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 spaceEwith 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) operatorA. It is known (cf. [4]) that various nonlocal boundary value problems for the parabolic equa- tions 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 thatAis a strongly positive operator, that is,−Ais 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
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 Cβ,γ0 (E) (0≤γ ≤β, 0< β <1)—the space obtained by completion of the space of all smoothE-valued functionsϕ(t)on[0,1]in the norm
ϕCβ,γ
0 (E)= max
0≤t≤1ϕ(t)E+ sup
0≤t<t+τ≤1
(t+τ)γϕ(t+τ)−ϕ(t)E
τβ , (1.3)
that is established in the following theorem.
Theorem1.1. LetAbe a strongly positive operator in a Banach spaceEand f (t)∈C0β,γ(E) (0≤γ≤β, 0< β <1). Then for the solutionv(t)inC0β,γ(E) of the initial value problem (1.2), the coercive inequality
v
C0β,γ(E)+AvCβ,γ
0 (E)+v
C(Eβ−γ)
≤Mv0E
β−γ+β−1(1−β)−1fCβ,γ
0 (E)
(1.4)
holds, whereM does not depend onβ,γ,v0, andf (t).
Theorem1.2. LetAbe a strongly positive operator in a Banach spaceEand f (t)∈C0β,γ(Eα−γ) (0≤γ ≤β≤α,0< α <1). Then for the solutionv(t)in Cβ,γ0 (Eα−γ)of the initial value problem (1.2), the coercive inequality
v
C0β,γ(Eα−β)+AvCβ,γ
0 (Eα−β)+vC(E
α−γ)
≤Mv0E
α−γ+α−1(1−α)−1fCβ,γ
0 (Eα−β)
(1.5)
holds, whereM does not depend onα,β,γ,v0, andf (t).
Here C(E) stands for the Banach space of all continuous functions ϕ(t) defined on[0,1]with values inEequipped with the norm
ϕC(E)= max
0≤t≤1ϕ(t)E, (1.6)
and the Banach space Eα (0< α <1)consists of thosev∈E for which the norm (see [3])
vEα =sup
z>0z1−αAexp{−zA}v
E+vE (1.7)
is finite.
A functionv(t)is called a solution of problem (1.1) inC0β,γ(E)if the fol- lowing conditions are satisfied:
(i) If the functionsv(t),Av(t)∈C0β,γ(E).
(ii) If the function v(t) satisfies the equation and the boundary condition (1.1).
From the existence of such solutions evidently follows thatf (t)∈C0β,γ(E) andµ∈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.
2. Coercive solvability inC0β,γ(E)
It is known that an operator Ais 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 inEwith exponentially decreasing norm whent → +∞, that is, the following estimates hold.
exp{−tA}
E→E≤Me−δt, tAexp{−tA}
E→E ≤M,
t >0, M >0, δ >0. (2.1) For a strongly positive operatorAwe have that
I−exp{−λA}−1E→E≤ M
1−e−δλ, 0< λ≤1. (2.2) Theorem2.1. LetAbe a strongly positive operator in a Banach spaceEand f (t)∈C0β,γ(E) (0≤γ≤β, 0< β <1). Then for the solutionv(t)inC0β,γ(E) of the boundary value problem (1.1), the coercive inequality
v
Cβ,γ0 (E)+AvCβ,γ
0 (E)
≤MAµ+f (λ)−f (0)
Eβ−γ+β−1(1−β)−1fCβ,γ
0 (E)
(2.3)
holds, whereM does not depend onβ,γ,µ, andf (t). Proof. From the strong positivity ofA, it follows that
v(t)=exp{−tA}
I−exp{−λA}−1 µ+
λ
0
exp
−(λ−s)A f (s)ds
+ t
0
exp
−(t−s)A f (s)ds
(2.4) for the solution of problem (1.1) in the space C0β,γ(E) (cf. [4]). Using this
formula, we have that v0 =f (0)−Au(0)
=
I−exp{−λA}−1 λ
0
Aexp
−(λ−s)A f (λ)−f (s) ds
−
I−exp{−λA}−1
Aµ+f (λ)−f (0)
=I1+I2,
(2.5)
where I1=
I−exp{−λA}−1 λ
0
Aexp
−(λ−s)A f (λ)−f (s)
ds, (2.6) I2= −
I−exp{−λA}−1
Aµ+f (λ)−f (0)
. (2.7)
Then the proof of Theorem 2.1 follows from the inequality (1.4) and the estimate
v0
Eβ−γ ≤MAµ+f (λ)−f (0)
Eβ−γ+β−1(1−β)−1fCβ,γ
0 (E)
. (2.8)
Using formula (2.7) and the estimates (2.1), (2.2) forz >0, we have that z1−β+γAe−zAI2
≤I−e−λA−1
E→E
z1−β+γAe−zA
Aµ+f (λ)−f (0)
E
+z1−β+γAe−(z+λ)A
E→Ef (λ)−f (0)
E
≤MAµ+f (λ)−f (0)
Eβ−γ+z1−β+γλβ−γ
z+λ fCβ,γ 0 (E)
≤MAµ+f (λ)−f (0)
Eβ−γ+fCβ,γ
0 (E)
.
