COERCIVE SOLVABILITY OF THE NONLOCAL
BOUNDARY VALUE PROBLEM FOR PARABOLIC
DIFFERENTIAL EQUATIONS
A. ASHYRALYEV, A. HANALYEV, AND P. E. SOBOLEVSKII
Received 26 March 2001
The nonlocal boundary value problem, v (t)+Av(t) = f (t) (0 ≤ t ≤ 1), v(0) =
v(λ)+µ (0 < λ ≤ 1), in an arbitrary Banach space E with the strongly positive
operator A, is considered. The coercive stability estimates in Hölder norms
for the solution of this problem are proved. The exact Schauder’s estimates in
Hölder norms of solutions of the boundary value problem on the range {0 ≤ t ≤
1, x ∈ Rn } for 2m-order multidimensional parabolic equations are obtained.
1. Introduction
We consider the following nonlocal boundary value problem for the differential
equation
v (t) + Av(t) = f (t)
(0 ≤ t ≤ 1),
v(0) = v(λ) + µ (0 < λ ≤ 1),
(1.1)
in an arbitrary Banach space with linear (unbounded) operator A. It is known
(cf. [4]) that various nonlocal boundary value problems for the parabolic equations can be reduced to the boundary value problem (1.1). We obtain the coercive
solvability of problem (1.1) in some function Banach space. The role played by
coercive inequalities in the study of boundary value problems for elliptic and
parabolic partial differential equations is well known (cf. [5, 6, 7]). Coercivity
inequalities for the solutions of an abstract differential equation of parabolic
type
v (t) + Av(t) = f (t) (0 ≤ t ≤ 1),
v(0) = v0 ,
(1.2)
were established in the various norms of Banach spaces by Sobolevskii P. E. and
Da Prato and their colleagues under the assumption that A is a strongly positive
operator, that is, −A is the generator of the analytic semigroup exp{−tA} (t ≥ 0)
Copyright © 2001 Hindawi Publishing Corporation
Abstract and Applied Analysis 6:1 (2001) 53–61
2000 Mathematics Subject Classification: 65N, 47D, 34B
URL: http://aaa.hindawi.com/volume-6/S1085337501000495.html
54
Coercive solvability of the nonlocal boundary value problem
of the linear bounded operators with exponentially decreasing norm (see [3]).
In [1], the coercive solvability of a Cauchy problem (1.2) was established in
β,γ
C0 (E) (0 ≤ γ ≤ β, 0 < β < 1)—the space obtained by completion of the
space of all smooth E-valued functions ϕ(t) on [0, 1] in the norm
(t + τ )γ ϕ(t + τ ) − ϕ(t)E
ϕC β,γ (E) = max ϕ(t)E + sup
, (1.3)
0
0≤t≤1
τβ
0≤t<t+τ ≤1
that is established in the following theorem.
Theorem 1.1. Let A be a strongly positive operator in a Banach space E and
β,γ
β,γ
f (t) ∈ C0 (E) (0 ≤ γ ≤ β, 0 < β < 1). Then for the solution v(t) in C0 (E)
of the initial value problem (1.2), the coercive inequality
v β,γ + Av β,γ + v C(Eβ−γ )
C0 (E)
C0 (E)
(1.4)
≤ M v0 E + β −1 (1 − β)−1 f C β,γ (E)
β−γ
0
holds, where M does not depend on β, γ ,
v0 ,
and f (t).
Theorem 1.2. Let A be a strongly positive operator in a Banach space E and
β,γ
f (t) ∈ C0 (Eα−γ ) (0 ≤ γ ≤ β ≤ α, 0 < α < 1). Then for the solution v(t) in
β,γ
C0 (Eα−γ ) of the initial value problem (1.2), the coercive inequality
v β,γ
+ AvC β,γ (E ) + v C(E )
C0 (Eα−β )
α−γ
α−β
0
(1.5)
≤ M v0 E + α −1 (1 − α)−1 f C β,γ (E )
α−γ
0
α−β
holds, where M does not depend on α, β, γ , v0 , and f (t).
