Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 5, Article 80, 2002
REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS TO INVERSE
HEAT SOURCE PROBLEMS
SABUROU SAITOH, VU KIM TUAN, AND MASAHIRO YAMAMOTO
D EPARTMENT OF M ATHEMATICS ,
FACULTY OF E NGINEERING ,
G UNMA U NIVERSITY,
K IRYU 376-8515 JAPAN .
ssaitoh@math.sci.gunma-u.ac.jp
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE ,
FACULTY OF S CIENCE ,
K UWAIT U NIVERSITY,
P.O. B OX 5969, S AFAT 13060, K UWAIT.
vu@mcc.sci.kuniv.edu.kw
G RADUATE S CHOOL OF M ATHEMATICAL S CIENCES ,
T HE U NIVERSITY OF T OKYO ,
3-8-1 KOMABA M EGURO
T OKYO 153-8914 JAPAN .
myama@ms.u-tokyo.ac.jp
Received 2 April, 2002; accepted 30 October, 2002
Communicated by S.S. Dragomir
A BSTRACT. We introduce reverse convolution inequalities obtained recently and at the same
time, we give new type reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determining f (t), 0 < t < T , in
the heat source of the heat equation ∂t u(x, t) = ∆u(x, t) + f (t)ϕ(x), x ∈ Rn , t > 0 from the
observation u(x0 , t), 0 < t < T , at a remote point x0 away from the support of ϕ. Under an
a priori assumption that f changes the signs at most N -times, we give a conditional stability of
Hölder type, as an example of applications.
Key words and phrases: Convolution, Heat source, Weighted convolution inequalities, Young’s inequality, Hölder’s inequality, Reverse Hölder’s inequality, Green’s function, Stability in inverse problems, Volterra’s equation,
Conditional stability of Hölder type, Analytic semigroup, Interpolation inequality, Sobolev inequality.
2000 Mathematics Subject Classification. Primary 44A35; Secondary 26D20.
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
029-02
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S ABUROU S AITOH , V U K IM T UAN , AND M ASAHIRO YAMAMOTO
1. I NTRODUCTION
For the Fourier convolution
Z
∞
(f ∗ g)(x) =
f (x − ξ)g(ξ) dξ,
−∞
the Young’s inequality
(1.1)
kf ∗ gkr ≤ kf kp kgkq ,
f ∈ Lp (R), g ∈ Lq (R),
r−1 = p−1 + q −1 − 1
(p, q, r > 0),
is fundamental. Note, however, that for the typical case of f, g ∈ L2 (R), the inequality (1.1)
does not hold. In a series of papers [12] – [16] (see also [5]) we obtained the following weighted
Lp (p > 1) norm inequality for convolution
Proposition 1.1. ([15]). For two nonvanishing functions ρj ∈ L1 (R) (j = 1, 2), the Lp (p > 1)
weighted convolution inequality
1
−1 (1.2)
((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) p ≤ kF1 kLp (R,|ρ1 |) kF2 kLp (R,|ρ2 |)
p
holds for Fj ∈ Lp (R, |ρj |) (j = 1, 2). Equality holds here if and only if
Fj (x) = Cj eαx ,
(1.3)
where α is a constant such that eαx ∈ Lp (R, |ρj |) (j = 1, 2). Here
Z ∞
p1
kF kLp (R,|ρ|) =
|F (x)|p |ρ(x)| dx .
−∞
Unlike the Young’s inequality, inequality (1.2) holds also in case p = 2.
Note that the proof of Proposition 1.1 is direct and fairly elementary. Indeed, we use only
Hölder’s inequality and Fubini’s theorem for exchanging the orders of integrals for the proof.
So, for various type convolutions, we can also obtain similar type convolution inequalities, see
[17] for various convolutions.
In many cases of interest, the convolution is given in the form
(1.4)
ρ2 (x) ≡ 1,
F2 (x) = G(x),
where G(x − ξ) is some Green’s function. Then inequality (1.2) takes the form
(1.5)
1− p1
k (F ρ) ∗ Gkp ≤ kρkp
kGkp kF kLp (R,|ρ|) ,
where ρ, F , and G are such that the right hand side of (1.5) is finite.
