxdt =
=
2π/ω∫
0
1∫
0
[
utϕt − gϕt + zxϕt − 2µzxϕ− 2µutϕ+ 2µgϕ
]
dxdt =
=
2π/ω∫
0
1∫
0
[
(ut + zx − g)ϕt − 2µ(zx + ut − g)ϕ
]
dxdt =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM FOR A LINEAR . . . 389
=
2π/ω∫
0
1∫
0
[
(ut + zx − g)(ϕt − 2µϕ)
]
dxdt.
Taking a constant ϕ, we get
2π/ω∫
0
1∫
0
(ut + zx − g) dxdt = 0. (3.12)
It follows that
ut + zx − g = 0 a.e. on (0, 1)×
(
0,
2π
ω
)
.
The first equality of (2.1) is therefore fulfilled. Furthermore, the system (2.1) implies that (u, z) ∈ Zγ .
The boundary conditions (2.2) and (2.3) follow from (1.3), (1.4), and (2.1).
To finish this part of the proof, it remains to note that the first two equalities in (3.11) follow by
Lemma 3.2, the third equality is straightforward, and the last one follows from (3.9) and the second
equality in (2.1).
(ii) Differentiating now the first equality of (2.1) with respect to t and the second with respect
to x, subtracting the resulting equations and then substituting zx from the first equation of (2.1), we
come to (1.2) with f = gt + 2µg. Hence (LTE ◦ α2) (u, z) = gt + 2µg. Moreover, by definitions of
L′WS and β2, we have L′WS(u, z) = (g, 0) and β2(g, 0) = gt + 2µg. The desired assertion follows.
Lemma 3.3 is proved.
We are prepared to formulate the main result of this section about the equivalence of the random
walk problem (1.5) – (1.7) and the telegraph problem (1.2) – (1.4).
Theorem 3.1. Suppose γ ≥ 2. Then
(i) The maps α : V γ
b → Uγb and β : W γ
d → H0,γ−1 defined by (3.5) are isomorphisms.
(ii) β ◦ LWS = LTE ◦ α.
The theorem follows directly from Lemmas 3.1 and 3.3.
4. Fredholm alternative. Here we prove the Fredholm alternative for the problem (1.2) – (1.4).
From Section 3 we know how the Fredholm and the index properties of the operators of the problems
(1.2) – (1.4) and (1.5) – (1.7) are related to each other. More precisely, the following lemma is true.
Lemma 4.1. Suppose γ ≥ 1 and µ 6= 0. Then
(i) dimker(LWS) = dimker(LTE),
(ii) dimker(L∗WS) = dimker(L∗TE).
The Fredholm solvability for the periodic-Neumann problem for the telegraph equation is now
a straightforward consequence of our Fredholm result for the corresponding correlated random walk
problem.
Theorem 4.1 ( [17], Theorem 1). Suppose γ ≥ 1 and µ 6= 0. Then we have:
(i) LWS is a Fredholm operator of index zero from V γ
b into W γ
d .
(ii) The image of LWS is the set of all (g, g) ∈W γ such that
2π/ω∫
0
1∫
0
g(x, t)(ṽ(x, t) + w̃(x, t)) dx dt = 0 for all (ṽ, w̃) ∈ ker(L̃WS). (4.1)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
390 I. KMIT
Remark 4.1. As it follows from the proof of [17] (Theorem 1), the kernel of the problem (1.5) –
(1.7) does not depend on γ ≥ 1 and, given γ > 1, all V 1
b -solutions to the problem (1.5) – (1.7) with
f ∈ W γ
d necessarily belong to V γ
b (a smoothing effect). This implies, in particular, that, if we have
(4.1) for f ∈W γ
d with γ ≥ 2, then we automatically have
2π/ω∫
0
1∫
0
∂st g(x, t)(ṽ(x, t) + w̃(x, t)) dx dt = 0 for all (ṽ, w̃) ∈ ker(L̃WS) and 0 ≤ s ≤ γ.
(4.2)
Note that the smoothing effect does not occur for the corresponding initial-boundary problem
with Neumann boundary conditions (see [7, 16, 20]).
We are prepared to formulate our main result.
