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 A comprehensive introduction to DOEs and historical references may be found in  and .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
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 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 , ,
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]).
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
Consider the following weighted spaces of functions:
with the mixed norm
and with the parameterized norm
Consider the BVP for DOE
In a similar way as in , Theorem 5.1], we obtain the following.
Theorem A1Let the following conditions be satisfied:
(3) Ais anR-positive operator inE.
By reasoning as in , Theorem 2.3], we obtain the following.
Theorem A2Let the following conditions be satisfied:
(2) Eis a UMD space andAis anR-positive operator inE;
Theorem A3Let the following conditions be satisfied:
(2) Eis a UMD space andAis anR-positive operator inE;
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
Moreover, under the substitution (3), problem (2) is transformed into the following non-degenerate problem:
Let us consider boundary value problem (4)-(5).
Theorem 1Let the following conditions be satisfied:
Define coefficients of local operators, i.e.,
Freezing coefficients in (7) obtain that
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:
Consequently, by using Theorem A2, from (12)-(13) we get
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
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
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 2 Let all conditions of Theorem 1 be satisfied. Then
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:
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:
Consider the BVP
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
Moreover, by virtue of embedding Theorem A3, we have the Ehrling-Nirenberg-Gagliardo type estimate
Then, by using perturbation elements, we obtain the assertion. □
From Theorem 2 and Theorem 3, we obtain the following result.
4 Singular degenerate parabolic DOE
Consider the following mixed problem for a parabolic DOE with parameter:
denotes the Sobolev space with a corresponding mixed norm (see  for a scalar case).
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:
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:
From Theorem 4 we obtain the following.
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.
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