Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 153, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
INVERSE COEFFICIENT PROBLEM FOR THE SEMI-LINEAR
FRACTIONAL TELEGRAPH EQUATION
HALYNA LOPUSHANSKA, VITALIA RAPITA
Abstract. We establish the unique solvability for an inverse problem for semilinear fractional telegraph equation
Dtα u + r(t)Dtβ u − ∆u = F0 (x, t, u, Dtβ u),
(x, t) ∈ Ω0 × (0, T ]
with regularized fractional derivatives Dtα u, Dtβ u of orders α ∈ (1, 2), β ∈ (0, 1)
with respect to time on bounded cylindrical domain. This problem consists
in the determination of a pair of functions: a classical solution u of the first
boundary-value problem for such equation, and an unknown continuous coefficient r(t) under the over-determination condition
Z
u(x, t)ϕ(x)dx = F (t), t ∈ [0, T ]
Ω0
with given functions ϕ and F .
1. Introduction
The existence and uniqueness theorems for fractional Cauchy problems were
proved in [2, 4, 5, 7, 11, 12, 13, 14, 15, 16, 24] and other works. The conditions of
classical solvability of the first boundary-value problem for equation
Dtβ u(x, t) − A(x, D)u(x, t) = F0 (x, t)
with regularized fractional derivative (see, for example, [4]) and some elliptic differential second order operator A(x, D) were obtained in [18] and [19].
Equations with fractional derivatives are applied in studying of anomalous diffusion and various processes in physics, mechanics, chemistry and engineering. The
telegraph fractional equations in theory of thermal stresses is considered, for example, in [21]. Inverse problems to such equations arise in many branches of science
and engineering. Some inverse boundary-value problems to diffusion-wave equation
with different unknown functions or parameters were investigated, for example, in
[1, 3, 6, 9, 17, 20, 22, 25]. In particular, the article [1] was devoted to determination of a source term for a time fractional diffusion equation with an integral type
over-determination condition.
2010 Mathematics Subject Classification. 35S15.
Key words and phrases. Fractional derivative; inverse boundary value problem;
over-determination integral condition; Green’s function; integral equation.
c
2015
Texas State University - San Marcos.
Submitted April 22, 2015. Published June 11, 2015.
1
2
H. LOPUSHANSKA, V. RAPITA
EJDE-2015/153
In this note we prove the existence and uniqueness of a classical solution (u, r)
of the inverse boundary-value problem
Dtα u + r(t)Dtβ u − ∆u = F0 (x, t, u, Dtβ u),
u(x, t) = 0,
(x, t) ∈ Ω0 × (0, T ],
(x, t) ∈ ∂Ω0 × [0, T ],
(1.1)
(1.2)
u(x, 0) = F1 (x),
x ∈ Ω̄0 ,
(1.3)
ut (x, 0) = F2 (x),
x ∈ Ω̄0 ,
(1.4)
Z
u(x, t)ϕ0 (x)dx = F (t),
t ∈ [0, T ]
(1.5)
Ω0
for regularized telegraph equation, where α ∈ (1, 2), β ∈ (0, 1), Ω0 is a bounded
domain in RN , N ≥ 3, with a boundary ∂Ω0 of class C 1+s , s ∈ (0, 1), F0 , F1 , F2 ,
F , ϕ0 – given functions. We shall use Green’s functions to prove the solvability of
this problem.
2. Definitions and auxiliary results
We shall use the notation: Ω1 = ∂Ω0 , Qi = Ωi × (0, T ], i = 0, 1, Q2 = Ω0 ,
D(Rm ) is a space of indefinitely differentiable functions with compact supports in
∂ k
Rm , m = 1, 2, . . . , D(Q0 ) = {v ∈ C ∞ (Q0 ) : ( ∂t
) v|t=T = 0, k = 0, 1, . . . },
0
m
0
D (R ) and D (Q0 ) are spaces of linear continuous functionals (generalized functions [23, p. 13-15]) over D(Rm ) and D(Q0 ), respectively, (f, ϕ) stands for the value
of f ∈ D0 (Rm ) on the test function ϕ ∈ D(Rm ) and also the value of f ∈ D0 (Q0 )
on ϕ ∈ D(Q0 ).
We denote by f ∗ g the convolution of generalized functions f and g, use the
function
(
θ(t)tλ−1
for λ > 0
Γ(λ)
fλ (t) =
0
f1+λ
(t) for λ ≤ 0,
where Γ(z) is the Gamma-function, and θ(t) is the Heaviside function. Note that
fλ ∗ fµ = fλ+µ .
(α)
Also note that the Riemann-Liouville derivative vt (x, t) of order α > 0 is defined
by
(α)
vt (x, t) = f−α (t) ∗ v(x, t)
and
h ∂ Z t v(x, τ )
1
v(x, 0) i
Dtα v(x, t) =
dτ
−
Γ(1 − α) ∂t 0 (t − τ )α
tα
(α)
= vt (x, t) − f1−α (t)v(x, 0),
for α ∈ (0, 1),
while
Dtα v(x, t) =
=
h∂
1
Γ(2 − α) ∂t
(α)
vt (x, t)
Z
0
t
vτ (x, τ )
vt (x, 0) i
dτ
−
(t − τ )α−1
(t − τ )α−1
− f1−α (t)v(x, 0) − f2−α (t)vt (x, 0)
for α ∈ (1, 2).
We denote Dt1 v = ∂v
∂t .
Let C(Q0 ), C(Q0 ), C[0, T ] be spaces of continuous functions on Q0 , Q0 and
[0, T ], respectively, C γ (Ω0 ) (C γ (Ω̄0 )) be a space of bounded continuous functions
on Ω0 (Ω̄0 ) satisfying Hölder continuity condition, C γ (Qi ) (C γ (Q̄i )), i = 0, 1, be a
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION
3
space of bounded continuous functions on Qi (Q̄i ) which for all t ∈ (0, T ] satisfies
Hölder continuity condition with respect to space variables, C γ (Q0 ×R2 ) be a space
of bounded continuous functions f (x, t, v, z) on Q0 × R2 which for all t ∈ (0, T ],
v, z ∈ R satisfies Hölder continuity condition with respect to space variables x ∈ Ω0 ,
Cβ (Q0 ) = {v ∈ C(Q0 ) : Dtβ v ∈ C(Q0 )},
Cβγ (Q0 ) = {v ∈ C γ (Q0 ) : Dtβ v ∈ C γ (Q0 )},
C2,α (Q0 ) = {v ∈ C(Q0 ) : ∆v, Dtα v ∈ C(Q0 )},
C2,α (Q̄0 ) = {v ∈ C2,α (Q0 ) : v, vt ∈ C(Q̄0 )}.
We use the following assumptions:
(A1) F0 ∈ C γ (Q0 × R2 ), γ ∈ (0, 1), |F0 (x, t, v, w)| ≤ A0 + B0 [|v|q + |w|p ],
|F0 (x, t, v1 , w1 ) − F0 (x, t, v2 , w2 )| ≤ D0 [|v1 − v2 |q + |w1 − w2 |p ]
(A2)
(A3)
(A4)
(A5)
for all v, w, v1 , v2 , w1 , w2 ∈ R where p, q ∈ [0, 2], A0 , B0 , D0 are some positive constants,
F1 ∈ C γ (Ω̄0 ), F1 |Ω1 = 0,
F2 ∈ C γ (Ω̄0 ),
F, Dβ F, Dα F ∈ C[0, T ], there exists f := inf t∈[0,T ] |Dβ F (t)| > 0,
ϕ0 ∈ C 2 (Ω̄0 ), ϕ0 |Ω1 = 0.
Definition 2.1. A pair of functions (u, r) ∈ C2,α (Q̄0 ) × C[0, T ] satisfying (1.1) on
Q0 and the conditions (1.2)-(1.5) is called a solution of the problem (1.1)-(1.5).
From (1.3), (1.4) and (1.5), it follows the necessary agreement conditions
Z
Z
F2 (x)ϕ0 (x)dx = F 0 (0).
(2.1)
F1 (x)ϕ0 (x)dx = F (0),
Ω0
Ω0
We introduce the operators
(α)
(Lv)(x, t) ≡ vt (x, t) − ∆v(x, t),
(x, t) ∈ Q0 ,
v ∈ D0 (Q0 ),
(Lreg v)(x, t) ≡ Dtα v(x, t) − ∆v(x, t),
(x, t) ∈ Q0 ,
v ∈ C2,α (Q̄0 ).
Definition 2.2. A vector-function (G0 (x, t, y, τ ), G1 (x, t, y), G2 (x, t, y)) is called a
Green’s vector-function of the problem
(Lreg u)(x, t) = g0 (x, t),
u(x, t) = 0,
u(x, 0) = g1 (x),
(x, t) ∈ Q0 ,
(x, t) ∈ Q̄1 ,
ut (x, 0) = g2 (x),
if for rather regular g0 , g1 , g2 the function
Z t Z
2 Z
X
u(x, t) =
dτ
G0 (x, t, y, τ )g0 (y, τ )dy +
0
Ω0
j=1
(2.2)
(2.3)
x ∈ Ω̄0 ,
(2.4)
Gj (x, t, y)gj (y)dy,
(2.5)
Ω0
for (x, t) ∈ Q0 , is the classical solution (in C2,α (Q̄0 )) of the first boundary-value
problem (2.2)–(2.4).
It follows from Definition 2.2 that
(LG0 )(x, t, y, τ ) = δ(x − y, t − τ ),
reg
(L
Gj )(x, t, y) = 0,
(x, t), (y, τ ) ∈ Q0 ,
(x, t) ∈ Q0 , y ∈ Ω0 , j = 1, 2,
4
H. LOPUSHANSKA, V. RAPITA
G1 (x, 0, y) = δ(x − y),
G2 (x, 0, y) = 0,
EJDE-2015/153
∂
G1 (x, 0, y) = 0,
∂t
∂
G2 (x, 0, y) = δ(x − y),
∂t
x, y ∈ Ω0
where δ is Dirac delta-function.
Lemma 2.3 ([15]). The following relations hold:
Z t
Gj (x, t, y) =
fj−α (τ )G0 (x, t, y, τ )dτ, (x, t) ∈ Q̄0 , y ∈ Ω0 , j = 1, 2.
0
Lemma 2.4. A Green’s vector-function of the first boundary-value problem (2.2)–
(2.4) exists.
The above lemma is proved following the strategy in [17, Lemma 2].
Theorem 2.5. If g0 ∈ C γ (Q0 ), γ ∈ (0, 1), gj ∈ C γ (Ω̄0 ), j = 1, 2, g1 |Ω1 = 0 then
there exists a unique solution u ∈ C2,α (Q̄0 ) of (2.2)–(2.4). It is defined by
u(x, t) = G0 g0 (x, t) + G1 g1 )(x, t) + G2 g2 )(x, t), (x, t) ∈ Q̄0
(2.6)
where
t
Z
(G0 g0 )(x, t) =
Z
dτ
0
G0 (x, t, y, τ )g0 (y, τ )dy,
Ω0
Z
(Gj gj )(x, t) =
Gj (x, t, y)gj (y)dy,
j = 1, 2.
Ω0
Proof. Taking Lemmas 2.3 and 2.4 into account, as in [7, 8, 10, 24] for the Cauchy
problems, we show that the function (2.6) belongs to C2,α (Q̄0 ) and satisfies the
problem (2.2)–(2.4). We use the estimates founded in [7, 11, 15, 24]:
C0∗
, |x − y|2 < 4(t − τ )α ,
(t − τ )|x − y|N −2
Cj∗ tj−1−α
, j = 1, 2, |x − y|2 < 4tα ,
|Gj (x, t, y)| ≤
|x − y|N −2
N
1
|x−y|2
Ĉ0 (t − τ )α−1 |x − y|2 1+ 2(2−α)
−c( 4a (t−τ )α ) 2−α
0
|G0 (x, t, y, τ )| ≤
e
|x − y|N
4(t − τ )α
α−1
C0 (t − τ )
≤
, |x − y|2 > 4(t − τ )α
|x − y|N
N
1
|x−y|2 2−α
Ĉj tj−1 |x − y|2 2(2−α)
Gj (x, t, y) ≤
e−c( 4tα )
N
α
|x − y|
4t
Cj tj−1
≤
, j = 1, 2, |x − y|2 > 4tα
|x − y|N
|G0 (x, t, y, τ )| ≤
where c, Cj∗ , Cj , Ĉj
(j = 0, 1, 2) are positive constants;
|Gj (x + ∆x, t + ∆t, y, τ ) − Gj (x, t, y, τ )| ≤ Aj (x, t, y, τ )[|∆x| + |∆t|α/2 ]γ ,
|Dtβ Gj (x + ∆x, t + ∆t, y, τ ) − Dtβ Gj (x, t, y, τ )|
≤ Aβ,j (x, t, y, τ )[|∆x| + |∆t|α/2 ]γ
∀(x, t), (x + ∆x, t + ∆t) ∈ Q̄0 , (y, τ ) ∈ Q̄j , j = 0, 1, 2
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION
5
with some 0 < γ < 1 where non-negative functions Aj (x, t, y, τ ), Aβ,j (x, t, y, τ ) have
the same kind of estimates as Gj (x, t, y, τ ), Dtβ Gj (x, t, y, τ ), j = 0, 1, 2, have, respectively, and Gj (x, t, y, τ ) = Gj (x, t, y), Aj (x, t, y, τ ) = Aj (x, t, y), Aβ,j (x, t, y, τ ) =
Aβ,j (x, t, y) for j = 1, 2. Note that for the general boundary-value problem to a
parabolic equation with partial derivatives the last properties of a Green’s vectorfunction were obtained in [10].
3. Existence and uniqueness theorems for the inverse problem
Now we prove the existence of a solution for the inverse problem (1.1)–(1.5).
It follows from the theorem 2.5 that under assumptions (A1), (A2), (A3) for a
given r ∈ C[0, T ] the solution u ∈ C2,α (Q̄0 ) of the first boundary-value problem
(1.1)–(1.4) satisfies
t
Z
u(x, t) = −
Z
r(τ )dτ
G0 (x, t, y, τ )Dτβ u(y, τ )dy
+
Ω0
β
h0 (x, t, u, Dt u)
(3.1)
0
+ h(x, t),
(x, t) ∈ Ω0
where
h0 (x, t, u, Dtβ u) =
t
Z
Z
0
h(x, t) =
G0 (x, t, y, τ )F0 (y, τ, u(y, τ ), Dτβ u(y, τ ))dy,
dτ
Ω0
2 Z
X
j=1
(3.2)
Gj (x, t, y)Fj (y)dy,
(x, t) ∈ Ω0 .
Ω0
Conversely, any solution u ∈ Cβγ (Q0 ) of (3.1) belongs to C2,α (Q̄0 ) and is the solution
of (1.1)–(1.5).
It follows from the equation (1.1) and assumption (A5) that
Z
Dtβ u(x, t)ϕ0 (x)dx
Dtα u(x, t)ϕ0 (x)dx + r(t)
Ω0
Z
Z Ω0
=
u(x, t)∆ϕ0 (x)dx +
F0 (x, t, u(x, t), Dtβ u(x, t))ϕ0 (x)dx,
Z
Ω0
t ∈ (0, T ].
Ω0
Using (1.5) we obtain
Dα F (t) + r(t)Dβ F (t)
Z
Z
=
u(x, t)∆ϕ0 (x)dx +
Ω0
F0 (x, t, u(x, t), Dtβ u(x, t))ϕ0 (x)dx
Ω0
(here Dα F (t) = Dtα F (t) and so on); that is, by using (A4),
r(t) =
hZ
Z
u(x, t)∆ϕ0 (x)dx +
F0 (x, t, u(x, t), Dtβ u(x, t))ϕ0 (x)dx
Ω0
Ω0
i
− Dα F (t) [Dβ F (t)]−1 , t ∈ (0, T ].
(3.3)
6
H. LOPUSHANSKA, V. RAPITA
EJDE-2015/153
Note that, for u ∈ Cβγ (Q0 ), the function r(t), defined by (3.3), belongs to C[0, T ].
By substituting the right-hand side of (3.3) into (3.1) in place of r(t) we obtain
Z
Z t
[Dβ F (τ )]−1 dτ
G0 (x, t, y, τ )
u(x, t) =
0
Ω0
hZ
×
F0 (z, τ, u(z, τ ), Dτβ u(z, τ ))ϕ0 (z)dz
(3.4)
Z Ω0
i
α
β
+
u(z, τ )∆ϕ0 (z)dz − D F (τ ) Dτ u(y, τ )dy
Ω0
+ h0 (x, t, u(x, t), Dtβ u(x, t)) + h(x, t),
(x, t) ∈ Q̄0 ,
where the functions h0 , h are defined by (3.2). We have reduced the problem (1.1)–
(1.5) to system (3.3), (3.4). Conversely, a pair (u, r) ∈ Cβγ (Q0 ) × C[0, T ] satisfying
(3.3) and (3.4) is a solution of the problem (1.1)-(1.5).
Thus, under assumptions (A1)–(A5) and (2.1), a pair of functions (u, r) ∈
C2,α (Q̄0 )×C[0, T ] is the solution of (1.1)-(1.5) if and only if the function u ∈ Cβγ (Q0 )
is the solution of (3.4), with r ∈ C[0, T ] defined by (3.3).
Theorem 3.1. Under the assumptions (A1)–(A5) and the condition (2.1), there
exists T ∗ ∈ (0, T ] (Q∗0 = Ω0 × (0, T ∗ ]) and a solution (u, r) ∈ C2,α (Q̄∗0 ) × C[0, T ∗ ]
of (1.1)-(1.5). The function u is the solution of (3.4), and r(t) is defined by (3.3).
Proof. From the previous conclusion, it is sufficient to prove the solvability of the
equation (3.4) in Cβγ (Q0 ). We shall use the Schauder principle. Let krkC[0,T ] =
maxt∈[0,T ] |r(t)|, and
n
kvkCβγ (Q0 ) = max
sup |v(x, t)|, sup |Dtβ v(x, t)|,
(x,t)∈Q0
(x,t)∈Q0
|v(x + ∆x, t) − v(x, t)|
,
sup
|∆x|γ
(x,t)∈Q0 ,|∆x|<1
|Dtβ v(x + ∆x, t) − Dtβ v(x, t)| o
,
|∆x|γ
(x,t)∈Q0 ,|∆x|<1
sup
Let R be some positive number, and
MR = MR (Q0 ) = {v ∈ Cβγ (Q0 ) : kvkCβγ (Q0 ) ≤ R}.
On MR we consider the operator
Z t
Z
(P v)(x, t) := −
[Dβ F (τ )]−1 dτ
0
G0 (x, t, y, τ )
Ω0
Z
v(z, τ )∆ϕ0 (z)dz
Ω0
+ F0 (z, τ, v(z, τ ), Dtβ v(z, τ ))ϕ0 (z)dz − Dtα F (τ ) Dτβ v(y, τ )dy
+ h0 (x, t, v, Dtβ v) + h(x, t),
(x, t) ∈ Ω0 , v ∈ MR
with the functions h0 , h defined by (3.2).
At the beginning we show the existence of R > 0, T ∗ ∈ (0, T ] and therefore
∗
MR = MR (Q∗0 ) such that P : MR∗ → MR∗ .
For v ∈ MR , (x, t) ∈ Q̄0 we find the estimates
|h0 (x, t, v, Dtβ v)|
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION
Z t Z
dτ
=
0
t
Z
≤
G0 (x, t, y, τ )F0 (y, τ, v(y, τ ), Dtβ v(y, τ ))dy Ω0
dτ
hZ
(y,τ )∈Ω0 :
|y−x|<2(t−τ )α/2
0
Z
+
(y,τ )∈Ω0 :
|y−x|>2(t−τ )α/2
t
Z
≤
dτ
(y,τ )∈Ω0 :
|y−x|<2(t−τ )α/2
Z
+
(y,τ )∈Ω0 :
|y−x|>2(t−τ )α/2
Z th
Z
|G0 (x, t, y, τ )| |F0 (y, τ, v(y, τ ), Dtβ v(y, τ ))|dy
|G0 (x, t, y, τ )| |F0 (y, τ, v(y, τ ), Dtβ v(y, τ ))|dy
hZ
0
7
i
C∗
A0 + |v(y, τ |q + |Dtβ v(y, τ ))|p dy
(t − τ )|y − x|N −2
i
C∗
β
q
p
A
+
|v(y,
τ
|
dy
+
|D
v(y,
τ
))|
0
t
(t − τ )1−α |y − x|N
2(t−τ )α/2
1
(t − τ )
rdr +
1
(t − τ )1−α
Z
diam Ω0
i
dr i h
dτ A0 + B0 (Rq + Rp )
0
2(t−τ )α/2 r
0
Z th
i
h
i
diam Ω0
q
p
≤ Ĉ
(t − τ )α−1 + (t − τ )α−1 ln
dτ
A
+
B
(R
+
R
)
0
0
(t − τ )α/2
0
α1
q
p
≤ k0 t A0 + B0 (R + R )
≤C
where C ∗ , C, Ĉ, k̂, k0 are positive constants, and α1 = α − %, with % an arbitrary
number in (0, 1). Also we have
Z
Gj (x, t, y)Fj (y)dy Ω0
Z
hZ
i
≤
G
(x,
t,
y)dy
+
Gj (x, t, y)dy kFj kC(Ω̄0 )
j
(y,τ )∈Ω :
(y,τ )∈Ω :
0
0
|y−x|<2tα/2
≤ c0
hZ
(y,τ )∈Ω0 :
|y−x|<2tα/2
Z
+
|y−x|>2tα/2
(y,τ )∈Ω0 :
|y−x|>2tα/2
h
≤ k̂j tj−1 1 +
Z
tj−1−α
dy
|x − y|N −2
i
1
N
|y−x|2
|y − x|2 2(2−α)
tj−1
−c( 4tα ) 2−α
dy
kFj kC(Ω̄0 )
e
|y − x|N
4tα
diam Ω0
r
N
2−α −1
t
αN
− 2(2−α)
2tα/2
≤ kj tj−1 kFj kC(Ω̄0 ) ,
e
−ĉ
r2
tα
2−α
1
i
dr · kFj kC(Ω̄0 )
j = 1, 2
where c0 , ĉ, k̂j , kj (j = 1, 2) are positive constants. Therefore,
X
|h(x, t)| ≤
kj tj−1 kFj kC(Ω̄0 ) , (x, t) ∈ Q̄0 .
j=1,2
Similarly,
Z
F0 (y, τ, v(y, τ ), Dtβ v(y, τ ))ϕ0 (z)dy Ω0
Z
≤ A0 + B0 (Rq + Rp )
|ϕ0 (z)|dz ∀(y, τ ) ∈ Q̄0
Ω0
8
H. LOPUSHANSKA, V. RAPITA
EJDE-2015/153
and therefore
Z
Z t
[Dβ F (τ )]−1 dτ
G0 (x, t, y, τ )F0 (z, τ, v(z, τ ), Dtβ v(z, τ ))ϕ0 (z)dz 0
Ω0
Z
k0 α1 ≤ t A0 + B0 (Rq + Rp )
|ϕ0 (z)|dz ∀(y, τ ) ∈ Q̄0 .
f
Ω0
Then, given R ≥ 1, we obtain the estimate
k0
|(P v)(x, t)| ≤ tα1 (c1 R + c2 R2 ) + H0 (t) ≤ q0 tα1 R2 + H0 , (x, t) ∈ Q̄0
f
where
Z
c1 =
dx · kDα F kC[0,T ] ,
Ω0
Z
Z
c2 = 2B0 f +
|ϕ0 (z)|dz +
|∆ϕ0 (z)|dz,
Ω0
Ω0
Z
1
|ϕ0 (z)|dz + k1 kF1 kC(Ω̄0 ) + k2 tkF2 kC(Ω̄0 ) ≤ H0 ,
H0 (t) = A0 k0 tα1 1 +
f Ω0
k0 (c1 + c2 )
q0 =
.
f
In the same way for v ∈ MR , (x, t) ∈ Q0 , |∆x| < 1 we obtain
|(P v)(x + ∆x, t) − (P v)(x, t)|
≤ q1 tα1 R2 + H1 ,
|∆x|γ
(x, t) ∈ Q0 .
Here and later qj , cj , Hj (j=1,2,. . . ) are positive numbers.
For every g ∈ C γ (Q0 ) we have
Z t
Z
(t − τ )−β dτ
β
Dt G0 g (x, t) =
G0 (x, t, y, τ )g(y, τ )dy,
Γ(1 − β)
0
Ω0
and, as previously, we find that
|Dtβ G0 g (x, t)| ≤ q̃0 tα1 −β kgkC(Q0 ) ,
(x, t) ∈ Q0
(x, t) ∈ Q0 , q̃0 = const > 0.
Furthermore,
Z t
Z
∂
(t − τ )−β dτ
Dtβ G1 F1 (x, t) =
G1 (x, τ, y)F1 (y)dy − f1−β (t)F1 (x)
∂t 0 Γ(1 − β)
Ω0
Z t
=
f1−β (t − τ ) G1 F1 τ (x, τ )dτ,
0
Z t
Z
∂
(t − τ )−β dτ
Dtβ G2 F2 (x, t) =
G2 (x, τ, y)F2 (y)dy − f2−β (t)F2 (x)
∂t 0 Γ(1 − β)
Ω0
Z t
=
f1−β (t − τ ) G2 F2 τ (x, τ )dτ.
0
Since Gj Fj
τ
(j = 1, 2) are continuous functions on Q̄0 , we have
β
Dt Gj Fj (x, t) ≤ c2+j t1−β , (x, t) ∈ Q̄0 , j = 1, 2.
So,
|Dtβ (P v)(x, t)| ≤ q2 tα2 R2 + H2 ,
(x, t) ∈ Q̄0
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION
9
and similarly
|Dtβ (P v)(x + ∆x, t) − Dtβ (P v)(x, t)|
≤ q3 tα2 R2 + H3 ,
|∆x|γ
(x, t) ∈ Q0 , |∆x| < 1
for all v ∈ MR with α2 = α1 − β and
H2 = c5 + c6 kF1 kC(Ω̄0 ) + c7 tkF2 kC(Ω̄0 ) ,
H3 = c8 + c9 kF1 kC(Ω̄0 ) + c10 kF2 kC(Ω̄0 ) .
As a result we obtain
kP vkCβγ (Q0 )
≤ max max{q0 tα1 R2 + H0 , q1 tα1 R2 + H1 , q2 tα2 R2 + H2 , q3 tα2 R2 + H3 }
t∈[0,T ]
≤ max [qR2 tα2 + H] ∀v ∈ MR , R ≥ 1
t∈[0,T ]
with q = max{q0 T β , q1 T β , q2 , q3 }, H = max{H0 , H1 , H2 , H3 }.
To show the inequality
max{qR2 tα2 + H, 1} ≤ R
∀t ∈ [0, T ∗ ], v ∈ MR∗
2 α2
0
(3.5)
α2
consider the function w(s) = qs t − s, s ≥ 0. We find w (s) = 2qt s − 1 and
prove that s0 = s0 (t) = [2qtα2 ]−1 is the point of the minimum of f (s). Then the
inequality
kP vkCβγ (Q∗0 ) ≤ R ∀v ∈ MR∗
is satisfied for some R ≥ max{1, H}, T ∗ = min{t∗ , T } if qtα2 s20 − s0 ≤ −H and
2qtα2 max{1, H} ≤ 1 for all t ∈ [0, t∗ ]. We have qtα2 s0 − s0 = − 4qt1α2 . There exists
t∗ > 0 such that 4qtα2 H ≤ 1 and 2qtα2 max{1, H} ≤ 1 for all t ∈ [0, t∗ ]. Then
t∗ = min{[
1 1/α2
1
]
,[
]1/α2 }.
4qH
2q max{1, H}
Note that (2.1) and (A4) imply kF1 kC(Ω̄0 ) > 0. We have proven the existence of
R ≥ 1, T ∗ > 0 such that P : MR∗ → MR∗ .
The operator P is continuous on
M̃R∗ = {v ∈ Cβ (Q∗0 ) : kvkCβ (Q∗0 ) = max{kvkC(Q∗0 ) , kDtβ vkC(Q∗0 ) } ≤ R};
thus, on MR∗ . Namely, for v1 , v2 ∈ M̃R∗ , (x, t) ∈ Q∗0 ,
|(P v1 )(x, t) − (P v2 )(x, t)|
Z
Z t
hZ
=
[Dβ F (τ )]−1 dτ
G0 (x, t, y, τ )
v2 (z, τ )∆ϕ0 (z)dz[Dτβ v2 (y, τ )
0
Ω0
Z Ω0
β
− Dτ v1 (z, τ )] −
[v1 (z, τ ) − v2 (z, τ )]∆ϕ0 (z)dz · Dτβ v1 (y, τ )
Ω0
Z
i
+
[F0 (z, τ, v1 (z, τ ), Dτβ v1 (z, τ ) − F0 (z, τ, v2 (z, τ ), Dτβ v2 (z, τ )]ϕ0 (z)dz dy
Ω0
+ h0 (z, τ, v1 (z, τ ), Dτβ v1 (z, τ ) − h0 (z, τ, v2 (z, τ ), Dτβ v2 (z, τ )
Z t
Z
G0 (x, t, y, τ )dy
≤ c11 sup
[Dβ F (τ )]−1 dτ
(x,t)∈Q∗
0
0
Ω0
10
H. LOPUSHANSKA, V. RAPITA
EJDE-2015/153
h
× kv2 kC(Q∗0 ) kDβ v1 − Dβ v2 kC(Q∗0 ) + kDβ v1 kC(Q∗0 ) kv1 − v2 kC(Q∗0 )
i
+ 2D0 kv1 − v2 kqC(Q∗ ) + 2D0 kDβ v1 − Dβ v2 kpC(Q∗ )
0
0
Z t Z
h
G0 (x, t, y, τ )dy kv1 − v2 kq ∗
dτ
+ 2D0 sup
C(Q )
(x,t)∈Q∗
0
0
0
Ω0
+ 2D0 kDβ v1 − Dβ v2 kpC(Q∗ )
i
0
≤ c12 (T ∗ )α1 Rkv1 − v2 kCβγ (Q∗0 ) .
Similarly, by using previous estimates,
|Dtβ (P v1 )(x, t) − Dtβ (P v2 )(x, t)| ≤ c13 (T ∗ )α2 Rkv1 − v2 kCβγ (Q∗0 )
for all v1 , v2 ∈ M̃R∗ , (x, t) ∈ Q̄∗ 0 .
The operator P is compact on M̃R∗ (and thus on MR∗ ): it was established earlier
the uniform boundedness of the set
P M̃R∗ := {(P v)(x, t), (x, t) ∈ Q∗0 : v ∈ M̃R∗ }
in addition, it follows from the properties of Green’s operators that for all ε > 0
there exists δ = δ(ε) such that for all (x, t) ∈ Q∗0 , |∆x| < δ, |∆t| < δ and for all
v ∈ M̃R∗
|(P v)(x + ∆x, t + ∆t) − (P v)(x, t)| < ε,
sup
(x,t)∈Q∗
0
sup
(x,t)∈Q∗
0
|Dtβ (P v)(x + ∆x, t + ∆t) − Dtβ (P v)(x, t)| < ε.
As a result, the operator P is equicontinuous on MR∗ . According to Schauder
principle there exists a solution u ∈ MR∗ of the equation (3.4).
Theorem 3.2. Assume that F0 ∈ C 1 (Q0 × R2 ) and is bounded, Dβ F ∈ C[0, T ]
and Dβ F (t) 6= 0, t ∈ [0, T ], ϕ satisfies the assumption (A5) and ϕ(x) 6= 0, x ∈ Ω0 .
Then a solution (u, r) ∈ C2,α (Q̄0 ) × C[0, T ] of the problem (1.1)-(1.5) is unique.
Proof. Take two solutions (u1 , r1 ), (u2 , r2 ) ∈ C2,α (Q̄0 ) × C[0, T ] of (1.1)-(1.5) and
substitute them into equation (1.1). For u = u1 − u2 , r = r1 − r2 we obtain the
equation
Dtα u = ∆u − r1 (t)Dtβ u − r(t)Dtβ u2 + F0 (x, t, u1 , Dtβ u1 ) − F0 (x, t, u2 , Dtβ u2 ).
By Hadamard lemma
F0 (x, t, u1 , Dtβ u1 ) − F0 (x, t, u2 , Dtβ u2 ) = F01 (x, t)u + F02 (x, t)Dtβ u
with some known functions F0j , j = 1, 2, which are continuous and bounded on
Q0 , depend on u1 , u2 , Dtβ u1 , Dtβ u2 . Then the previous equation becomes
Dtα u + r1 (t) − F02 (x, t) Dtβ u = ∆u + F01 (x, t)u − r(t)Dtβ u2 , (x, t) ∈ Q0 .