Theorem 4.2. Suppose γ ≥ 2 and µ 6= 0. Then we have:
(i) LTE is a Fredholm operator of index zero from Uγb into H0,γ−1.
(ii) The image of LTE is the set of all f ∈ H0,γ−1 such that
2π/ω∫
0
1∫
0
f(x, t)u(x, t) dx dt = 0 for all u ∈ ker(L̃TE). (4.3)
Theorem 4.2 follows directly from Theorems 3.1 and 4.1 and Lemma 4.1 (see also Remark 4.1).
Remark 4.2. Like the problem (1.5) – (1.7), one can observe a similar smoothing effect for the
problem (1.2) – (1.4): The kernel of the problem does not depend γ and, given γ > 2, all U2
b -solutions
to problem (1.2) – (1.4) with f ∈ H0,γ−1, necessarily belong to Uγb .
Remark 4.3. Since the set of Fredholm operators is open, the conclusion of Theorem 4.2 sur-
vives under sufficiently small perturbations of the operator LTE ∈ L
(
Uγ ;H0,γ−1). The theorem
remains true, if, instead of LTE , we consider the operator of the problem (1.8), (1.3), (1.4) with ν
and α such that ν(x, t)ut ∈ H0,γ , ν(x, t) is sufficiently small perturbation of a nonzero constant,
α(x, t)u ∈ H0,γ , and α(x, t) is sufficiently small. The Fredholm property of the slightly perturbed
problem is fulfilled independently of whether or not the perturbed problems (1.2) – (1.4) and (1.5) –
(1.7) remain equivalent.
1. Ahmed E., Hassan S. Z. On diffusion in some biological and economic systems // Z. Naturforsch. A. – 2000. – 55. –
669 S.
2. Alonso J. M., Mawhin J., Ortega R. Bounded solutions of second order semilinear evolution equations and applications
to the telegraph equation // J. math. pures et appl. – 1999. – 78, № 1. – P. 49 – 63.
3. Bereanu C. Periodic solutions of the nonlinear telegraph eqations with bounded nonlinearities // J. Math. Anal. and
Appl. – 2008. – 343. – P. 758 – 762.
4. Chow S.-N., Hale J. K. Methods of bifurcation theory // Grundlehren math. Wiss. – New York; Berlin: Springer,
1982. – 251. – 515 S.
5. Fucik S., Mawhin J. Generalized periodic solutions of nonlinear telegraph equations // Nonlinear Anal., Theory,
Methods, Appl. – 1978. – 2. – P. 609 – 617.
6. Hadeler K. P. Reaction transport systems // Math. Inspired by Biology / Eds V. Capasso, O. Diekmann. – Springer,
1998. – P. 95 – 150.
7. Hillen T. Existence theory for correlated random walks on bounded domains // Can. Appl. Math. Q. – 2010. – 18,
№ 1. – P. 1 – 40.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
FREDHOLM SOLVABILITY OF A PERIODIC NEUMANN PROBLEM FOR A LINEAR . . . 391
8. Hillen T. A Turing model with correlated random walk // J. Math. Biol. – 1996. – 35, № 1. – P. 49 – 72.
9. Horsthemke W. Spatial instabilities in reaction random walks with direction-independent kinetics // Phys. Rev. E. –
1990. – 60. – P. 2651 – 2663.
10. Jeffrey A. Applied partial differential equations: An introduction. – New York: Acad. Press, 2002. – 394 p.
11. Kac M. A stochastic model related to the telegrapher’s equation // Rocky Mountain J. Math. – 1956. – 4. – P. 497 – 509.
12. Kielhöfer H. Bifurcation theory. An introduction with applications to PDEs // Appl. Math. Sci. – New York; Berlin:
Springer, 2004. – 156. – 346 p.
13. Kim Wan Se. Double-periodic boundary value problem for nonlinear dissipative hyperbolic equations // J. Math. Anal.
and Appl. – 1990. – 145, № 1. – P. 1 – 16.
14. Kim Wan Se. Multiple doubly-periodic solutions of semilinear dissipative hyperbolic equations // J. Math. Anal. and
Appl. – 1996. – 197, № 3. – P. 735 – 748.