Inequality (1.5) enables us to estimate the output function
Z ∞
(1.6)
F (ξ)ρ(ξ)G(x − ξ) dξ
−∞
in terms of the input function F in the related differential equation. We are also interested in
the reverse type inequality for (1.5), namely, we wish to estimate the input function F by means
of the output (1.6). This kind of estimates is important in inverse problems. One estimate is
obtained by using the following famous reverse Hölder inequality
Proposition 1.2. ([18], see also [10, p. 125–126]). For two positive functions f and g satisfying
(1.7)
J. Inequal. Pure and Appl. Math., 3(5) Art. 80, 2002
0<m≤
f
≤M <∞
g
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C ONVOLUTION I NEQUALITIES AND I NVERSE H EAT S OURCE P ROBLEMS
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on the set X, and for p, q > 1, p−1 + q −1 = 1,
Z
p1 Z
1q
mZ
1
1
(1.8)
f dµ
gdµ
≤ Ap,q
f p g q dµ,
M
X
X
X
if the right hand side integral converges. Here
1
Ap,q (t) = p
− p1 − 1q
q
t− pq (1 − t)
p1 1q .
1
1
1 − tp
1 − tq
Then, by using Proposition 1.2 we obtain, as in the proof of Proposition 1.1, the following
Proposition 1.3. ([16]). Let F1 and F2 be positive functions satisfying
1
(1.9)
1
0 < m1p ≤ F1 (x) ≤ M1p < ∞,
1
1
0 < m2p ≤ F2 (x) ≤ M2p < ∞,
p > 1,
x ∈ R.
Then for any positive continuous functions ρ1 and ρ2 , we have the reverse Lp –weighted convolution inequality
−1
1
m1 m2
(1.10) Ap,q
kF1 kLp (R,ρ1 ) kF2 kLp (R,ρ2 ) ≤ ((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) p −1 .
M1 M2
p
Inequality (1.10) should be understood in the sense that if the right hand side is finite, then
so is the left hand side, and in this case the inequality holds.
In formula (1.10) replacing ρ2 by 1, and F2 (x − ξ) by G(x − ξ), and integrating with respect
to x from c to d we arrive at the following inequality
(1.11)
p−1 Z ∞
Z d−ξ
n
m o−p Z ∞
p
Ap,q
ρ(ξ) dξ
F (ξ) ρ(ξ) dξ
Gp (x)dx
M
−∞
−∞
c−ξ
p
Z d Z ∞
≤
F (ξ) ρ(ξ) G(x − ξ) dξ dx,
c
−∞
if positive continuous functions ρ, F , and G satisfy
(1.12)
1
1
0 < m p ≤ F (ξ)G(x − ξ) ≤ M p ,
x ∈ [c, d],
ξ ∈ R.
Inequality (1.11) is especially important when G(x − ξ) is a Green’s function. We gave various
concrete examples in [16] from the viewpoint of stability in inverse problems.
2. R EMARKS FOR R EVERSE H ÖLDER I NEQUALITIES
In connection with Proposition 1.2 which gives Proposition 1.3, Izumino and Tominaga [8]
consider the upper bound of
X p1 X 1q
X
bqk − λ
ak b k
apk
for λ > 0, for p, q > 1 satisfying p1 + 1q = 1 and for positive numbers {ak }nk=1 and {bk }nk=1 ,
in detail. In their different approach, they showed that the constant Ap,q (t) in Proposition 1.2 is
best possible in a sense. Note that the proof of Proposition 1.2 is quite involved. In connection
with Proposition 1.2 we note that the following version whose proof is surprisingly simple
Theorem 2.1. In Proposition 1.2, replacing f and g by f p and g q , respectively, we obtain the
reverse Hölder type inequality
1q 1 Z
Z
p1 Z
m − pq
p
q
g dµ
≤
(2.1)
f dµ
f gdµ.
M
X
X
X
J. Inequal. Pure and Appl. Math., 3(5) Art. 80, 2002
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4
Proof. Since
S ABUROU S AITOH , V U K IM T UAN , AND M ASAHIRO YAMAMOTO
fp
gq
1
p
≤ M , g ≥ M − q f q , therefore
p
1
1
f g ≥ M − q f 1+ q = M − q f p
and so,
Z
(2.2)
p1
p1
Z
1
pq
≤M
f gdµ .
f dµ
p
1
p
q
On the other hand, since m ≤ fgq , f ≥ m p g p , hence
Z
Z
Z
q
1
1
1+
f gdµ ≥ m p g p dµ = m p g q dµ,
and so,
1q
Z
≥m
f gdµ
1
pq
1q
g dµ .
Z
q
Combining with (2.2), we have the desired inequality
Z
1q
Z
p1
Z
1q
p1 Z
−1
1
q
p
f dµ
g dµ
≤ M pq
f gdµ m pq
f gdµ
Z
m −1
pq
=
f gdµ.
