Abstract
The boundary value problems for singular degenerate arbitrary order differentialoperator equations with variable coefficients are considered. The uniform coercivity properties of ordinary and partial differential equations with small parameters are derived in abstract spaces. It is shown that corresponding differential operators are positive and also are generators of analytic semigroups. In application, wellposedeness of the Cauchy problem for an abstract parabolic equation and systems of parabolic equations are studied in mixed spaces. These problems occur in fluid mechanics and environmental engineering.
MSC: 34G10, 35J25, 35J70.
Keywords:
differentialoperator equations; degenerate equations; semigroups of operators; Banachvalued function spaces; coercive problems; operatorvalued Fourier multipliers; interpolation of Banach spaces0 Introduction
Boundary value problems (BVPs) for differentialoperator equations (DOEs) in Hvalued (Hilbert space valued) function spaces have been studied extensively by many researchers (see, e.g., [114] and the references therein). A comprehensive introduction to DOEs and historical references may be found in [6] and [14]. The maximal regularity properties for DOEs have been studied, e.g., in [3,1019].
In this work, singular degenerate BVPs for arbitrary order DOEs with parameters are considered. This problem has numerous applications. The parameterdependent BVPs occur in different situations of fluid mechanics and environmental engineering etc.
In Section 2, the BVP for the following singular degenerate ordinary DOE with a small parameter:
is considered, where
t is a small parameter, is a complexvalued function, , is a principal, and are subordinate linear operators in a Banach space E. Several conditions for the uniform coercivity and the resolvent estimates for this problem are given in abstract spaces. We prove that the problem has a unique solution for , , with sufficiently large and the following uniform coercive estimate holds:
where
In Section 3, the partial DOE with small parameters
is considered in a mixed space, where are complexvalued functions, A and are linear operators in E, λ is a complex and are positive parameters, G is an ndimensional rectangular domain, . Here we prove that for , with sufficiently large , this problem has a unique solution u that belongs to the Sobolev space with a mixed p norm and the following coercive uniform estimate holds:
where
In Section 4, the uniform wellposedeness of the mixed problem for the following singular degenerate abstract parabolic equation:
is obtained. Particularly, the above problem occurs in atmospheric dispersion of pollutants and evolution models for phytoremediation of metals from soils. In application, particularly, by taking , , , , we consider the mixed problem for the system of the following parabolic equations with parameters:
which arises in phytoremediation process, where are realvalued functions and are data. The maximal regularity properties of this problem are studied. Note that the maximal regularity properties for undegenerate DOEs were investigated, e.g., in [110,1416,19,20]. Regular degenerate DOEs in Banach spaces were treated in [1113,15,1719,21]. It should be noted that contrary to these results, here highorder singular degenerated BVPs with small parameters are considered. Moreover, principal coefficients depend on space variables. The proofs are based on abstract harmonic analysis, operator theory, interpolation of Banach spaces, theory of semigroups of linear operators, microlocal analysis, embedding and trace theorems in vectorvalued SobolevLions spaces.
1 Notations and background
Let , be a positive measurable function on a domain . denotes the space of strongly measurable Evalued functions that are defined on Ω with the norm
For , the space will be denoted by .
The Banach space E is called a UMDspace if the Hilbert operator
is bounded in , (see, e.g., [22]). UMD spaces include, e.g., , spaces and Lorentz spaces , .
Let C denote the set of complex numbers and
A linear operator A is said to be φpositive in a Banach space E with bound if is dense on E and for any , , where I is the identity operator in E, is the space of bounded linear operators in E. Sometimes will be written as and denoted by . It is known [23], Section 1.15.1] that a positive operator A has welldefined fractional powers . Let denote the space with the norm
Let and be two Banach spaces continuously embedded in a locally convex space. By , , , we denote the interpolation spaces obtained from by the Kmethod [23], Section 1.3.2].
Let denote the space of Evalued uniformly bounded continuous functions on the domain .
Let N denote the set of natural numbers and be a sequence of independent symmetric valued random variables on (see [22]). A set is called uniform Rbounded with respect to h (see, e.g., [16]) if there is a constant C independent of such that for all and , ,
The smallest C for which the above estimate holds is called an Rbound of the collection and is denoted by .
A φpositive operator A is said to be Rpositive in E if the set , , is Rbounded.
Note that for Hilbert spaces , , all normbounded sets are Rbounded (see, e.g., [16]). Therefore, in Hilbert spaces all positive operators are Rpositive. If A is a generator of a contraction semigroup on , , or A has the bounded imaginary powers with , in , then those operators are Rpositive (e.g., see [16], Section 4.3]).
The operator , is said to be φpositive in E uniformly with respect to if is independent of t, is dense in E and for all , , where M does not depend on t.
