Singular degenerate problems occurring in biosorption process

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 Lp 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 Lp spaces. These problems occur in fluid mechanics and environmental engineering. MSC:34G10, 35J25, 35J70.


Introduction
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., [-] 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 A comprehensive introduction to DOEs and historical references may be found in [] and [].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 [, -].
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 , the BVP for the following singular degenerate ordinary DOE with a small parameter: t is a small parameter, a(x) is a complex-valued function, γ > , A = A(x) is a principal, A  = A  (x) and A  = A  (x) are subordinate linear operators in a Banach space E. http://www.boundaryvalueproblems.com/content/2013/1/30 Several conditions for the uniform coercivity and the resolvent estimates for this problem are given in abstract L p -spaces. We prove that the problem has a unique solution u ∈ W [] p,γ (, ; E(A), E) for f ∈ L p (, ; E), | arg λ| ≤ ϕ,  ≤ ϕ < π with sufficiently large |λ| and the following uniform coercive estimate holds: In Section , the partial DOE with small parameters  [i] x k u L p (G;E) + Au L p (G;E) ≤ C f L p (G;E) , where In Section , 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 E = R  , A(x) = [a ij (x)], u = (u  , u  , u  ), i, j = , , , we consider the mixed problem for the system of the following parabolic equations with parameters:

Notations and background
. , x n ) be a positive measurable function on a domain ⊂ R n . L p,γ ( ; E) denotes the space of strongly measurable E-valued functions that are defined on with the norm is bounded in L p (R, E), p ∈ (, ∞) (see, e.g., []). UMD spaces include, e.g., L p , l p spaces and Lorentz spaces L pq , p, q ∈ (, ∞). Let C denote the set of complex numbers and where I is the identity operator in E, B(E) is the space of bounded linear operators in E. Sometimes A + λI will be written as A + λ and denoted by A λ . It is known [, Section ..] that a positive operator A has well-defined fractional powers Let E  and E  be two Banach spaces continuously embedded in a locally convex space. By The smallest C for which the above estimate holds is called an R-bound of the collection W h and is denoted by Note that for Hilbert spaces H  , H  , 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 L q ,  ≤ q ≤ ∞, or A has the bounded imaginary powers with A it B(E) ≤ Ce ν|t| , ν < π  in E ∈ UMD, then those operators are R-positive (e.g., see [, Section .]).
The operator Let E  and E be two Banach spaces and E  be continuously and densely embedded into E. Let m be a positive integer. W m p,γ (a, b; E  , E) denotes an E  -valued function space defined by Let t be a positive parameter. We define a parameterized norm in W m p,γ (a, b; E  , E) as follows: Let l k be positive integers, l = (l  , l  , . . . , l n ), t k be positive parameters and t = (t  , t  , . . . , t n ).
Consider the following weighted spaces of functions: k u ∈ L p (G; E) , http://www.boundaryvalueproblems.com/content/2013/1/30 with the mixed norm and with the parameterized norm respectively. Consider the BVP for DOE In a similar way as in [, Theorem .], we obtain the following.
Theorem A  Let the following conditions be satisfied: () α ki , δ kj are complex numbers, α km k = , t is a small positive parameter and Then problem () for f ∈ L p (, ; E) and | arg λ| ≤ ϕ with sufficiently large |λ| has a unique solution u ∈ W [m] p,γ (, ; E(A), E). Moreover, the following uniform coercive estimate holds: By reasoning as in [, Theorem .], we obtain the following.
Theorem A  Let the following conditions be satisfied: holds for all u ∈ W m p,γ (, ∞; E(A), E) and  < h ≤ h  < ∞. http://www.boundaryvalueproblems.com/content/2013/1/30 Let Theorem A  Let the following conditions be satisfied: . . , α n ) and l = (l  , l  , . . . , l n ) are n-tuples of a nonnegative integer such that , the following uniform estimate holds:

