Abstract
In the present paper, the wellposedness of the initial value problem for the delay differential equation , ; () in an arbitrary Banach space E with the unbounded linear operators A and in E with dense domains is studied. Two main theorems on wellposedness of this problem in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.
MSC: 35G15.
Keywords:
delay parabolic equations; wellposedness; fractional spaces; coercive stability estimates1 Introduction
The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [113] and the references therein) and insight has developed over the last three decades. The theory of stability and coercive stability of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations (see [1419]). It is well known that various initialboundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form
in an arbitrary Banach space E with the unbounded linear operators A and in E with dense domains . Let A be a strongly positive operator, i.e. −A is the generator of the analytic semigroup () of the linear bounded operators with exponentially decreasing norm when . That means the following estimates hold:
for some , . Let be closed operators.
A function is called a solution of the problem (1) if the following conditions are satisfied:
(i) is continuously differentiable on the interval . The derivative at the endpoint is understood as the appropriate unilateral derivative.
(ii) The element belongs to for all , and the function is continuous on the interval .
(iii) satisfies the equation and the initial condition (1).
A solution of the initial value problem (1) is said to be coercive stable (wellposed) if
for every t, . We are interested in studying the coercive stability of solutions of the initial value problem under the assumption that
holds for every . We have not been able to obtain the estimate (3) in the arbitrary Banach space E. Nevertheless, we can establish the analog of estimates (3) where the space E is replaced by the fractional spaces () under an assumption stronger than (4). The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained.
The present paper is organized as follows. Section 1 is introduction. In Section 2, two main theorems on wellposedness of the initial value problem (1) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of the initialboundary value problem for delay parabolic equations are obtained. Finally, Section 4 is our conclusion.
2 Theorems on wellposedness
The strongly positive operator A defines the fractional spaces () consisting of all for which the following norms are finite:
We consider the initial value problem (1) for delay differential equations of parabolic type in the space of all continuous functions defined on the segment with values in a Banach space . First, we consider the problem (1) when and commute, i.e.
Theorem 2.1Assume that the condition
holds for every, whereMis the constant from (2). Then for everyt, , , we have the following coercive stability estimate:
wheredoes not depend onand. Here, we putwhen.
Proof It is clear that
where is the solution of the problem
and is the solution of the problem
First, we consider the problem (9). Using the formula
the semigroup property, condition (5), and the estimates (2), (6), we obtain
for every t, and λ, . This shows that
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality
is true for t, , for some n. Letting , we have
Using the estimate (12), we obtain
for every t, , and λ, . This shows that
is true for every . Applying (9), the triangle inequality, condition (5), and the estimates (6) and (13), we get
for every . Second, we consider the problem (10). To prove the theorem it suffices to establish the following stability inequality:
for the solution of the problem (10) for every t, , . Using the formula
the semigroup property, and the definition of the spaces , we obtain
for every t, and λ, . This shows that
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , , for some n. Using the formula
the semigroup property, the definition of the spaces , the estimate (2), and condition (6), we obtain
for every t, , and λ, . This shows that
for every t, , . Applying equation (9), the triangle inequality, and condition (5) and estimates (6) and (19), we get
for every . This result completes the proof of Theorem 2.1. □
Now, we consider the problem (1) when
for some . Note that A is a strongly positive operator in a Banach spaces E iff its spectrum lies in the interior of the sector of angle φ, , symmetric with respect to the real axis, and if on the edges of this sector, and and outside it the resolvent is the subject to the bound
for some . First of all let us give lemmas from the paper [18] that will be needed in the sequel.
Lemma 2.1For anyzon the edges of the sector,
and
and outside it the estimate
holds for any. Here and in the futureMandare the same constants of the estimates (2) and (20).
Lemma 2.2Let for allthe operatorwith domain which coincide withadmit a closurebounded inE. Then for allthe following estimate holds:
Suppose that
holds for every. Here and in the futureεis some constant, .
The application of Lemmas 2.1 and 2.2 enables us to establish the following fact.
Theorem 2.2Assume that the condition
holds for every. Then for everythe coercive stability estimate (7) holds.
Proof In a similar manner as in the proof of Theorem 2.1 we establish estimates for the solution of the problems (9) and (10), separately. First, we consider the problem (9). Let and λ, . Then using (11), we have
where
Using the estimates (2), (20), and condition (22), we obtain
for every t, and λ, . Now let us estimate . By Lemma 2.1 and using the estimate (21), we obtain
for every t, and λ, . Using the triangle inequality, we obtain
for every t, and λ, . This shows that
for every t, . In a similar manner as with Theorem 2.1 applying mathematical induction, one can easily show that it is true for every t. Therefore, to prove the theorem it suffices to establish the coercive stability inequality (15) for the solution of the problem (10). Now, we consider the problem (10). Exactly in the same manner, using (16), the semigroup property, and the definition of the spaces , we obtain (15) for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , for some n. Using (18) and the semigroup property, we write
where
Using the estimate (2) and condition (22), we obtain
for every t, , , and λ, . Now let us estimate . By Lemma 2.2 and using the estimate (2) and condition (21), we obtain
for every t, , and λ, . Using the triangle inequality and estimates for all , , we obtain
for every t, , and λ, . This shows that
for every t, , . This result completes the proof of Theorem 2.2. □
Note that these abstract results are applicable to the study of stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study the structure of for space operators in Banach spaces. The structure of for some space differential and difference operators in Banach spaces has been investigated (see [2030]). In Section 3, applications of Theorem 2.1 to the study of the coercive stability of initialboundary value problem for delay parabolic equations are given.
3 Applications
First, we consider the initialboundary value problem for one dimensional delay differential equations of parabolic type
where , , , are given sufficiently smooth functions and is a sufficiently large number. We will assume that . The problem (23) has a unique smooth solution. This allows us to reduce the initialboundary value problem (23) to the initial value problem (1) in Banach space with a differential operator defined by the formula
with domain . Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.1Assume that
Then for allthe solutions of the initialboundary value problem (23) satisfy the following coercive stability estimates:
whereis not dependent onand. Hereis the space of functions satisfying a Hölder condition with the indicator.
The proof of Theorem 3.1 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in (see [31,32]) and on Theorem 3.2 on the structure of the fractional space for .
Theorem 3.2For, the norms of the spaceand the Hölder spaceare equivalent[21].
Second, we consider the initial nonlocal boundary value problem for one dimensional delay differential equations of parabolic type,
where , , , are given sufficiently smooth functions and is a sufficiently large number. We will assume that . The problem (26) has a unique smooth solution. This allows us to reduce the initialboundary value problem (26) to the initial value problem (1) in Banach space with a differential operator defined by the formula
with domain . Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.3Assume that condition (25) holds. Then for allthe solutions of the initialboundary value problem (26) satisfy the following coercive stability estimates:
The proof of Theorem 3.3 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in (see [6]) and on Theorem 3.4 on the structure of the fractional space for .
Theorem 3.4For, the norms of the spaceand the Hölder spaceare equivalent[6].
Third, we consider the initial value problem on the range
for 2mth order multidimensional delay differential equations of parabolic type,
where , , , and are sufficiently smooth functions and is a sufficiently large number. We will assume that the symbol
of the differential operator of the form
acting on functions defined on the space , satisfies the inequalities
for , where . The problem (28) has a unique smooth solution. This allows us to reduce the initial value problem (28) to the initial value problem (1) in Banach space E with a strongly positive operator defined by (29). Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.5Assume that condition (25) holds. Then for allthe solutions of the initialboundary value problem (28) satisfy the following coercive stability estimates:
wheredoes not depend onand. Hereis the space of functions satisfying a Hölder condition with the indicator.
The proof of Theorem 3.5 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in , and on the equivalence of the norms in the spaces and when [20,23].
4 Conclusion
In the present paper, two theorems on the wellposedness of the initial value problem for the delay parabolic differential equations with unbounded operators acting on delay terms in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
This work is supported by Trakya University Scientific Research Projects Unit (Project No: 201091).
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