15. Kim Wan Se. Multiplicity results for periodic solutions of semilinear dissipative hyperbolic equations with coercive
nonlinear term // J. Korean Math. Soc. – 2001. – 38, № 4. – P. 853 – 881.
16. Kmit I. Smoothing solutions to initial-boundary problems for first-order hyperbolic systems // Appl. Anal. – 2011. –
90, № 11. – P. 1609 – 1634.
17. Kmit I., Recke L. Fredholm Alternative for periodic-Dirichlet problems for linear hyperbolic systems // J. Math. Anal.
and Appl. – 2007. – 335, № 1. – P. 355 – 370.
18. Kmit I., Recke L. Fredholmness and smooth dependence for linear hyperbolic periodic-Dirichlet problems // J.
Different. Equat. – 2012. – 252, № 2. – P. 1962 – 1986.
19. Li Y. Positive doubly-periodic solutions of nonlinear telegraph equations // Nonlinear Anal. – 2003. – 55. – P. 245 – 254.
20. Lyulko N. A. The increasing smoothness properties of solutions to some hyperbolic problems in two independent
variables // Sib. Electron. Math. Reprts. – 2010. – 7. – P. 413 – 424.
21. Markovich P. A., Ringhofer C. A., Schmeister C. Semiconductor equations. – New York: Springer, 1990. – 258 p.
22. Mawhin J., Ortega R., Robles-PГ c©rez A. M. A maximum principle for bounded solutions of the telegraph equations
and applications to nonlinear forcings // J. Math. Anal. and Appl. – 2000. – 251, № 2. – P. 695 – 709.
23. Taylor G. I. Diffusion by discontinuous movements // Proc. London Math. Soc. – 1920. – P. 25 – 70.
24. Vejvoda O. et al. Partial differential equations: time-periodic solutions. – Sijthoff Noordhoff, 1981. – 358 p.
Received 01.02.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 3
|
| id | umjimathkievua-article-2425 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:10Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cc/c9201ffbd6ffac5130387117c3bb03cc.pdf |
| spelling | umjimathkievua-article-24252020-03-18T19:15:16Z Fredholm solvability of a periodic Neumann problem for a linear telegraph equation Фредгольмовiсть перiодичної задачi неймана для лiнiйного телеграфного рiвняння Kmit, I. Ya. Кміть, І. Я. We investigate a periodic problem for the linear telegraph equation $$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$ with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions. This result is stable under small perturbations of the equation where p becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions of this problem possess smoothing properties. Дослiджується перiодична задача для лiнiйного телеграфного рiвняння $$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$ з крайовими умовами Неймана. Доведено, що оператор задачi моделюється фредгольмовим оператором нульового iндексу у шкалi просторiв Соболєва перiодичних функцiй. Цей результат є стiйким щодо малих збурень рiвняння, де µ стає змiнною i розривною або з’являється додатковий член нульового порядку. Також показано, що розв’язки задачi мають властивiсть пiдвищення гладкостi Institute of Mathematics, NAS of Ukraine 2013-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2425 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 3 (2013); 381-391 Український математичний журнал; Том 65 № 3 (2013); 381-391 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2425/1617 https://umj.imath.kiev.ua/index.php/umj/article/view/2425/1618 Copyright (c) 2013 Kmit I. Ya. |
| spellingShingle | Kmit, I. Ya. Кміть, І. Я. Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title | Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title_alt | Фредгольмовiсть перiодичної задачi неймана для лiнiйного телеграфного рiвняння |
| title_full | Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title_fullStr | Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title_full_unstemmed | Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title_short | Fredholm solvability of a periodic Neumann problem for a linear telegraph equation |
| title_sort | fredholm solvability of a periodic neumann problem for a linear telegraph equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2425 |
| work_keys_str_mv | AT kmitiya fredholmsolvabilityofaperiodicneumannproblemforalineartelegraphequation AT kmítʹíâ fredholmsolvabilityofaperiodicneumannproblemforalineartelegraphequation AT kmitiya fredgolʹmovistʹperiodičnoízadačinejmanadlâlinijnogotelegrafnogorivnânnâ AT kmítʹíâ fredgolʹmovistʹperiodičnoízadačinejmanadlâlinijnogotelegrafnogorivnânnâ |