Let and E be two Banach spaces and be continuously and densely embedded into E. Let m be a positive integer. denotes an valued function space defined by
Let t be a positive parameter. We define a parameterized norm in as follows:
Let , , denote the space of all psummable Evalued functions with a mixed norm (see, e.g., [24], Section 8] for scalar case), i.e., the space of all measurable Evalued functions f defined on G, for which
Let be positive integers, , be positive parameters and .
Consider the following weighted spaces of functions:
with the mixed norm
and with the parameterized norm
respectively.
Consider the BVP for DOE
where , , are complex numbers and , ; A is a possible unbounded operator in E.
In a similar way as in [17], Theorem 5.1], we obtain the following.
Theorem A1Let the following conditions be satisfied:
(1) , are complex numbers, , tis a small positive parameter and;
(3) Ais anRpositive operator inE.
Then problem (1) forandwith sufficiently largehas a unique solution. Moreover, the following uniform coercive estimate holds:
By reasoning as in [17], Theorem 2.3], we obtain the following.
Theorem A2Let the following conditions be satisfied:
(2) Eis a UMD space andAis anRpositive operator inE;
(3) there exists a bounded linear extension operator fromto.
Then the embeddingis continuous and for, the uniform estimate
Let
Theorem A3Let the following conditions be satisfied:
(2) Eis a UMD space andAis anRpositive operator inE;
(3) andarentuples of a nonnegative integer such that
(4) there exists a bounded linear extension operator fromto.
Then the embeddingis continuous. Moreover, there is a constantsuch that for, the following uniform estimate holds:
2 Singular degenerate DOEs with parameter
Consider the BVP for the following differentialoperator equation with parameter:
on the domain , where t is a positive parameter and λ is a complex parameter; , are complex numbers and , , is a complexvalued function on ; and are linear operators in a Banach space E and . Note that
A function satisfying equation (2) a.e. on is said to be the solution of equation (2) on .
Remark 1
Let
Under the substitution (3), spaces and are mapped isomorphically onto weighted spaces
respectively, where
Moreover, under the substitution (3), problem (2) is transformed into the following nondegenerate problem:
in the weighted space , where , , , are again denoted by , , , γ after the substitution (3), respectively.
Let us consider boundary value problem (4)(5).
Theorem 1Let the following conditions be satisfied:
(1) , is a positive uniformly bounded continuous function on;
(3) isRpositive inEuniformly with respect toand, ;
(4) for any, there is a positivesuch that
Then problem (4)(5) has a unique solutionforandwith sufficiently large. Moreover, the following uniform coercive estimate holds:
Proof Let be bounded intervals in and correspond to a partition of unit that functions are smooth on , and . Then, for all , we have , where . For , from (4) we obtain
where
Since a is uniformly bounded on for all small , there is a large such that for all . Let
Cover by finitely many intervals such that
Define coefficients of local operators, i.e.,
and
for each . Then, for all and , we get
Freezing coefficients in (7) obtain that
where
Since functions have compact supports in (9), if we extend on the outsides of , we obtain BVPs with constant coefficients
Let denote Evalued weighted norms with respect to domains . Let be such that . Then, by virtue of Theorem A_{1}, we obtain that problem (11) has a unique solution and for and sufficiently large , the following estimate holds:
Theorem A_{2} implies that for all , there is a continuous function such that
Consequently, by using Theorem A_{2}, from (12)(13) we get
Then, by using the equality and by virtue of (15) for , we have
Let be a solution of problem (4)(5). For , we have
By Theorem A_{2}, by virtue of (16) and (17) for sufficiently large , we have
Consider the operator in generated by problem (4)(5), i.e.,
Estimate (18) implies that problem (4)(5) has only a unique solution and the operator has an invertible operator in its rank space. We need to show that this rank space coincides with the space . We consider the smooth functions with respect to the partition of the unit on that equals one on , where and . Let us construct for all j the function that is defined on the regions and satisfies problem (4)(5). Problem (4)(5) can be expressed in the form
Consider operators in generated by BVPs (19). By virtue of Theorem A_{1} for all , for and sufficiently large , we have
Extending zero on the outside of in equalities (20) and passing substitutions , we obtain operator equations with respect to
By virtue of Theorem A_{2}, by estimate (20), in view of the smoothness of the coefficients of , for and sufficiently large , we have , where ε is sufficiently small. Consequently, equations (21) have unique solutions
Moreover,
Whence, are bounded linear operators from X to . Thus, we obtain that the functions
are the solutions of equations (21). Consider the linear operator in X such that
It is clear from the constructions and estimate (20) that operators are bounded linear from X to and
Therefore, is a bounded linear operator from X to X. Let denote the operator in generated by BVP (4)(5). Then the act of to gives , where is a linear combination of and . By virtue of Theorem A_{2}, estimate (22) and in view of the expression , we obtain that operators are bounded linear from X to and . Therefore, there exists a bounded linear invertible operator . Whence, we obtain that for all , BVP (4)(5) has a unique solution