Here C(E) stands for the Banach space of all continuous functions ϕ(t)
defined on [0, 1] with values in E equipped with the norm
ϕC(E) = max ϕ(t)E ,
(1.6)
0≤t≤1
and the Banach space Eα (0 < α < 1) consists of those v ∈ E for which the
norm (see [3])
(1.7)
vEα = sup z1−α A exp{−zA}v E + vE
z>0
is finite.
β,γ
A function v(t) is called a solution of problem (1.1) in C0 (E) if the following conditions are satisfied:
β,γ
(i) If the functions v (t), Av(t) ∈ C0 (E).
A. Ashyralyev et al.
55
(ii) If the function v(t) satisfies the equation and the boundary condition
(1.1).
β,γ
From the existence of such solutions evidently follows that f (t) ∈ C0 (E)
and µ ∈ D(A).
In the present paper, the coercive inequalities in the norms of the same spaces
for the solutions of the boundary value problem (1.1) are obtained. The exact
Schauder’s estimates in Hölder norms of the solution of the boundary value
problem on the range {0 ≤ t ≤ 1, x ∈ Rn } for 2m-order multidimensional
parabolic equations are obtained.
β,γ
2. Coercive solvability in C0 (E)
It is known that an operator A is strongly positive in E if and only if −A is
the generator of the analytic semigroup exp{−tA} (t ≥ 0) of the linear bounded
operators in E with exponentially decreasing norm when t → +∞, that is, the
following estimates hold.
exp{−tA}
≤ Me−δt , t A exp{−tA}E→E ≤ M,
E→E
(2.1)
t > 0, M > 0, δ > 0.
For a strongly positive operator A we have that
I − exp{−λA} −1 E→E
≤
M
,
1 − e−δλ
0 < λ ≤ 1.
(2.2)
Theorem 2.1. Let A be a strongly positive operator in a Banach space E and
β,γ
β,γ
f (t) ∈ C0 (E) (0 ≤ γ ≤ β, 0 < β < 1). Then for the solution v(t) in C0 (E)
of the boundary value problem (1.1), the coercive inequality
v β,γ + Av β,γ
C0 (E)
C0 (E)
(2.3)
≤ M Aµ + f (λ) − f (0)E + β −1 (1 − β)−1 f C β,γ (E)
β−γ
0
holds, where M does not depend on β, γ , µ, and f (t).
Proof. From the strong positivity of A, it follows that
λ
−1
v(t) = exp{−tA} I − exp{−λA}
exp − (λ − s)A f (s)ds
µ+
0
t
+
exp − (t − s)A f (s)ds
0
(2.4)
β,γ
for the solution of problem (1.1) in the space C0 (E) (cf. [4]). Using this
56
Coercive solvability of the nonlocal boundary value problem
formula, we have that
v0 = f (0) − Au(0)
−1
= I − exp{−λA}
λ
A exp − (λ − s)A f (λ) − f (s) ds
0
−1
− I − exp{−λA} Aµ + f (λ) − f (0)
(2.5)
= I 1 + I2 ,
where
−1
I1 = I − exp{−λA}
λ
0
−1
I2 = − I − exp{−λA}
A exp − (λ − s)A f (λ) − f (s) ds,
Aµ + f (λ) − f (0) .
(2.6)
(2.7)
Then the proof of Theorem 2.1 follows from the inequality (1.4) and the
estimate
−1
−1
v Aµ + f (λ) − f (0)
≤
M
+
β
(1
−
β)
f
. (2.8)
β,γ
0 E
E
C (E)
β−γ
β−γ
0
Using formula (2.7) and the estimates (2.1), (2.2) for z > 0, we have that
z1−β+γ Ae−zA I2 −1 ≤ I − e−λA E→E z1−β+γ Ae−zA Aµ + f (λ) − f (0) E
+ z1−β+γ Ae−(z+λ)A E→E f (λ) − f (0)E
≤ M Aµ + f (λ) − f (0)E
z1−β+γ λβ−γ
f C β,γ (E)
β−γ
0
z+λ
≤ M Aµ + f (λ) − f (0)E + f C β,γ (E) .