M
3. N EW R EVERSE C ONVOLUTION I NEQUALITIES
In reverse convolution inequality (1.10), similar type inequalities for m1 = m2 = 0 are
also important as we see from our example in Section 4. For these, we obtain a new reverse
convolution inequality.
Theorem 3.1. Let p ≥ 1, δ > 0, 0 ≤ α < T , and f, g ∈ L∞ (0, T ) satisfy
0 ≤ f, g ≤ M < ∞,
(3.1)
0 < t < T.
Then
(3.2)
kf kLp (α,T ) kgkLp (0,δ) ≤ M
2p−2
p
Z
T +δ
Z
α
t
p1
f (s)g(t − s)ds dt .
α
In particular, for
Z
(f ∗ g)(t) =
t
f (t − s)g(s)ds,
0<t<T
0
and for α = 0, we have
kf kLp (0,T ) kgkLp (0,δ) ≤ M
2p−2
p
1
kf ∗ gkLp 1 (0,T +δ) .
Proof. Since 0 ≤ f, g ≤ M for 0 ≤ t ≤ T , we have
Z t
Z t
p
p
(3.3)
f (s) g(t − s) ds =
f (s)p−1 g(t − s)p−1 f (s)g(t − s)ds
α
α
Z t
2p−2
≤M
f (s)g(t − s)ds.
α
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C ONVOLUTION I NEQUALITIES AND I NVERSE H EAT S OURCE P ROBLEMS
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Hence
Z
T +δ
Z
t
p
p
f (s) g(t − s) ds dt ≤ M
α
2p−2
Z
α
Z
α
On the other hand, we have
Z
Z T +δ Z t
p
p
f (s) g(t − s) ds dt =
α
T +δ
T +δ
f (s)g(t − s)ds dt.
α
T +δ
g(t − s) dt f (s)p ds
α
s
Z T +δ Z T +δ−s
p
=
g(η) dη f (s)p ds
α
0
Z T Z T +δ−s
p
≥
g(η) dη f (s)p ds
α
0
Z T Z δ
p
≥
g(η) dη f (s)p ds
α
Z
t
p
α
=
0
p
kf kLp (α,T ) kgkpLp (0,δ) .
Thus the proof of Theorem 3.1 is complete.
4. A PPLICATIONS TO I NVERSE S OURCE H EAT P ROBLEMS AND R ESULTS
We consider the heat equation with a heat source:
(4.1)
∂t u(x, t) = ∆u(x, t) + f (t)ϕ(x),
u(x, 0) = 0,
(4.2)
x ∈ Rn , t > 0
x ∈ Rn .
We assume that ϕ is a given function and satisfies


ϕ ≥ 0, 6≡ 0
in Rn ,




ϕ has compact support, ϕ ∈ C ∞ (Rn ), if n ≥ 4 and
(4.3)




 ϕ ∈ L (Rn ),
if n ≤ 3.
2
Our problem is to derive a conditional stability in the determination of f (t), 0 < t < T , from
the observation
u(x0 , t),
(4.4)
0 < t < T,
where x0 6∈ supp ϕ.
We are interested only in the case of x0 6∈ supp ϕ, because in the case where x0 is in the
interior of supp ϕ, the problem can be reduced to a Volterra integral equation of the second
kind by differentiation in t formula (4.8) stated below. Moreover x0 6∈ supp ϕ means that our
observation (4.4) is done far from the set where the actual process is occuring, and the design
of the observation point is easy.
Let
|x|2
1
(4.5)
K(x, t) = √ n e− 4t , x ∈ Rn , t > 0.
(2 πt)
Then the solution u to (4.1) and (4.2) is represented by
Z tZ
(4.6)
u(x, t) =
K(x − y, t − s)f (s)ϕ(y)dyds, x ∈ Rn , t > 0
0
Rn
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6
S ABUROU S AITOH , V U K IM T UAN , AND M ASAHIRO YAMAMOTO
(e.g., Friedman [6]). Therefore, setting
Z
(4.7)
µx0 (t) =
K(x0 − y, t)ϕ(y)dy,
t > 0,
Rn
we have
Z
(4.8)
u(x0 , t) ≡ hx0 (t) =
t
µx0 (t − s)f (s)ds,
0 < t < T,
0
which is a Volterra integral equation of the first kind with respect to f . Since
dk µx0
(t) = (∆k ϕ)(x0 ) = 0, k ∈ N ∪ {0}
t↓0 dtk
by x0 6∈ supp ϕ (e.g., [6]), the equation (4.8) cannot be reduced to a Volterra equation of the
second kind by differentiating in t. Hence, even though, for any m ∈ N, we take the C m -norms
for data h, the equation (4.8) is ill-posed, and we cannot expect a better stability such as of
Hölder type under suitable a priori boundedness.