Then, by using the above representation and in view of Theorem A_{1}, we obtain the assertion of Theorem 1. □
Result 1 Theorem 1 implies that the operator has a resolvent for and the following estimate holds:
Let denote the operator in generated by BVP (2). By virtue of Theorem 1 and Remark 1, we obtain the following result.
Result 2 Let all conditions of Theorem 1 be satisfied. Then
(a) problem (2) has a unique solution for and sufficiently large . Moreover, the following uniform coercive estimate holds:
(b) has a resolvent operator for and
Theorem 2Let all conditions of Theorem 1 hold. Then the operatoris uniformlyRpositive in, alsois a generator of an analytic semigroup.
Proof By virtue of Theorem 1, we obtain that for , BVP (4)(5) has a unique solution expressed in the form
where are local operators generated by problems (7)(8) and , are uniformly bounded operators in . In a similar way as in [1,11,17], we obtain that operators are Rpositive. Then, by using the above representation and by virtue of Kahane’s contraction principle, the product and additional properties of the collection of Rbounded operators (see, e.g., [16], Lemma 3.5, Proposition 3.4), we obtain the assertions. □
3 Singular degenerate anisotropic equation with parameters
Consider the following degenerate BVP with parameters:
where and are linear operators in a Banach space E,
are complexvalued functions on G, are complex numbers, are positive and λ is a complex parameter.
Note that BVP (24) is degenerated with different speeds on different directions in general.
The main result of this section is the following.
Theorem 3Assume the following conditions hold:
(1) Eis a UMD space, isRpositive inEuniformly with respect toand, ;
(2) for any, there is a positivesuch that
(4) are continuous positive functions on.
Then, for, and sufficiently large, problem (24) has a unique solutionuthat belongs toand the following coercive uniform estimate holds:
Proof
Consider the BVP
where are boundary conditions of type (24) on . By virtue of Result 2, problem (26) has a unique solution for all , and sufficiently large . Moreover, the following coercive uniform estimate holds:
Let us now consider in the BVP on the domain
where . Since , then problem (27) can be expressed in the following way:
where B is the differential operator in generated by problem (26), i.e.,
By virtue of [22], for provided . Moreover, by virtue of Theorem 2, the operator B is Rpositive in . Hence, by Result 2, we get that problem (27) has a unique solution
for , and sufficiently large , and coercive uniform estimate (25) holds. By continuing this process for , we obtain that the following problem:
for , and sufficiently large , has a unique solution and the following coercive uniform estimate holds:
Moreover, by virtue of embedding Theorem A_{3}, we have the EhrlingNirenbergGagliardo type estimate
Let denote the operator generated by problem (28) and
By using estimate (29), we obtain that there is a such that
Then, by using perturbation elements, we obtain the assertion. □
From Theorem 2 and Theorem 3, we obtain the following result.
Result 3 Let all conditions of Theorem 3 hold for and . Then the operator is uniformly Rpositive in , it also is a generator of an analytic semigroup.
4 Singular degenerate parabolic DOE
Consider the following mixed problem for a parabolic DOE with parameter:
where , , G, , are defined as in Section 3, .
For , , will denote the space of all Evalued summable functions with a mixed norm. Analogously,
denotes the Sobolev space with a corresponding mixed norm (see [24] for a scalar case).
Let denote a differential operator generated by (28) for .
Theorem 4Let all conditions of Theorem 3 hold forand. Then, forand sufficiently large, problem (30) has a unique solution belonging toand the following coercive estimate holds:
Proof Problem (30) can be expressed as the following Cauchy problem:
Result 3 implies that the operator is Rpositive in . By [23], Section 1.14], is a generator of an analytic semigroup in F. Then, by virtue of [20], Theorem 4.2], we obtain that for problem (31) has a unique solution belonging to and the following estimate holds:
The above estimate proves the hypothesis to be true. □
5 Cauchy problem for infinite systems of degenerate parabolic equations with small parameters
Consider the infinity systems of BVP for the degenerate anisotropic parabolic equation:
where N is finite or infinite natural number, , , G, , , , d are defined as in Sections 3 and 4, are real functions and
From Theorem 4 we obtain the following.
Theorem 5Let, , , , and. Then forand sufficiently large, problem (32) has a unique solutionthat belongs to the spaceand the following coercive uniform estimate holds:
Proof Let and A be infinite matrices such that
It is clear that the operator A is Rpositive in . Problem (32) can be rewritten as problem (30). Then, from Theorem 4, we obtain the assertion. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The results to proving the uniform coercivity properties of ordinary and partial differential equations with small parameters in abstract spaces, the showing that corresponding differential operators are positive and also are generators of analytic semigroups and wellposedeness of Cauchy problem for abstract parabolic equation and systems of parabolic equations are studied in mixed spaces due to VS.
The applications of these abstract problems to concrete mathematics and engineering problem belongs to AS.
Both authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
References