β−γ
Therefore
I 2 E
β−γ
+
0
≤ M Aµ + f (λ) − f (0)E
β−γ
(2.9)
+ f C β,γ (E) .
(2.10)
0
Now we estimate I1 in the norm Eβ−γ . Using formula (2.6) and the estimates
(2.1), (2.2), for any z > 0 we obtain
z1−β+γ Ae−zA I1 E
λ
2 −(z+λ−s) −λA −1 1−β+γ
A e
f (λ) − f (s) ds
≤ I −e
z
E→E
E→E
E
0
λ
≤ Mz1−β+γ
0
(λ − s)β ds
(z + λ − s)2 λγ
f C β,γ (E) .
0
(2.11)
A. Ashyralyev et al.
57
If z ≤ λ, then
z
λ
1−β+γ
0
(λ − s)β ds
≤ z1−β
(z + λ − s)2 λγ
λ
0
1
ds
.
≤
2−β
1−β
(z + λ − s)
(2.12)
If z > λ, then
λ
z1−β+γ
0
λ
(λ − s)β ds
ds
−β+γ −γ
λ
≤
z
(z + λ − s)2 λγ
(λ
−
s)1−β
0
β−γ
1 λ
1
=
< .
β z
β
(2.13)
Therefore, for any z > 0, we have that
z
λ
1−β+γ
0
1
(λ − s)β ds
.
≤
β(1 − β)
(z + λ − s)2 λγ
(2.14)
From the last estimate and (2.11), it follows that
I1 Eβ−γ
≤
M
f C β,γ (E) .
0
β(1 − β)
(2.15)
Using the estimates (2.10), (2.15), and the triangle inequality, we obtain the
estimate (2.8). Theorem 2.1 is then proved.
β,γ
Remark 2.2. Note that the spaces of smooth functions C0 (E) in which coercive solvability has been established depend on the parameters β and γ . However, the constants in the coercive inequality (2.3) depend only on β. Hence, γ
can be chosen freely in [0, β], which increases the number of function spaces
in which problem (1.1) is coercively solvable. For example, it is important
that problem (1.1) is coercively solvable in the Hölder space without a weight
(γ = 0).
β,γ
3. Coercive solvability in C0 (Eα−β )
Theorem 3.1. Let A be a strongly positive operator in a Banach space E and
β,γ
f (t) ∈ C0 (Eα−γ ) (0 ≤ γ ≤ β ≤ α, 0 < α < 1). Then for the solution v(t) in
β,γ
C0 (Eα−γ ) of the boundary value problem (1.1), the coercive inequality
v β,γ
+ AvC β,γ (E )
C0 (Eα−β )
α−β
0
(3.1)
≤ M Aµ + f (λ) − f (0)E + α −1 (1 − α)−1 f C β,γ (E )
α−γ
holds, where M does not depend on β, γ , α, and f (t).
0
α−β
58
Coercive solvability of the nonlocal boundary value problem
Proof. The proof of Theorem 3.1 follows from the inequality (1.5) and the
estimate
1
Aµ + f (λ) − f (0)
v f
. (3.2)
+
β,γ
0 Eα−γ ≤ M
Eα−γ
C0 (Eα−β )
α(1 − α)
Using formula (2.7) and the estimates (2.1), (2.2) for any z > 0, we have that
z1−α+γ Ae−zA I2 E
≤ I − e−λA E→E z1−α+γ Ae−zA Aµ + f (λ) − f (0) E
+ z1−α+γ Ae−(z+λ)A f (λ) − f (0) E
≤ M1 Aµ + f (λ) − f (0)E
α−γ
≤ M1 Aµ + f (λ) − f (0)E
α−γ
z1−α+γ f (λ) − f (0)
Eα−β
(z + λ)1−α+β
z1−α+γ λβ−γ
.
