The boundary value problems for singular degenerate arbitrary order differential-operator 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, well-posedeness 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:differential-operator equations; degenerate equations; semigroups of operators; Banach-valued function spaces; coercive problems; operator-valued Fourier multipliers; interpolation of Banach spaces
Boundary value problems (BVPs) for differential-operator equations (DOEs) in H-valued (Hilbert space valued) function spaces have been studied extensively by many researchers (see, e.g., [1-14] and the references therein). A comprehensive introduction to DOEs and historical references may be found in  and . The maximal regularity properties for DOEs have been studied, e.g., in [3,10-19].
In this work, singular degenerate BVPs for arbitrary order DOEs with parameters are considered. This problem has numerous applications. The parameter-dependent 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 complex-valued 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:
In Section 3, the partial DOE with small parameters
is considered in a mixed space, where are complex-valued functions, A and are linear operators in E, λ is a complex and are positive parameters, G is an n-dimensional 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:
In Section 4, the uniform well-posedeness 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 real-valued 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 [1-10,14-16,19,20]. Regular degenerate DOEs in Banach spaces were treated in [11-13,15,17-19,21]. It should be noted that contrary to these results, here high-order 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 vector-valued Sobolev-Lions spaces.
1 Notations and background
Let , be a positive measurable function on a domain . denotes the space of strongly measurable E-valued functions that are defined on Ω with the norm
For , the space will be denoted by .
The Banach space E is called a UMD-space if the Hilbert operator
is bounded in , (see, e.g., ). 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 φ-positive 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 , Section 1.15.1] that a positive operator A has well-defined 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 K-method , Section 1.3.2].
Let denote the space of E-valued 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 ). A set is called uniform R-bounded with respect to h (see, e.g., ) 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 R-bound of the collection and is denoted by .
A φ-positive operator A is said to be R-positive in E if the set , , is R-bounded.
Note that for Hilbert spaces , , all norm-bounded sets are R-bounded (see, e.g., ). Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on , , or A has the bounded imaginary powers with , in , then those operators are R-positive (e.g., see , Section 4.3]).
The operator , is said to be φ-positive 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 p-summable E-valued functions with a mixed norm (see, e.g., , Section 8] for scalar case), i.e., the space of all measurable E-valued 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
Consider the BVP for DOE
where , , are complex numbers and , ; A is a possible unbounded operator in E.
In a similar way as in , Theorem 5.1], we obtain the following.
Theorem A1Let the following conditions be satisfied:
(1) , are complex numbers, , tis a small positive parameter and ;
(2) Eis a UMD space, , ;
(3) Ais anR-positive operator inE.
Then problem (1) for and with sufficiently large has a unique solution . Moreover, the following uniform coercive estimate holds:
By reasoning as in , Theorem 2.3], we obtain the following.
Theorem A2Let the following conditions be satisfied:
(1) , , , ;
(2) Eis a UMD space andAis anR-positive operator inE;
(3) there exists a bounded linear extension operator from to .
Then the embedding is continuous and for , the uniform estimate
holds for all and .
Theorem A3Let the following conditions be satisfied:
(1) , , ;
(2) Eis a UMD space andAis anR-positive operator inE;
(3) and aren-tuples of a nonnegative integer such that
(4) there exists a bounded linear extension operator from to .
Then the embedding is continuous. Moreover, there is a constant such that for , the following uniform estimate holds:
2 Singular degenerate DOEs with parameter
Consider the BVP for the following differential-operator equation with parameter:
on the domain , where t is a positive parameter and λ is a complex parameter; , are complex numbers and , , is a complex-valued 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 .
Under the substitution (3), spaces and are mapped isomorphically onto weighted spaces
Moreover, under the substitution (3), problem (2) is transformed into the following non-degenerate 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 ;
(2) Eis a UMD space, , , and ;
(3) isR-positive inEuniformly with respect to and , ;
(4) for any , there is a positive such that
Then problem (4)-(5) has a unique solution for and with 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
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.,
for each . Then, for all and , we get
Freezing coefficients in (7) obtain that
Since functions have compact supports in (9), if we extend on the outsides of , we obtain BVPs with constant coefficients
Let denote E-valued weighted -norms with respect to domains . Let be such that . Then, by virtue of Theorem A1, we obtain that problem (11) has a unique solution and for and sufficiently large , the following estimate holds:
Theorem A2 implies that for all , there is a continuous function such that
Consequently, by using Theorem A2, from (12)-(13) we get
Choosing , from (14) we have
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 A2, 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 A1 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 A2, 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
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 A2, 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 A1, 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 operator is uniformlyR-positive in , also is 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 R-positive. Then, by using the above representation and by virtue of Kahane’s contraction principle, the product and additional properties of the collection of R-bounded operators (see, e.g., , 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 complex-valued 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, isR-positive inEuniformly with respect to and , ;
(2) for any , there is a positive such that
(3) , , , ;
(4) are continuous positive functions on .
Then, for , and sufficiently large , problem (24) has a unique solutionuthat belongs to and the following coercive uniform estimate holds:
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 , for provided . Moreover, by virtue of Theorem 2, the operator B is R-positive 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 A3, we have the Ehrling-Nirenberg-Gagliardo 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 R-positive 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 E-valued -summable functions with a mixed norm. Analogously,
denotes the Sobolev space with a corresponding mixed norm (see  for a scalar case).
Let denote a differential operator generated by (28) for .
Theorem 4Let all conditions of Theorem 3 hold for and . Then, for and sufficiently large , problem (30) has a unique solution belonging to and the following coercive estimate holds:
Proof Problem (30) can be expressed as the following Cauchy problem:
Result 3 implies that the operator is R-positive in . By , Section 1.14], is a generator of an analytic semigroup in F. Then, by virtue of , Theorem 4.2], we obtain that for problem (31) has a unique solution belonging to and the following estimate holds:
Since , by Theorem 3 we have
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 for and sufficiently large , problem (32) has a unique solution that belongs to the space and the following coercive uniform estimate holds:
Proof Let and A be infinite matrices such that
It is clear that the operator A is R-positive in . Problem (32) can be rewritten as problem (30). Then, from Theorem 4, we obtain the assertion. □
The authors declare that they have no competing interests.
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 well-posedeness 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.
Dedicated to Professor Hari M Srivastava.
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 differential-operator equations in UMD Banach spaces. Semigroup Forum. 79(1), 22–54 (2009). Publisher Full Text
Shakhmurov, VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Theory Methods Appl.. 73, 2383–2397 (2010). Publisher Full Text
Shakhmurov, VB: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl.. 292(2), 605–620 (2004). Publisher Full Text
Weis, L: Operator-valued 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 space-valued functions. Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, pp. 270–286. Wadsworth, Belmont Chicago, 1981. (1983)