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; α ki , δ ki are complex numbers and a.e. on (, ) is said to be the solution of equation () on (, ).
Freezing coefficients in () obtain that where Let · G j ,p,γ denote E-valued weighted L p -norms with respect to domains G j . Let ϕ j be such that  ∈ σ j . Then, by virtue of Theorem A  , we obtain that problem () has a unique solution u j and for | arg λ| ≤ ϕ and sufficiently large |λ|, the following estimate holds: Theorem A  implies that for all ε > , there is a continuous function C(ε) such that Consequently, by using Theorem A  , from ()-() we get Then, by using the equality u(y) = ∞ j= u j (y) and by virtue of () for u ∈ W m p,γ (-∞, ; E(A), Let u ∈ W m p,γ (-∞, ; E(A), E) be a solution of problem ()-(). For | arg λ| ≤ ϕ, we have By Theorem A  , by virtue of () and () for sufficiently large |λ|, we have Estimate () implies that problem ()-() has only a unique solution and the operator O λ has an invertible operator in its rank space. We need to show that this rank space http://www.boundaryvalueproblems.com/content/2013/1/30 coincides with the space X = L p,γ (-∞, ; E). We consider the smooth functions g j = g j (y) with respect to the partition of the unit ϕ j = ϕ j (y) on (-∞, ) that equals one on supp ϕ j , where supp g j ⊂ G j and |g j (y)| < . Let us construct for all j the function u j that is defined on the regions j = (-∞, ) ∩ G j and satisfies problem ()-(). Problem ()-() can be expressed in the form Consider operators O jtλ in L p,γ (G j ; E) generated by BVPs (). By virtue of Theorem A  for all f ∈ L p,γ (G j ; E), for | arg λ| ≤ ϕ and sufficiently large |λ|, we have Extending u j zero on the outside of supp ϕ j in equalities () and passing substitutions u j = O - jλ υ j , we obtain operator equations with respect to υ j By virtue of Theorem A  , by estimate (), in view of the smoothness of the coefficients of K jλt , for | arg λ| ≤ ϕ and sufficiently large |λ|, we have K jλt < ε, where ε is sufficiently small. Consequently, equations () have unique solutions Moreover, Whence, [I -K jλt ] - g j are bounded linear operators from X to L p,γ (G j ; E). Thus, we obtain that the functions are the solutions of equations (). Consider the linear operator (U t + λ) in X such that Therefore, (U t + λ) is a bounded linear operator from X to X. Let L t denote the operator in L p,γ generated by BVP ()-(). Then the act of (L t + λ) to u = ∞ j= ϕ j U jtλ f gives (L t + λ)u = f + ∞ j= jtλ f , where jλ is a linear combination of U jλ and d dy U jλ . By virtue of Theorem A  , estimate () and in view of the expression jλ , we obtain that operators jλ are bounded linear from X to L p,γ (G j ; E) and jλt < ε. Therefore, there exists a bounded linear invertible operator (I + ∞ j= jtλ ) - . Whence, we obtain that for all f ∈ X, BVP ()-() has a unique solution Then, by using the above representation and in view of Theorem A  , we obtain the assertion of Theorem .
Result  Theorem  implies that the operator L t has a resolvent (L t + λ) - for | arg λ| ≤ ϕ and the following estimate holds: Let G t denote the operator in L p (, ; E) generated by BVP (). By virtue of Theorem  and Remark , we obtain the following result.

Result  Let all conditions of Theorem  be satisfied. Then
(a) problem () has a unique solution u ∈ W [m] p,γ (, ; E(A), E) for f ∈ L p (, ; E) and sufficiently large |λ|. Moreover, the following uniform coercive estimate holds: Theorem  Let all conditions of Theorem  hold. Then the operator L t is uniformly Rpositive in L p (, ; E), also L t is a generator of an analytic semigroup.
Proof By virtue of Theorem , we obtain that for f ∈ L p (, ; E), BVP ()-() has a unique solution expressed in the form where O jλt = O j t + λ are local operators generated by problems ()-() and K jtλ , jtλ are uniformly bounded operators in L p (, ; E). In a similar way as in [, , ], we obtain that operators O jt 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 ., Proposition .), we obtain the assertions.

Singular degenerate anisotropic equation with parameters
Consider the following degenerate BVP with parameters: where A(x) and A α (x) are linear operators in a Banach space E, a k are complex-valued functions on G, α kji are complex numbers, t k are positive and λ is a complex parameter. Note that BVP () is degenerated with different speeds on different directions in general.
The main result of this section is the following.
Theorem  Assume the following conditions hold: Proof Consider the BVP where L k are boundary conditions of type () on (, b  ). By virtue of Result , problem () has a unique solution u ∈ W , | arg λ| ≤ ϕ and sufficiently large |λ|. Moreover, the following coercive uniform estimate holds: Let us now consider in can be expressed in the following way: where B is the differential operator in L p  (, b  ; E) generated by problem (), i.e., By virtue of [], L p  (, b  ; E) ∈ UMD for p  ∈ (, ∞) provided E ∈ UMD. Moreover, by virtue of Theorem , the operator B is R-positive in L p  (, b  ; E). Hence, by Result , we get that problem () has a unique solution for f ∈ L p  ,p  (G  ; E), | arg λ| ≤ ϕ and sufficiently large |λ|, and coercive uniform estimate () holds. By continuing this process for k = n, we obtain that the following problem: Moreover, by virtue of embedding Theorem A  , we have the Ehrling-Nirenberg-Gagliardo type estimate Let Q t denote the operator generated by problem () and By using estimate (), we obtain that there is a δ ∈ (, ) such that Then, by using perturbation elements, we obtain the assertion.
From Theorem  and Theorem , we obtain the following result.
Result  Let all conditions of Theorem  hold for ϕ > π  and A α = . Then the operator Q t is uniformly R-positive in L p (G; E), it also is a generator of an analytic semigroup.

Singular degenerate parabolic DOE
Consider the following mixed problem for a parabolic DOE with parameter: where t k , a k (x), G, G kb , m kj are defined as in Section , d > . Forp = (p  , p  , . . . , p n , p  ), + = R + × G, Lp( + ; E) will denote the space of all E-valued p-summable functions with a mixed norm. Analogously, denotes the Sobolev space with a corresponding mixed norm (see [] for a scalar case).
Let Q t denote a differential operator generated by () for λ = .
Theorem  Let all conditions of Theorem  hold for A β =  and ϕ > π  . Then, for f ∈ Lp( + ; E) and sufficiently large d > , problem () has a unique solution belonging to W , [m] p,α Result  implies that the operator Q t is R-positive in F = L p (G; E). By [, Section .], Q t is a generator of an analytic semigroup in F. Then, by virtue of [, Theorem .], we obtain that for f ∈ L p  (R + ; F) problem () has a unique solution belonging to W  p  (R + ; D(Q t ), F) and the following estimate holds: du dy L p  (R + ;F) + Q t u L p  (R + ;F) ≤ C f L p  (R + ;F) .
The above estimate proves the hypothesis to be true.

Cauchy problem for infinite systems of degenerate parabolic equations with small parameters
Consider the infinity systems of BVP for the degenerate anisotropic parabolic equation: