Journal of Inequalities in Pure and
Applied Mathematics
ON AN INTEGRAL INEQUALITY WITH A KERNEL SINGULAR
IN TIME AND SPACE
volume 4, issue 4, article 82,
2003.
NASSER-EDDINE TATAR
King Fahd University of Petroleum and Minerals
Department of Mathematical Sciences
Dhahran 31261, Saudi Arabia.
EMail: tatarn@kfupm.edu.sa
Received 07 March, 2003;
accepted 01 August, 2003.
Communicated by: D. Bainov
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
030-03
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Abstract
In this paper we deal with a nonlinear singular integral inequality which arises
in the study of partial differential equations. The integral term is non local in
time and space and the kernel involved is also singular in both the time and
the space variable. The estimates we prove may be used to establish (global)
existence and asymptotic behavior results for solutions of the corresponding
problems.
On an Integral Inequality with a
Kernel Singular in Time and
Space
2000 Mathematics Subject Classification: 42B20, 26D07, 26D15.
Key words: Nonlinear integral inequality, Singular kernel.
Nasser-eddine Tatar
The author would like to thank King Fahd University of Petroleum and Minerals for its
financial support.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
8
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1.
Introduction
We consider the following integral inequality
Z Z t
F (s)ϕm (s, y)
dyds,
(1.1) ϕ(t, x) ≤ k(t, x) + l(t, x)
1−β |x − y|n−α
Ω 0 (t − s)
x ∈ Ω, t > 0,
where Ω is a domain in Rn (n ≥ 1) (bounded or possibly equal to Rn ), the
functions k(t, x), l(t, x) and F (t) are given positive continuous functions in t.
The constants 0 < α < n, 0 < β < 1 and m > 1 will be specified below.
This inequality arises in the theory of partial differential equations, for example, when treating the heat equation with a source of polynomial type

 ut (t, x) = ∆u(t, x) + um (t, x), x ∈ Rn , t > 0, m > 1
On an Integral Inequality with a
Kernel Singular in Time and
Space
Nasser-eddine Tatar
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 u(0, x) = u (x), x ∈ Rn .
0
If we write the (weak) solution using the fundamental solution G(t, x) of the
heat equation
ut (t, x) = ∆u(t, x),
namely,
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Z tZ
Z
G(t, x − y)u0 (y)dy +
u(t, x) =
Rn
0
G(t − s, x − y)um (s, y)dyds
Rn
and take into account the Solonnikov estimates of this fundamental solution (see
[14] for instance), then one is led to an inequality of type (1.1).
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The features of this inequality, which make it difficult to deal with, are the
singularities of the kernel in both the space and time variables. It is also non
integrable with respect to the time variable. The standard methods one can find
in the literature (see the recent books by Bainov and Simeonov [1] and Pachpatte
[12] and the references therein) concerning regular and/or summable kernels
cannot be applied in our situation. Indeed, these methods are based on estimates
involving the value of the kernels at zero and/or some Lp −norms of the kernels.
In contrast, there are very few papers dealing explicitly with singular kernels
similar to ours. Let us point out, however, some works concerning integral
equations with singularities in time. In Henry [4, Lemmas 7.1.1 and 7.1.2], a
similar inequality to (1.1) with only the integral with respect to time, namely
Z t
ψ(t) ≤ a(t) + b (t − s)β−1 sγ−1 ψ(s)ds, β > 0, γ > 0
0
i.e. the linear case (m = 1) has been treated. The case m > 1 has been
considered by Medved in [9], [10]. More precisely, the following inequality
Z t
ψ(t) ≤ a(t) + b(t) (t − s)β−1 sγ−1 F (s)ψ m (s)ds, β > 0, γ > 0
0
was discussed. The result was used to prove a global existence and an exponential decay result for a parabolic Cauchy problem with a source of power type
and a time dependent coefficient, namely

 ut + Au = f (t, u), u ∈ X,
 u(0) = u ∈ X
0
On an Integral Inequality with a
Kernel Singular in Time and
Space
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with kf (t, u)k ≤ tκ η(t) kukm
α , m > 1, κ ≥ 0, where A is a sectorial operator
(see [4]) and k·kα stands for the norm of the fractional space X α associated to
the operator A (see also [11]). This, in turn, has been improved and extended to
integro-differential equations and functional differential equations by M. Kirane
and N.-E. Tatar in [6] (see also N.-E. Tatar [13] and S. Mazouzi and N.-E. Tatar
[7, 8] for more general abstract semilinear evolution problems).