+
f
β,γ
C0 (Eα−β )
(z + λ)1−α+β
(3.3)
+
Since
z1−α+γ λβ−γ
≤ 1,
(z + λ)1−α+β
(3.4)
we have that
z1−α+γ Ae−zA I2 E ≤ M1 Aµ + f (λ) − f (0)E
for any z > 0, and it follows that
Aµ + f (λ) − f (0)
I2 ≤
M
1
E
E
α−γ
α−γ
α−γ
+ f C β,γ (E
0
+ f C β,γ (E
0
α−β )
,
(3.5)
α−β )
.
(3.6)
Now we estimate I1 in the norm Eα−γ . Using formula (2.6) and the estimates
(2.1), (2.2) for any z > 0, we obtain
z1−α+γ Ae−zA I1 E
λ
2 −(z+λ−s) −λA −1 1−α+γ
ds
A e
f
(λ)
−
f
(s)
≤ I −e
z
E→E
E
0
≤ M1 z1−α+γ
λ
1
f (λ) − f (s)
ds
Eα−β
(z + λ − s)2−α+β
λ
(λ − s)β ds
f C β,γ (E ) .
α−β
0
(z + λ − s)2−α+β λγ
0
≤ M1 z1−α+γ
0
(3.7)
A. Ashyralyev et al.
59
If z ≤ λ, then
z
λ
1−α+γ
0
z1−α+γ
(λ − s)β ds
≤
λγ
(z + λ − s)2−α+β λγ
λ
0
ds
(z + λ − s)2−α
(3.8)
zα
1
.
≤
≤
(1 − α)λα
1−α
If z ≥ λ, then
λ
z1−α+γ
0
z−α+γ
(λ − s)β ds
≤
λγ
(z + λ − s)2−α+β λγ
λ
0
λα−γ
1
ds
=
≤ .
αzα−γ
α
(λ − s)1−α
(3.9)
Therefore, for any z > 0 we have that
λ
z1−α+γ
0
1
(λ − s)β ds
.
≤
2−α+β
γ
α(1 − α)
(z + λ − s)
λ
(3.10)
From the last estimate and (3.7), it follows that
I1 Eα−γ
≤
M
f C β,γ (E ) .
α−β
0
α(1 − α)
(3.11)
Using the estimates (3.6), (3.11), and the triangle inequality, we obtain the
estimate (3.2). This completes the proof of Theorem 3.1.
Remark 3.2. Using this approach we can obtain the same results for solutions
of the general boundary value problem
v (t) + Av(t) = f (t) (0 ≤ t ≤ 1),
v(0) =
p
ci v ti + µ,
(3.12)
i=1
where 0 < t1 < t2 < · · · < tp ≤ 1, if the operator I −
bounded inverse in E.
p
i=1 ci e
−ti A
has a
4. Applications
We consider the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ Rn } for
2m-order multidimensional differential equations of parabolic type
∂v(t, x)
∂ |τ | v(t, x)
+
aτ (x) τ1
+ δv(t, x) = f (t, x) (0 ≤ t ≤ 1),
∂t
∂x1 · · · ∂xnτn
|τ |=2m
0 < λ ≤ 1, x ∈ Rn ,
|τ | = τ1 + · · · + τn ,
(4.1)
where ar (x), f (y, x) are given sufficiently smooth functions and δ > 0 is a
v(0, x) = v(λ, x) + µ(x),
60
Coercive solvability of the nonlocal boundary value problem
sufficiently large number. We will assume that the symbol
B(ξ ) =
ar (x)(iξ )r1 · · · (iξ )rn
(4.2)
|τ |=2m
of the differential operator of the form
B=
|r|=2m
ar (x)
∂ |r|
,
∂x1r1 · · · ∂xnrn
acting on functions defined on the space Rn , satisfies the inequalities
0 < M1 |ξ |2m ≤ B(ξ ) ≤ M2 |ξ |2m < ∞,
(4.3)
(4.4)
for ξ = 0.