In Cannon and Esteva [3], an estimate of logarithmic type is proved: let n = 1 and ϕ = ϕ(x)
be the characteristic function of an interval (a, b) ⊂ R. Set
(
)
2 df
d
f
(4.9)
VM = f ∈ C 2 [0, ∞); f (0) = 0, ,
≤M .
dt dt2 C[0,∞)
C[0,∞)
lim
Let x0 6∈ (a, b). Then, for T > 0, there exists a constant C = C(M, a, b, x0 ) > 0 such that
(4.10)
|f (t)| ≤
C
,
| log ku(x0 , ·)kL2 (0,∞) |2
0 ≤ t ≤ T,
for all f ∈ VM . The stability rate is logarithmic and worse than any rate of Hölder type:
ku(x0 , ·)kαL2 (0,∞) for any α > 0. For (4.10), the condition f ∈ VM prescribes a priori information and (4.10) is called conditional stability within the admissible set VM . The rate of
conditional stability heavily depends on the choice of admissible sets and an observation point
x0 . As for other inverse problems for the heat equation, we can refer to Cannon [2], Cannon
and Esteva [4], Isakov [7] and the references therein.
We arbitrarily fix M > 0 and N ∈ N. Let
(4.11)
U = {f ∈ C[0, T ]; kf kC[0,T ] ≤ M, f changes the signs at most N -times}.
We take U as an admissible set of unknowns f . Then, within U, we can show an improved
conditional stability of Hölder type:
Theorem 4.1. Let ϕ satisfy (4.3), and x0 6∈ supp ϕ. We set

4


, n ≤ 3,
4−n
(4.12)
p>


1,
n ≥ 4.
Then, for an arbitrarily given δ > 0, there exists a constant C = C(x0 , ϕ, T, p, δ, U) > 0 such
that
(4.13)
1
pN
kf kLp (0,T ) ≤ Cku(x0 , ·)kL1 (0,T +δ)
for any f ∈ U.
We will see that limδ→0 C = ∞ and, in order to estimate f over the time interval (0, T ), we
have to observe u(x0 , ·) over a longer time interval (0, T + δ).
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C ONVOLUTION I NEQUALITIES AND I NVERSE H EAT S OURCE P ROBLEMS
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Remark 4.2. In the case of n ≥ 4, we can relax the regularity of ϕ to H α (Rn ) with some
α > 0, but we will not go into the details. In the case of n ≤ 3, if we assume that ϕ ∈ C ∞ (Rn )
in (4.3), then in Theorem 4.1 we can take any p > 1.
Remark 4.3. As a subset of U, we can take, for example,
PN = {f ; f is a polynomial whose order is at most N and kf kC[0,T ] ≤ M }.
The condition f ∈ U is quite restrictive at the expense of the practically reasonable estimate of
Hölder type (4.4).
Remark 4.4. The a priori boundedness kf kC[0,T ] ≤ M is necessary for the stability.
Example 4.1. Let n = 1, p > 2 and

0
|x| < r, |x| > R,


√
(4.14)
ϕ(x) =
π


, r < |x| < R.
R−r
We set x0 = 0. Then, by (4.7), we have
1
µ0 (t) = √
2 πt
(4.15)
Z
y2
e− 4t ϕ(y)dy,
r<|y|<R
so that
R2
r2
1
1
√ e− 4t ≤ µ0 (t) ≤ √ e− 4t ,
t
t
(4.16)
t > 0.
We choose fn as
1
1
fn (t) = √ e− nt ,
t
(4.17)
t > 0, n ∈ N.
Then fn does not change the signs in (0, T ) and limn→∞ max0≤t≤T |fn (t)| = ∞. The corresponding solution un (x, t) of (4.1) – (4.2) with fn is estimated as follows:
Z t
Z t
r2
1
1
1
√
|un (0, t)| = µ0 (t − s)fn (s)ds ≤
e− 4(t−s) √ e− ns ds,
s
t−s
0
0
and so
Z
0
T
T
Z
t
r2
1
1 −1
− 4(t−s)
√
√ e ns ds dt
|un (0, t)|dt ≤
e
s
t−s
0
0
Z T Z T
2
r
1
1
1
− 4(t−s)
√
dt √ e− ns ds
=
e
s
t−s
0
s
Z T Z T
1 − r2
1
≤
√ e 4η dη √ ds
η
s
0
0
Z
T
√
1 − r2
=2 T
√ e 4η dη.