Here, we shall combine the techniques in [9, 10], based on the application
of Lemma 2.1 and the use of Lemma 2.3 below, with the Hardy-LittlewoodSobolev inequality (see Lemma 2.2) to prove our result. We will give sufficient
conditions yielding boundedness by continuous functions, exponential decay
and polynomial decay of solutions to the integral inequality (1.1).
The paper is organized as follows. In the next section we prepare some
notation and lemmas needed in the proofs of our results. Section 3 contains the
statement and proof of our result and two corollaries giving sufficient conditions
for the exponential decay and the polynomial decay. Finally we point out that
our results hold (a fortiori) for weakly singular kernels in time.
On an Integral Inequality with a
Kernel Singular in Time and
Space
Nasser-eddine Tatar
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2.
Preliminaries
In this section we introduce some material necessary for our results. We will
use the usual Lp -space with its norm k·kp .
Lemma 2.1. (Young inequality) We have, for
1
r
=
1
ρ
+ 1q , the inequality
kf gkr ≤ kf kρ · kgkq .
Lemma 2.2. Let α ∈ [0, 1) and β ∈ R. There exists a positive constant C =
C(α, β) such that

Ceβt ,
if β > 0;




Z t

−α βs
C(t + 1), if β = 0;
s e ds ≤

0




C,
if β < 0.
Lemma 2.3. (Hardy-Littlewood-Sobolev inequality)
Let u ∈ Lp (Rn ) (p > 1), 0 < γ < n and nγ > 1 − p1 , then (1/ |x|γ ) ∗
u ∈ Lq (Rn ) with 1q = nγ + p1 − 1. Also the mapping from u ∈ Lp (Rn ) into
(1/ |x|γ ) ∗ u ∈ Lq (Rn ) is continuous.
See [5, Theorem 4.5.3, p. 117].
Lemma 2.4. Let a(t), b(t), K(t), ψ(t) be nonnegative, continuous functions
on the interval I = (0, T ) (0 < T ≤ ∞), ω : (0, ∞) → R be a continuous,
On an Integral Inequality with a
Kernel Singular in Time and
Space
Nasser-eddine Tatar
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nonnegative and nondecreasing function, ω(0) = 0, ω(u) > 0 for u > 0 and let
A(t) = max0≤s≤t a(s), B(t) = max0≤s≤t b(s). Assume that
Z t
ψ(t) ≤ a(t) + b(t)
K(s)ω(ψ(s))ds, t ∈ I.
0
Then
ψ(t) ≤ W
−1
Z t
W (A(t)) + B(t)
K(s)ds , t ∈ (0, T1 ),
0
where
Z
v
W (v) =
v0
W
−1
dσ
, v ≥ v0 > 0,
ω(σ)
is the inverse of W and T1 > 0 is such that
Z t
W (A(t)) + B(t)
K(s)ds ∈ D(W −1 )
0
See [2] (or [5]) for the proof.
Lemma 2.5. If δ, ν, τ > 0 and z > 0, then
Z z
1−ν
z
(z − ζ)ν−1 ζ δ−1 e−τ ζ dζ ≤ M (ν, δ, τ ) ,
0
See [6] for the proof of this lemma.
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for all t ∈ (0, T1 ).
where M (ν, δ, τ ) = max (1, 21−ν ) Γ (δ) 1 +
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Kernel Singular in Time and
Space
δ
ν
τ −δ .
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3.
Estimation
In this section we state and prove our result on boundedness and also present an
exponential and a polynomial decay result.
Theorem 3.1. Assume that the constants α, β and m are such that 0 < α < βn,
0 < β < 1 and m > 1.