It follows from the boundary condition that
u(x, t) = 0,
x ∈ Ω̄1 ,
t ∈ [0, T ],
and from the initial conditions
u(x, 0) = 0,
ut (x, 0) = 0,
x ∈ Ω̄0 .
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 11
Let G∗0 (x, t, y, τ ) be the main Green’s function of the first boundary-value problem for the equation
Dtα u + r1 (t) − F02 (x, t) Dtβ u = ∆u + F01 (x, t)u.
Then the function u(x, t) satisfies the equation
Z t Z
dτ
G∗0 (x, t, y, τ )Dτβ u2 (y, τ )r(τ )dy,
u(x, t) = −
0
(x, t) ∈ Q̄0 .
Ω0
It follows from the over-determination condition (1.5) that
Z h
β
r(t)D F (t) =
u(x, t)∆ϕ0 (x)
Ω0
i
+ F01 (x, t)u(x, t) + F02 (x, t)Dtβ u(x, t) ϕ0 (x) dx,
and then u(x, t) satisfies the equation
Z
Z t
h
dτ
K ∗ (x, t, τ ) F01 (z, τ )ϕ0 (z) + ∆ϕ0 (z) u(z, τ )dz
u(x, t) =
β
0 D F (τ ) Ω0
i
+ F02 (z, τ )ϕ0 (z)Dτβ u(z, τ ) dz, (x, t) ∈ Q̄0
(3.6)
(3.7)
where
Z
∗
G∗0 (x, t, y, τ )Dτβ u2 (y, τ )dy
K (x, t, τ ) =
Ω0
is continuous with respect to x and integrable in time function.
If F01 (z, τ ) = F02 (z, τ ) = 0, (z, τ ) ∈ Q0 then by the uniqueness of the solution of
this linear second type Volterra integral equation we obtain u(x, t) = 0, (x, t) ∈ Q0 .
Then it follows from (3.6) that r(t)Dβ F (t) = 0, t ∈ [0, T ]. Since Dβ F (t) 6= 0 on
[0, T ] (under the assumptions of this theorem), it follows that r(t) ≡ 0 for t ∈ [0, T ].
Assume that
|F01 (z, τ )| + |F02 (z, τ )| =
6 0, (z, τ ) ∈ Q0 .
(3.8)
If F02 (z, τ ) = 0, (z, τ ) ∈ Q0 then by the uniqueness of the solution of the linear
second type Volterra integral equation (3.7) with integrable kernel
K ∗ (x, t, τ ) F01 (z, τ )ϕ0 (z) + ∆ϕ0 (z)
we obtain, as in previous case, that u(x, t) = 0, (x, t) ∈ Q0 and then, from (3.6),
that r(t) ≡ 0, t ∈ [0, T ].
In the general case denote
V (x, t) = F01 (x, t)ϕ0 (x) + ∆ϕ0 (x) u(x, t) + F02 (x, t)ϕ0 (x)Dtβ u(x, t).
Then (3.7) implies
Z t
V (x, t) =
0
+
dτ
Dβ F (τ )
Z
F01 (x, t)ϕ0 (x) + ∆ϕ0 (x) K ∗ (x, t, τ )V (z, τ )dz
Ω0
F02 (x, t)ϕ0 (x)Dtβ
Z
0
t
dτ
Dβ F (τ )
Z
K ∗ (x, t, τ )V (z, τ )dz;
Ω0
that is,
Z
V (x, t) =
t
Z
dτ
0
K(x, t, z, τ )V (z, τ )dz,
Ω0
(x, t) ∈ Q̄0 ,
12
H. LOPUSHANSKA, V. RAPITA
EJDE-2015/153
where
K(x, t, z, τ )
h
1
F02 (x, t)ϕ0 (x)(t − τ )−β i ∗
F
(x,
t)ϕ
(x)
+
∆ϕ
(x)
+
K (x, t, τ ).
01
0
0
Dβ F (τ )
Γ(1 − β)
By the uniqueness of the solution of this linear second type Volterra integral equation with integrable kernel K(x, t, z, τ ) we obtain V (x, t) = 0, (x, t) ∈ Q0 .
Note that (3.8) implies
=
|F01 (x, t)ϕ0 (x) + ∆ϕ0 (x)| + |F02 (x, t)ϕ0 (x)| =
6 0,
(x, t) ∈ Q0 .
Note also that
Dtβ u(x, t) = 0 ⇐⇒ f−β (t) ∗ u(x, t) = 0
⇐⇒ fβ (t) ∗ f−β (t) ∗ u(x, t) = 0
⇐⇒ u(x, t) = 0, (x, t) ∈ Q0 ,
for the function u(x, t) satisfying zero initial conditions.
Then it follows from the previous results that u(x, t) = 0, (x, t) ∈ Q0 and from
(3.6), by the assumptions of this theorem, we obtain r(t) ≡ 0, t ∈ [0, T ].
In separate case F01 (x, t)ϕ0 (x) + ∆ϕ0 (x) = 0 for all (x, t) ∈ Q0 we may put
V1 (x, t) = F02 (x, t)ϕ0 (x)Dtβ u(x, t) and, as before, obtain the linear second type
Volterra integral equation
Z t Z
V1 (x, t) =
dτ
K1 (x, t, z, τ )V1 (z, τ )dz, (x, t) ∈ Q̄0
0
Ω0
with integrable kernel
K1 (x, t, z, τ ) = F02 (x, t)ϕ0 (x)K ∗ (x, t, τ )(t − τ )−β Γ(1 − β)Dβ F (τ ).
As before, from here we obtain V1 (x, t) = 0, (x, t) ∈ Q0 and, as in the general case,
since F02 (x, t)ϕ0 (x) 6= 0 on Q0 , conclude that u(x, t) = 0, (x, t) ∈ Q0 and r(t) ≡ 0,
t ∈ [0, T ].
