Abstract
The coercive properties of degenerate abstract convolution-elliptic equations are investigated. Here we find sufficient conditions that guarantee the separability of these problems in $L_{p}$ spaces. It is established that the corresponding convolution-elliptic operator is positive and is also a generator of an analytic semigroup. Finally, these results are applied to obtain the maximal regularity properties of the Cauchy problem for a degenerate abstract parabolic equation in mixed $L_{\mathbf{p}}$ norms, boundary value problems for degenerate integro-differential equations, and infinite systems of degenerate elliptic integro-differential equations.
Highlights
The coercive properties of degenerate abstract convolution-elliptic equations are investigated
The main aim of the present paper is to study the following degenerate elliptic convolutiondifferential operator equations (CDOEs): aα ∗ D[α]u + (A + λ) ∗ u = f (x)
One of the main features of the present work is that the convolution equations are degenerate on some points of R =(–∞, ∞)
Summary
Proposition A Let E be a UMD space and γ ∈ Ap. Assume h is a set of operator functions from Cn(Rn\{ }; B(E)) depending on the parameter h ∈ Q ∈ R and there is a positive constant K such that sup R |ξ ||β|Dβ h(ξ ) : ξ ∈ Rn\{ }, βk ∈ { , } ≤ K. Holds and λ ∈ Sφ with φ ∈ [ , π ), where φA + φ + φ < π , the operator functions σi(ξ , λ) are uniformly bounded, i.e., σi(ξ , λ) B(E) ≤ C, i = , ,. The operator functions |ξ ||β|Dβ σi(ξ , λ), i = , , , are uniformly bounded
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