(2.9) Therefore
I2
Eβ−γ ≤MAµ+f (λ)−f (0)
Eβ−γ+fCβ,γ 0 (E)
. (2.10)
Now we estimateI1 in the normEβ−γ. Using formula (2.6) and the estimates (2.1), (2.2), for anyz >0 we obtain
z1−β+γAe−zAI1E
≤I−e−λA−1
E→Ez1−β+γ λ
0
A2e−(z+λ−s)
E→Ef (λ)−f (s)
Eds
≤Mz1−β+γ λ
0
(λ−s)βds
(z+λ−s)2λγfCβ,γ
0 (E).
(2.11)
Ifz≤λ,then z1−β+γ
λ
0
(λ−s)βds
(z+λ−s)2λγ ≤z1−β λ
0
ds
(z+λ−s)2−β ≤ 1
1−β. (2.12) Ifz > λ,then
z1−β+γ λ
0
(λ−s)βds
(z+λ−s)2λγ ≤z−β+γλ−γ λ
0
ds (λ−s)1−β
= 1 β
λ z
β−γ
< 1 β.
(2.13)
Therefore, for anyz >0, we have that z1−β+γ
λ
0
(λ−s)βds
(z+λ−s)2λγ ≤ 1
β(1−β). (2.14)
From the last estimate and (2.11), it follows that I1
Eβ−γ ≤ M
β(1−β)fCβ,γ
0 (E). (2.15)
Using the estimates (2.10), (2.15), and the triangle inequality, we obtain the
estimate (2.8).Theorem 2.1is then proved.
Remark 2.2. Note that the spaces of smooth functionsC0β,γ(E)in which coer- cive solvability has been established depend on the parametersβandγ. How- ever, the constants in the coercive inequality (2.3) depend only onβ. 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).
3. Coercive solvability inC0β,γ(Eα−β)
Theorem3.1. LetAbe a strongly positive operator in a Banach spaceEand f (t)∈C0β,γ(Eα−γ) (0≤γ ≤β≤α,0< α <1). Then for the solutionv(t)in Cβ,γ0 (Eα−γ)of the boundary value problem (1.1), the coercive inequality
v
Cβ,γ0 (Eα−β)+AvCβ,γ
0 (Eα−β)
≤MAµ+f (λ)−f (0)
Eα−γ+α−1(1−α)−1fCβ,γ
0 (Eα−β)
(3.1)
holds, whereM does not depend onβ,γ,α, andf (t).
Proof. The proof of Theorem 3.1 follows from the inequality (1.5) and the estimate
v0E
α−γ ≤MAµ+f (λ)−f (0)E
α−γ+ 1
α(1−α)fCβ,γ
0 (Eα−β)
. (3.2)
Using formula (2.7) and the estimates (2.1), (2.2) for anyz >0, we have that z1−α+γAe−zAI2
E
≤I−e−λAE→E
z1−α+γAe−zA
Aµ+f (λ)−f (0)E +z1−α+γAe−(z+λ)A
f (λ)−f (0)
E
≤M1
Aµ+f (λ)−f (0)E
α−γ+ z1−α+γ
(z+λ)1−α+βf (λ)−f (0)E
α−β
≤M1
Aµ+f (λ)−f (0)
Eα−γ+z1−α+γλβ−γ
(z+λ)1−α+βfCβ,γ 0 (Eα−β)
.
(3.3) Since
z1−α+γλβ−γ
(z+λ)1−α+β ≤1, (3.4)
we have that
z1−α+γAe−zAI2E ≤M1
Aµ+f (λ)−f (0)E
α−γ+fCβ,γ
0 (Eα−β)
, (3.5) for anyz >0, and it follows that
I2
Eα−γ ≤M1
Aµ+f (λ)−f (0)
Eα−γ+fCβ,γ 0 (Eα−β)
. (3.6)
Now we estimateI1in the normEα−γ. Using formula (2.6) and the estimates (2.1), (2.2) for anyz >0, we obtain
z1−α+γAe−zAI1E
≤I−e−λA−1E→Ez1−α+γ λ
0
A2e−(z+λ−s)
f (λ)−f (s)
Eds
≤M1z1−α+γ λ
0
1
(z+λ−s)2−α+βf (λ)−f (s)
Eα−βds
≤M1z1−α+γ λ
0
(λ−s)βds
(z+λ−s)2−α+βλγfCβ,γ
0 (Eα−β).
(3.7)
Ifz≤λ,then z1−α+γ
λ
0
(λ−s)βds
(z+λ−s)2−α+βλγ ≤z1−α+γ λγ
λ
0
ds (z+λ−s)2−α
≤ zα
(1−α)λα ≤ 1 1−α.