The similar result holds for the inverse problem on determination of a pair of
functions (u, b): a solution u of the first (or second) boundary-value problem for
the equation
Dtα u = ∆u + b(t)u = F0 (x, t), (x, t) ∈ Q0
and an unknown coefficient b(t) under the same over-determinating condition (1.2).
We may study the cases N = 1, 2 in the same way.
Acknowledgments. The authors are grateful to Prof. Mokhtar Kirane for useful
discussions.
References
[1] T. S. Aleroev, M. Kirane, S. A. Malik; Determination of a source term for a time fractional diffusion equation with an integral type over-determination condition, Electronic J. of
Differential Equations, 2013 (2013), no. 270, 1–16.
[2] V. V. Anh, N. N. Leonenko; Spectral analysis of fractional kinetic equations with random
datas, 104 (2001), no. 5/6, 1349-1387.
[3] J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki; Uniqueness in an inverse problem for
a one-dimentional fractional diffusion equation, Inverse problems, 25 (2000), 1–16.
[4] M. M. Djrbashian, A. B. Nersessyan; Fractional derivatives and Cauchy problem for differentials of fractional order, Izv. AN Arm. SSR. Matamatika, 3 (1968), 3-29.
EJDE-2015/153
INVERSE PROBLEM FOR THE FRACTIONAL TELEGRAPH EQUATION 13
[5] Jun Sheng Duan; Time- and space-fractional partial differential equations, J. Math. Phis.,
46 (2005), 013504.
[6] Mahmoud M. El-Borai; On the solvability of an inverse fractional abstract Cauchy problem,
LJRRAS, 4 (2010), 411–415.
[7] S. D. Eidelman, S. D. Ivasyshen, A. N. Kochubei; Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser Verlag, Basel-BostonBerlin, 2004.
[8] A. Friedman; Partial differential equations of parabolic type, Englewood Cliffs, N. J., PrenticeHall, 1964.
[9] Y. Hatano, J. Nakagawa, Sh. Wang, M. Yamamoto; Determination of order in fractional
diffusion equation, Journal of Math-for-Industry, 5A, (2013), 51–57.
[10] S. D. Ivasyshen; Green matrix of parabolic boundary value problems, Vyshcha shkola, Kiev,
1990.
[11] A. N. Kochubei; Fractional-order diffusion, Differential Equations, 26 (1990), 485-492.
[12] A. N. Kochubei, S. D. Eidelman; Equations of one-dimentional fractional-order diffusion,
Dop. NAS of Ukraine, 12 (2003), 11-16.
[13] H. Lopushanska, A. Lopushansky, E. Pasichnyk; The Cauchy problem in a space of generalized
functions for the equations possessing the fractional time derivative, Sib. Math. J., 52 (2011),
no. 6, 1288-1299.
[14] H. P. Lopushanska, A. O. Lopushansky; Space-time fractional Cauchy problem in spaces of
generalized functions, Ukrainian Math. J., 64 (2013), no. 8, 1215-1230. doi: 10.1007/s11253013-0711-z (translaited from Ukr. Mat. Zhurn. 64 (2012), 8, 1067-1080)
[15] A. O. Lopushansky; Solution of fractional Cauchy problem in spaces of generalized functions,
Visnyk Lviv. Univ. Ser. Mech.-Math., 77 (2012), 132-144.
[16] A. O. Lopushansky; The Cauchy problem for an equation with fractional derivatives in Bessel potential spaces, Sib. Math. J., 55 (2014), no. 6, 1089-1097 - DOI:
10.1134/30037446614060111.
[17] A. O. Lopushansky, H. P. Lopushanska; One inverse boundary value problem for diffusionwave equation with fractional derivarive, Ukr. math. J., 66 (2014), no. 5, 655-667.
[18] Yu. Luchko; Boundary value problem for the generalized time-fractional diffusion equation
of distributed order, Fract. Calc. Appl. Anal., 12 (2009), no 4, 409-422.
[19] M. M. Meerschaert, Nane Erkan, P. Vallaisamy; Fractional Cauchy problems on bounded
domains, Ann. Probab., 37 (2009), 979-1007.
[20] J. Nakagawa, K. Sakamoto, M. Yamamoto; Overview to mathematical analysis for fractional
diffusion equation – new mathematical aspects motivated by industrial collaboration, Journal
of Math-for-Industry, 2A (2010), 99–108.
[21] Y. Povstenko; Theories of thermal stresses based on space-time-fractional telegraph equations,
Computers and Mathematics with Applications, 64 (2012), 3321 –3328.
[22] W. Rundell, X. Xu and L. Zuo; The determination of an unknown boundary condition in
fractional diffusion equation, Applicable Analysis, 1 (2012), 1–16.
[23] G. . Shilov; Mathimatical Analyses. Second special curse, Nauka, Moskow, 1965.
[24] A. A. Voroshylov, A. A. Kilbas; Conditions of the existence of classical solution of the Cauchy
problem for diffusion-wave equation with Caputo partial derivative, Dokl. Ak. Nauk., 414
(2007), no. 4, 1-4.
[25] Y. Zhang, X. Xu; Inverse source problem for a fractional diffusion equation, Inverse problems,
27 (2011), 1–12.
Halyna Lopushanska
Department of Differential Equations, Ivan Franko National University of Lviv, Lviv,
Ukraine
E-mail address: lhp@ukr.net
Vitalia Rapita
Department of Differential Equations, Ivan Franko National University of Lviv, Lviv,
Ukraine
E-mail address: vrapita@gmail.com