(m−1)n m
n
(i) If Ω = R , then for any r satisfying max
, β < r < mn
, we have
α
α
On an Integral Inequality with a
Kernel Singular in Time and
Space
kϕ(t, x)kr ≤ Up,r,ρ (t)
with
Nasser-eddine Tatar
Up,r,ρ (t) = 2
m(p−1)
r
1
K(t) p
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× 1 − 2m(p−1) (m − 1)C1p−1 C2p K(t)m−1 L(t)eεpt
Z
×
t
−εps
e
p
F (s)ds
Contents
m
(1−m)r
,
0
where K(t) = max0≤s≤t kk(s, ·)kpr , L(t) = max0≤s≤t kl(s, ·)kpρ , p = mr
nr
and ρ = αr−(m−1)n
for some ε > 0. Here C1 and C2 are the best constants
in Lemma 2.2 and Lemma 2.3, respectively. The estimation is valid as long
as
Z t
1
m−1
εpt
(3.1) K(t)
L(t)e
e−εps F p (s)ds ≤ m(p−1) (m − 1)C1p−1 C2p .
2
0
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(ii) If Ω is bounded, then
kϕ(t, x)kr̃ ≤ Up,r,ρ (t)
n
(but
for any r̃ ≤ r where p, r and ρ are as in (i). If moreover, r < nβ−α
not necessarily r > (m−1)n
) that is m
< r < min mn
, n , then this
α
β
α nβ−α
estimation holds for any
1
β
<p≤
r
m
provided that ρ >
nr
.
n−(nβ−α)r
Proof. (i) Suppose that Ω = Rn . By the Minkowski inequality and the Young
inequality (Lemma 2.1), we have
(3.2)
kϕ(t, x)kr ≤ kk(t, x)kr
Z
+ kl(t, x)kρ Z
Rn
for r, ρ and q such that
1
r
1
q
=
1
+ 1q .
ρ
α
and p0
n
Let p be such that p1 = +
the Hölder inequality, we see that
0
t
F (s)ϕm (s, y)
dsdy
n−α
(t − s)1−β |x − y|
q
1
p
its conjugate i.e.
+
1
p0
= 1. Using
0
Z
≤
0
t
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(t − s)β−1 F (s)ϕm (s, y)ds
(3.3)
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t
Z
On an Integral Inequality with a
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p10 Z t
p1
−εps p
mp
(β−1)p0 εp0 s
(t − s)
e ds
e
F (s)ϕ (s, y)ds ,
0
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εs
−εs
for any positive constant ε. We have multiplied by e ·e
Hölder inequality.
before applying
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nr
Choosing q = (nm−αr)
, we see that p = mr . By our assumption on r it is
easy to see that q > 1, p > 1 and 1 + (β − 1)p0 > 0. Therefore, we may
apply Lemma 2.2 to get
Z t
0
0
0
(t − s)(β−1)p eεp s ds < C1 eεp t .
0
Hence, inequality (3.3) becomes
Z
t
β−1
(t − s)
m
1
p0
εt
Z
F (s)ϕ (s, y)ds ≤ C1 e
p1
t
−εps
e
p
F (s)ϕ
mp
(s, y)ds
.
0
0
It follows that
Z Z
(3.4) Nasser-eddine Tatar
t
F (s)ϕm (s, y)
dsdy
n−α
1−β
n
|x − y|
R
0 (t − s)
q
Z Z
p1
t
1
dy
0
e−εps F p (s)ϕmp (s, y)ds
≤ C1p eεt n−α .
Rn
|x − y|
0
q
As r <
(3.5)
mn
,
α
On an Integral Inequality with a
Kernel Singular in Time and
Space
we may apply the results in Lemma 2.3 to obtain
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Z Z
p1
t
dy
e−εps F p (s)ϕmp (s, y)ds
n−α Rn
|x − y|
0
q
Z
p1 t
−εps p
mp
≤ C2 e
F (s)ϕ (s, ·)ds ,
0
p
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with p as above and C2 is the best constant in the result of Lemma 2.3.