(3.8)
Ifz≥λ,then z1−α+γ λ
0
(λ−s)βds
(z+λ−s)2−α+βλγ ≤z−α+γ λγ
λ
0
ds
(λ−s)1−α = λα−γ αzα−γ ≤1
α. (3.9) Therefore, for anyz >0 we have that
z1−α+γ λ
0
(λ−s)βds
(z+λ−s)2−α+βλγ ≤ 1
α(1−α). (3.10) From the last estimate and (3.7), it follows that
I1E
α−γ ≤ M
α(1−α)fCβ,γ
0 (Eα−β). (3.11) Using the estimates (3.6), (3.11), and the triangle inequality, we obtain the estimate (3.2). This completes the proof ofTheorem 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 i=1
civ ti
+µ, (3.12)
where 0 < t1 < t2 < ··· < tp ≤1, if the operator I−p
i=1cie−tiA has a bounded inverse inE.
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)
∂t +
|τ|=2m
aτ(x) ∂|τ|v(t,x)
∂x1τ1···∂xnτn+δv(t,x)=f (t,x) (0≤t≤1), v(0,x)=v(λ,x)+µ(x), 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
sufficiently large number. We will assume that the symbol
B(ξ)=
|τ|=2m
ar(x)(iξ)r1···(iξ)rn (4.2)
of the differential operator of the form
B=
|r|=2m
ar(x) ∂|r|
∂x1r1···∂xnrn, (4.3) acting on functions defined on the spaceRn, satisfies the inequalities
0< M1|ξ|2m≤B(ξ)≤M2|ξ|2m<∞, (4.4) forξ=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 spaceEwith a strongly positive operatorA=B+δIdefined by (4.3). We give inTheorem 4.1a number of corollaries to Theorems2.1and3.1.
Theorem 4.1. The solutions of the boundary value problem (4.1) satisfy the following coercive inequalities:
∂v
∂t Cβ,γ
0 (Cε(Rn))+
|τ|=2m
∂|τ|v
∂x1τ1···∂xnτn
Cβ,γ 0 (Cε(Rn))
≤M(ε)
|τ|=2m
aτ(x) ∂|τ|µ(x)
∂x1τ1···∂xnτn−f (λ,x)+f (0,x)
C2m(β−γ )+ε(Rn)
+ 1
β(1−β)fCβ,γ
0 (Cε(Rn))
,
0<2m(β−γ )+ε <1,0≤γ ≤β,0< β <1, ∂v
∂t
C0β,γ(C2m(α−β)(Rn))+
|τ|=2m
∂|τ|v
∂x1τ1···∂xnτn
Cβ,γ0 (C2m(α−β)(Rn))
≤M(α,β,γ )
|τ|=2m
aτ(x) ∂|τ|µ(x)
∂x1τ1···∂xnτn−f (λ,x)+f (0,x)
C2m(β−γ )(Rn)
+ 1
β(1−β)fCβ,γ
0 (C2m(α−β)(Rn))
,
0<2m(α−γ )+ε <1, 0≤γ≤β, 0< α <1, (4.5)
whereM(ε)do not depend onβ,γ,α,µ(x), andf (t,x)andM(α,β,γ )do not depend onµ(x), andf (t,x). HereCε(Rn)is the space of functions satisfying a Hölder condition with the indicatorε∈(0,1).
The proof of Theorem 4.1 is based on Theorems 2.1 and 3.1, the strong positivity of the operatorAinCε(Rn),the coercive inequality for the solution of the resolvent equation of the elliptic operatorAinCε(Rn), and equivalent of the norms in the spacesEβ=Eβ(A,C(Rn))andC2mβ(Rn)when 0< β <1/2m (see [2,3]).
References
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[2] A. O. Ashyralyev and P. E. Sobolevskii,Coercive stability of a multidimensional difference elliptic equation of2m-th order with variable coefficients, Investiga- tions in the Theory of Differential Equations (Russian), Minvuz Turkmen. SSR, Ashkhabad, 1987, pp. 31–43.CMP 1 009 419.
[3] , Well-Posedness of Parabolic Difference Equations, Birkhäuser Verlag, Basel, 1994.MR 95j:65094.
[4] S. G. Krein,Linear Differential Equations in a Banach Space, Nauka, Moscow, 1966 (Russian).
[5] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,Linear and Quasi- linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Rhode Island, 1967.MR 39#3159b.
Zbl 174.15403.
[6] O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and Quasilinear Elliptic Equa- tions, Academic Press, New York, 1968.MR 39#5941. Zbl 164.13002.
[7] 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:[email protected]
A. Hanalyev: Department of Applied Mathematics, Turkmen State University, Ashgabat, Turkmenistan
E-mail address:[email protected]
P. E. Sobolevskii: Institute of Mathematics, Hebrew University, Jerusalem, Israel
E-mail address:[email protected]
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