η
0
Z
Next
Z
T
p
fn (t) dt = n
0
J. Inequal. Pure and Appl. Math., 3(5) Art. 80, 2002
p
−1
2
Z
nT
p
p
e− η η − 2 dη.
0
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S ABUROU S AITOH , V U K IM T UAN , AND M ASAHIRO YAMAMOTO
Therefore, for any γ ∈ (0, 1), we have
p1
R
R
p1
1
T
nT − p − p
− p1
p
2
η η 2 dη
f
(t)
dt
n
e
n
0
0
R
γ ≥ √ R
γ −→ ∞ as n −→ ∞
2
T
T 1 −r
4η
√
u
(0,
t)dt
2 T 0 η e dη
n
0
by p > 2. Hence the stability of the type (4.4) is impossible for p > 2.
Remark 4.5. For our stability, the finiteness of changes of signs is essential. In fact, we take
(4.18)
fn (t) = cos nt,
0 ≤ t ≤ T, n ∈ N.
Then fn oscillates very frequently and we cannot take any finite partition of (0, T ) where the
condition on signs in (4.1) holds true. We note that we can take M = 1, that is, kfn kC[0,T ] ≤ 1
for n ∈ N. We denote the solution to (4.1) – (4.2) for f = fn by un (x, t). Then
Z t
un (x0 , t) =
µx0 (t − s)fn (s)ds
0
Z t
=
µx0 (s)fn (t − s)ds
0
Z t
Z t
= cos nt
µx0 (s) cos nsds − sin nt
µx0 (s) sin nsds.
0
0
By µx0 ∈ L1 (0, T ), the Riemann-Lebesgue lemma yields limn→∞ un (x0 , t) = 0 for all t ∈
[0, T + δ]. Moreover we readily see that
Z T +δ
|un (x0 , t)| ≤
µx0 (s)ds < ∞,
n ∈ N, 0 ≤ t ≤ T + δ.
0
Therefore, by the Lebesgue convergence theorem, we can conclude that
lim kun (x0 , ·)kL1 (0,T +δ) = 0.
n→∞
For n = 1, we can choose p = 2 in Theorem 4.1. We have
Z T
sin 2nT
T
fn (t)2 dt = +
,
2
4n
0
so that limn→∞ kfn kL2 (0,T ) 6= 0. Thus any stability cannot hold for fn , n ∈ N.
5. P ROOF OF T HEOREM 4.1
Suppose that f changes the signs at 0 < t1 < t2 < ... < tI = T, I ≤ N . Without loss
of generality, we may assume that f ≥ 0 on (0, t1 ). Since f ∈ U, we see that f satisfies
(3.1). Meanwhile since µx0 (t) is positive and bounded, for some constant B > 0, Bµx0 (t)
satisfies (3.1). We apply Theorem 3.1 on (0, t1 ), setting α = 0 and g(t) = Bµx0 (t). Setting
1
C1 = B 1− p kµx0 kLp (0,δ) (C1 > 0), we obtain
(5.1)
kf kLp (0,t1 ) ≤ C1−1 M
2p−2
p
1
ku(x0 , ·)kLp 1 (0,t1 +δ) .
Next we will prove
1
(5.2)
|u(x0 , t1 )| ≤ C2 ku(x0 , ·)kLp 1 (0,t1 +δ) ,
where the constant C2 > 0 depends on ϕ, T, δ, p. Henceforth the constants Cj > 0, j ≥ 2, are
independent of the choice of 0 < t1 < t2 < · · · < tN < T .
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C ONVOLUTION I NEQUALITIES AND I NVERSE H EAT S OURCE P ROBLEMS
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Proof of (5.2). Let L2 (Rn ) be the usual L2 -space with the norm k · k and let −A be the operator
defined by
x ∈ Rn ,
(−Au)(x) = ∆u(x),
(5.3)
D(A) = H 2 (Rn ).
Then −A generates an analytic semigroup e−tA , t > 0 and, by the definition of H 2` (Rn ) and
the interpolation inequality (e.g., Lions and Magenes [9]), we see that
(DA` ) = H 2` (Rn ),
kukH 2` (Rn ) ≤ C3 (`)kA` uk,
u ∈ D(A` ).