From (3.2), (3.4) and (3.5) it appears that
kϕ(t, x)kr
Z
p1 t
≤ kk(t, x)kr + C3 eεt kl(t, x)kρ e−εps F p (s)ϕmp (s, ·)ds 0
p
or
(3.6)
On an Integral Inequality with a
Kernel Singular in Time and
Space
kϕ(t, x)kr
Z
εt
≤ kk(t, x)kr + C3 e kl(t, x)kρ
−εps
e
Rn
p1
t
Z
p
F (s)ϕ
mp
(s, x)dsdx
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,
0
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1
p0
where C3 = C1 C2 . Inequality (3.6) can also be written as
Contents
kϕ(t, x)kr
εt
Z
t
−εps
≤ kk(t, x)kr + C3 e kl(t, x)kρ
e
F
p
(s) kϕ(s, x)kmp
mp
1
p
ds
.
0
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kϕ(t, x)kr
≤ kk(t, x)kr + C3 eεt kl(t, x)kρ
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Observe that by our choice of p we have r = mp. It follows that
(3.7)
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Z
0
t
e−εps F p (s) kϕ(s, x)krr ds
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p1
.
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Applying the algebraic inequality
(a + b)p ≤ 2p−1 (ap + bp ), a, b ≥ 0, p > 1,
we deduce from (3.7) that
(3.8)
kϕ(t, x)kpr
p−1
≤2
kk(t, x)kpr
εpt
+ C4 e
kl(t, x)kpρ
Z
t
e−εps F p (s) kϕ(s, ·)krr ds,
0
where C4 =
form
2p−1 C3p .
Let us put ψ(t) =
kϕ(t, x)kr/m
,
r
then (3.8) takes the
On an Integral Inequality with a
Kernel Singular in Time and
Space
Nasser-eddine Tatar
ψ(t) ≤ 2p−1 kk(t, ·)kpr + C4 eεpt kl(t, x)kpρ
Z
t
e−εps F p (s)ψ m (s)ds.
0
1
By the Lemma 2.4 with ω(u) = um , W (v) = 1−m
(v 1−m − v01−m ) and
1
W −1 (z) = (1 − m)z + v01−m 1−m , we conclude that
Z t
−1
p−1
εpt
−εps p
ψ(t) ≤ W
W 2 K(t) + C4 e L(t)
e
F (s)ds
0
m−1 εpt
p−1
≤ 2 K(t) 1 − (m − 1)C4 2p−1 K(t)
e L(t)
Z
×
t
e−εps F p (s)ds
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1
1−m
,
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0
where K(t) and L(t) are as in the statement of the theorem.
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(ii) If Ω is bounded, then the first part in the assertion (ii) of the theorem
follows from the argument in (i) by an extension procedure and the applin
cation of the embedding Lr (Ω) ⊂ Lr̃ (Ω) for r̃ ≤ r. Now if r < nβ−α
n
nr
we choose q such that r < q < nβ−α
and ρ > n−(nβ−α)r
. The Young
np
inequality is therefore applicable. Also as 1 < q < n−αp , the HardyLittlewood-Sobolev inequality (Lemma 2.3) applies (see also [3, p. 660]
when Ω is bounded).
In what follows, in order to simplify the statement of our next results, we
define v := r or r̃ according to the cases (i) or (ii) in Theorem 3.1, respectively.
Corollary 3.2. Suppose that the hypotheses of Theorem 3.1 hold. Assume
further that k(t, x) and l(t, x) decay exponentially in time, that is k(t, x) ≤
e−k̃t k̄(x) and l(t, x) ≤ e−l̃t ¯l(x) for some positive constants k̃ and ˜l. Then
ϕ(t, x) is also exponentially decaying to zero i.e.,
(3.9)
kϕ(t, x)kv ≤ C5 e−µt , t > 0
for some positive constants C5 and µ provided that
Z ∞
m−1 1
p
k̄ (x)
¯l(x)
F
(s)ds
≤
(m − 1)C6p−1 C2p ,
r
ρ
m(p−1)
2
0
where C6 is the best constant in Lemma 2.2 (third estimation) and the other
constants are as in (i) and (ii) of Theorem 3.1.
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Proof. From the inequality (1.1) we have
Z Z
−k̃t
−l̃t ¯
(3.10)
ϕ(t, x) ≤ e k̄(x) + e l(x)
Ω
t
0
F (s)ϕm (s, y)
dsdy.
(t − s)1−β |x − y|n−α
After multiplying by e−mµt · emµt , where µ = min{k̃, ˜l}, we use the Hölder
inequality to get
Z
t
(t − s)β−1 F (s)ϕm (s, y)ds
0
Z
≤
t
(β−1)p0 −mp0 µt
(t − s)
e
p1
p10 Z t
p
mpµt mp
F (s)e
ϕ (s, y)ds .
ds
As in the proof of Theorem 3.1, 0 < (1 − β)p0 < 1. If C6 is the best constant in
Lemma 2.2, then we may write
t
(t − s)β−1 F (s)ϕm (s, y)ds ≤ C6
(3.11)
0
Z
t
p1
F p (s)ϕmp (s, y)ds .
0
Z Z
Ω
0
t
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From (3.10) and (3.11) it appears that
eµt ϕ(t, x) ≤ k̄(x) + C6 ¯l(x)
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0
0
Z
On an Integral Inequality with a
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p1
p
mpµt mp
F (s)e
ϕ (s, y)ds
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.
|x − y|n−α
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r
Taking the L −norm and applying the Minkowski inequality and then the Young
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inequality (Lemma 2.1), we find
eµt kϕ(t, x)kr ≤ k̄ (x)r + C6 ¯l(x)ρ
Z Z
p1
t
dyds
×
F p (s)empµt ϕmp (s, y)ds
n−α ,
Ω
|x − y|
0
q
with
1
r
=
1
ρ
+ 1q . Applying Lemma 2.3, we arrive at
On an Integral Inequality with a
Kernel Singular in Time and
Space
µt
e kϕ(t, x)kr
Z
p1 t
p
mpµt mp
k̄
¯
≤ (x) r + C2 C6 l(x) ρ F (s)e
ϕ (s, y)ds 0
Nasser-eddine Tatar
p
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or
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(3.12) eµt kϕ(t, x)kr
≤ k̄ (x)r + C2 C6 ¯l(x)ρ
t
Z
F p (s)empµt kϕ(s, x)kmp
r ds
p1
.
0
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(3.13) eµpt kϕ(t, x)kpr
≤2
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Taking both sides of (3.12) to the power p, we obtain
p−1
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k̄ (x)p + C7 ¯l(x)p
r
ρ
Z
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t
p
mpµt
F (s)e
0
kϕ(s, x)kmp
r
ds.
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Next, putting
χ(t) := eµpt kϕ(t, x)kpr ,
the inequality (3.13) may be written as
p−1
χ(t) ≤ 2
k̄ (x)p + C7 ¯l(x)p
r
ρ
Z
t
F p (s)χm (s)ds.
0
The rest of the proof is essentially the same as that of Theorem 3.1.
In the following corollary we consider the somewhat more general inequality
Z Z
(3.14) ϕ(t, x) ≤ k(t, x) + l(t, x)
Ω
0
t
sδ F (s)ϕm (s, y)
dyds,
(t − s)1−β |x − y|n−α
x ∈ Ω, t > 0
for some specified δ.
On an Integral Inequality with a
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Corollary 3.3. Suppose that the hypotheses of Theorem 3.1 hold. Assume further that k(t, x) ≤ t−k̂ k̄(x) and 1 + δp0 − mp0 min{k̂, 1 − β} > 0. Then any
ϕ(t, x) satisfying (3.14) is also polynomially decaying to zero
kϕ(t, x)kv ≤ C8 t−ω , C8 , ω > 0
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provided that
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k̄ (x)m−1 L(t)
r
Z
t
eεps F p (s)ds ≤
0
where C9 is the best constant in Lemma 2.5.
1
2m(p−1)
(m − 1)C9p−1 C2p
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Proof. Let us consider the inequality
Z Z
−k̂
ϕ(t, x) ≤ t k̄(x) + l(t, x)
(3.15)
Ω
0
t
sδ F (s)ϕm (s, y)
dyds.
(t − s)1−β |x − y|n−α
Multiplying by s−m min{k̂,1−β} e−εs · sm min{k̂,1−β} eεs , we obtain
Z t
(3.16)
(t − s)β−1 sδ F (s)ϕm (s, y)ds
0
Z
≤
t
(β−1)p0 δp0 −mp0 min{k̂,1−β} −εp0 s
(t − s)
s
e
p10
On an Integral Inequality with a
Kernel Singular in Time and
Space
ds
0
Z
p1
t
×
s
mp min{k̂,1−β} pεs
e
p
F (s)ϕ
mp
(s, y)ds
.
Nasser-eddine Tatar
0
As 1 + δp0 − mp0 min{k̂, 1 − β} > 0, we may apply Lemma 2.5 to the first term
in the right hand side of (3.16) to get
Z t
(3.17)
(t − s)β−1 sδ F (s)ϕm (s, y)ds
0
≤ M tβ−1
Z
t
smp min{k̂,1−β} epεs F p (s)ϕmp (s, y)ds
p1
.
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Using (3.17) we infer from inequality (3.15) that
tmin{k̂,1−β} ϕ(t, x) ≤ k̄(x) + M l(t, x)
p1
Z Z t
mp min{k̂,1−β} pεs p
mp
×
s
e F (s)ϕ (s, y)ds
Ω
0
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Page 17 of 20
dy
.
|x − y|n−α
J. Ineq. Pure and Appl. Math. 4(4) Art. 82, 2003
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Next, after using the Hardy-Littlewood-Sobolev inequality and defining
φ(t, x) := tp min{k̂,1−β} kϕ(t, x)kpr ,
we proceed as in Theorem 3.1 to find a (uniform) bound for φ(t, x).
Remark 3.1. The investigation of (1.1) with a weakly singular kernel in time,
that is
Z Z t −γ(t−s)
e
F (s)ϕm (s, y)
ϕ(t, x) ≤ k(t, x) + l(t, x)
dyds, γ > 0
1−β |x − y|n−α
Ω 0 (t − s)
is simpler since this kernel is summable (in time).
On an Integral Inequality with a
Kernel Singular in Time and
Space
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References
[1] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications,
Kluwer Academic Publishers, New York, 1992.
[2] G. BUTLER AND T. ROGERS, A generalization of a lemma of Bihari and
applications to pointwise estimates for integral equations, J. Math. Anal.
Appl. 33(1) (1971), 77–81.
[3] R.E. EDWARDS, Functional Analysis: Theory and Applications, New
York, Holt, Rinehart and Winston, 1965.
[4] D. HENRY, Geometric Theory of Semilinear Parabolic Equations,
Springer-Verlag, Berlin, Heidelberg, New York, 1981.
[5] L. HÖRMANDER, The Analysis of Linear Partial Differential Operator,
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[6] M. KIRANE AND N.-E. TATAR, Global existence and stability of some
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[7] S. MAZOUZI AND N.-E. TATAR, Global existence for some integrodifferential equations with delay subject to nonlocal conditions, Zeit. Anal.
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[8] S. MAZOUZI AND N.-E. TATAR, An improved exponential decay result for some semilinear integrodifferential equations, to appear in Arch.
Math.
On an Integral Inequality with a
Kernel Singular in Time and
Space
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[9] M. MEDVED’, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl., 214 (1997),
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[10] M. MEDVED’, Singular integral inequalities and stability of semilinear
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[12] B.G. PACHPATTE, Inequalities for Differential and Integral Equations,
in Mathematics in Science and Engineering Vol. 197, Academic Press,
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[13] N.-E. TATAR, Exponential decay for a semilinear problem with memory,
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[14] V.A. SOLONNIKOV, Existence of solutions of non-stationary linearized
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213–317 [English trans: AMS Trans. Ser. II (1975), 1–116].
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Kernel Singular in Time and
Space
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