Problem (4.1) has a unique smooth solution. This allows us to reduce the
boundary value problem (4.1) to the boundary value problem (1.1) in the Banach
space E with a strongly positive operator A = B +δI defined by (4.3). We give
in Theorem 4.1 a number of corollaries to Theorems 2.1 and 3.1.
Theorem 4.1. The solutions of the boundary value problem (4.1) satisfy the
following coercive inequalities:
∂v ∂ |τ | v
∂t β,γ ε n +
∂x τ1 · · · ∂x τn β,γ ε n
n
C0 (C (R )) |τ |=2m
C0 (C (R ))
1

|τ | µ(x)
∂

≤ M(ε) aτ (x) τ1
τn − f (λ, x) + f (0, x)
∂x1 · · · ∂xn
|τ |=2m
2m(β−γ )+ε n
C
(R )

1
f C β,γ (C ε (Rn ))  ,
+
0
β(1 − β)
0 < 2m(β − γ ) + ε < 1, 0 ≤ γ ≤ β, 0 < β < 1,
|τ |
∂v τ∂ v τ +
∂x 1 · · · ∂x n β,γ 2m(α−β) n
∂t β,γ 2m(α−β) n
n C0 (C
C0 (C
(R )) |τ |=2m
(R ))
1

|τ
|
∂ µ(x)

≤ M(α, β, γ ) aτ (x) τ1
τn −f (λ, x)+f (0, x)
∂x1 · · · ∂xn
|τ |=2m
2m(β−γ ) n
C
(R )

1
f C β,γ (C 2m(α−β) (Rn ))  ,
+
0
β(1 − β)
0 < 2m(α − γ ) + ε < 1, 0 ≤ γ ≤ β, 0 < α < 1,
(4.5)
A. Ashyralyev et al.
61
where M(ε) do not depend on β, γ , α, µ(x), and f (t, x) and M(α, β, γ ) do not
depend on µ(x), and f (t, x). Here C ε (Rn ) is the space of functions satisfying
a Hölder condition with the indicator ε ∈ (0, 1).
The proof of Theorem 4.1 is based on Theorems 2.1 and 3.1, the strong
positivity of the operator A in C ε (Rn ), the coercive inequality for the solution of
the resolvent equation of the elliptic operator A in C ε (Rn ), and equivalent of the
norms in the spaces Eβ = Eβ (A, C(Rn )) and C 2mβ (Rn ) when 0 < β < 1/2m
(see [2, 3]).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
A. O. Ashyralyev, Coercive solvability of parabolic equations in spaces of smooth
functions, Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk
(1989), no. 3, 3–13. MR 90i:35150. Zbl 677.34005.
A. O. Ashyralyev and P. E. Sobolevskii, Coercive stability of a multidimensional
difference elliptic equation of 2m-th order with variable coefficients, Investigations in the Theory of Differential Equations (Russian), Minvuz Turkmen. SSR,
Ashkhabad, 1987, pp. 31–43. CMP 1 009 419.
, Well-Posedness of Parabolic Difference Equations, Birkhäuser Verlag,
Basel, 1994. MR 95j:65094.
S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow,
1966 (Russian).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs,
vol. 23, American Mathematical Society, Rhode Island, 1967. MR 39#3159b.
Zbl 174.15403.
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. MR 39#5941. Zbl 164.13002.
M. I. Vishik, A. D. Myshkis, and O. A. Oleinik, Partial differential equations,
Mathematics in USSR in the Last 40 Years, 1917–1957 (Russian), Fizmatgiz,
Moscow, 1959, pp. 563–599.
A. Ashyralyev: Department of Mathematics, Fatih University, Istanbul,
Turkey
Current address: International Turkmen-Turkish University, Ashgabat,
Turkmenistan
E-mail address: charyar@go.com
A. Hanalyev: Department of Applied Mathematics, Turkmen State University,
Ashgabat, Turkmenistan
E-mail address: aashyr@fatih.edu.tr
P. E. Sobolevskii: Institute of Mathematics, Hebrew University, Jerusalem,
Israel
E-mail address: pavels@math.hiji.ac.il