Moreover
kA` e−tA k ≤ C3 (`)t−`
(5.4)
and
t
Z
e−(t−s)A f (s)ϕ(·)ds,
u(t) = u(·, t) =
(5.5)
t>0
0
(e.g., Pazy [11]). By the Sobolev inequality: H 2` (Rn ) ⊂ L∞ (Rn ) if 4` > n (e.g., Adams [1]),
we have D(A` ) ⊂ L∞ (Rn ) and
kukL∞ (Rn ) ≤ C4 (`)kA` uk,
(5.6)
u ∈ D(A` ).
Case: n ≤ 3.
We can take
`=
(5.7)
n
+ ε0 < 1 with a sufficiently small ε0 > 0.
4
Let q > 1 satisfy p1 + 1q = 1. Since p >
sufficiently small such that
4
,
4−n
we have q < n4 , therefore we can choose ε0 > 0
q` < 1.
(5.8)
Hence, by (5.4), (5.5) and the Hölder inequality, we obtain
Z t1
`
(5.9)
kA u(t1 )k ≤
|f (s)|kA` e−(t1 −s)A ϕkds
0
Z t1
≤ C5
(t1 − s)−` |f (s)|ds
0
t1
Z
≤ C5
−q`
(t1 − s)
1q Z
t1
p1
|f (s)| ds .
p
ds
0
0
Consequently, by (5.8), we have
`
kA u(t1 )k ≤ C5
t1−q`
1
1 − q`
! 1q
kf kLp (0,t1 ) ≤ C5
T
1 − q`
1q
kf kLp (0,t1 ) .
Therefore (5.1) and (5.6) yield (5.2).
J. Inequal. Pure and Appl. Math., 3(5) Art. 80, 2002
http://jipam.vu.edu.au/
10
S ABUROU S AITOH , V U K IM T UAN ,
AND
M ASAHIRO YAMAMOTO
Case: n ≥ 4.
By ϕ ∈ C0∞ (Rn ), we have ϕ ∈ D(A` ) for ` ∈ N. Therefore the estimate of kA` u(t)k is simpler
than in (5.9):
Z t1
−(t1 −s)A `
kA` u(t1 )k = f
(s)e
A
ϕds
0
Z t1
≤
|f (s)|ke−(t1 −s)A kkA` ϕkds
0
≤ C60 kf kL1 (0,t1 ) ≤ C6 kf kLp (0,t1 ) .
Thus (5.1) and (5.6) complete the proof of (5.2).
Next we will estimate kf kLp (t1 ,t2 ) . By (4.8), we have
Z t
(5.10)
−u(x0 , t) = −u(x0 , t1 ) +
µx0 (t − s)(−f (s))ds,
t1 ≤ t ≤ t2 .
t1
Taking α = t1 , T = t2 in Theorem 3.1, we obtain
Z t2 +δ
p1
2p−2
−1
kf kLp (t1 ,t2 ) ≤ M p C1
(5.11)
| − u(x0 , t) + u(x0 , t1 )|dt
t1
≤M
2p−2
p
1
C1−1 (ku(x0 , ·)kL1 (t1 ,t2 +δ) + T |u(x0 , t1 )|) p .
Therefore we apply (5.2), and
(5.12) kf kLp (t1 ,t2 ) ≤ C1−1 M
2p−2
p
C1−1 M
2p−2
p
1
2p−2
1
1
1
2
ku(x0 , ·)kLp 1 (t1 ,t2 +δ) + C1−1 M p T p C2p ku(x0 , ·)kLp 1 (0,t1 +δ)
1
1
− 12
2p−2
2
p
−1
ku(x0 , ·)kL1 (tp1 ,t2 +δ) ku(x0 , ·)kLp 1 (t1 ,t2 +δ)
≤ C1 M p
1
1
1
2
p
p
+ T p C2 ku(x0 , ·)kL1 (0,t1 +δ)
≤
0
((T M )
p−1
p2
1
p
1
p
1
p2
+ T C2 )ku(x0 , ·)kL1 (0,t2 +δ) .
Here, since u(x0 , t) is bounded, we take a positive M 0 such that |u(x0 , t)| ≤ M 0 . By (5.1) and
(5.12), we can estimate kf kLp (0,t2 ) . Continuing this argument until tI = T , we can complete
the proof of Theorem 4.1.
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