Agarwal, RP, Bohner, R, Shakhmurov, VB: Maximal regular boundary value problems in Banachvalued weighted spaces. Bound. Value Probl.. 1, 9–42 (2005)

Ashyralyev, A: On wellposedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim.. 24(1 & 2), 1–15 (2003)

Dore, C, Yakubov, S: Semigroup estimates and non coercive boundary value problems. Semigroup Forum. 60, 93–121 (2000). Publisher Full Text

Favini, A, Shakhmurov, V, Yakubov, Y: Regular boundary value problems for complete second order elliptic differentialoperator equations in UMD Banach spaces. Semigroup Forum. 79(1), 22–54 (2009). Publisher Full Text

Goldstain, JA: Semigroups of Linear Operators and Applications, Oxford University Press, Oxford (1985)

Krein, SG: Linear Differential Equations in Banach Space, Am. Math. Soc., Providence (1971)

Lunardi, A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (2003)

Lions, JL, Magenes, E: Nonhomogenous Boundary Value Problems, Mir, Moscow (1971)

Sobolevskii, PE: Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR. 57(1), 27–40 (1964)

Prüss, J: Maximal regularity for evolution equations in spaces. Monopoli. (2002)

Shakhmurov, VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Theory Methods Appl.. 73, 2383–2397 (2010). Publisher Full Text

Shakhmurov, VB: Imbedding theorems and their applications to degenerate equations. Differ. Equ.. 24(4), 475–482 (1988)

Shakhmurov, VB: Coercive boundary value problems for regular degenerate differentialoperator equations. J. Math. Anal. Appl.. 292(2), 605–620 (2004). Publisher Full Text

Yakubov, S, Yakubov, Y: DifferentialOperator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000)

Amann, H: Linear and QuasiLinear Equations, Birkhäuser, Basel (1995)

Denk, R, Hieber, M, Prüss, J: Rboundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788) (2003)

Shakhmurov, VB: Degenerate differential operators with parameters. Abstr. Appl. Anal.. 2006, 1–27 (2007)

Shakhmurov, VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl.. 327(2), 1182–1201 (2006)

Shakhmurov, VB: Nonlinear abstract boundary value problems in vectorvalued function spaces and applications. Nonlinear Anal., Theory Methods Appl.. 67(3), 745–762 (2006)

Weis, L: Operatorvalued Fourier multiplier theorems and maximal regularity. Math. Ann.. 319, 735–758 (2001). Publisher Full Text

Agarwal, R, O’Regan, D, Shakhmurov, VB: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl.. 338, 970–983 (2008). Publisher Full Text

Burkholder, DL: A geometrical condition that implies the existence certain singular integral of Banach spacevalued functions. Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, pp. 270–286. Wadsworth, Belmont Chicago, 1981. (1983)

Triebel, H: Interpolation Theory, Function Spaces, Differential Operators, NorthHolland, Amsterdam (1978)

Besov, OV, Ilin, VP, Nikolskii, SM: Integral Representations of Functions and Embedding Theorems, Nauka, Moscow (1975)