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1
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Well-posedness of second order degenerate differential equations in vector-valued function spaces

100%
Bu S.
Studia Mathematica
|
2013
|
tom 214
|
nr 1
1-16
EN
Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu')(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu')(0) = (Mu')(2π), in Lebesgue-Bochner spaces $L^{p}(𝕋,X)$, periodic Besov spaces $B_{p,q}^{s}(𝕋,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^{s}(𝕋,X)$, where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M). Our results generalize the previous results of W. Arendt and S. Q. Bu when $M = I_{X}$.
2
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Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

63%
Bu S., Fang Y.
Studia Mathematica
|
2008
|
tom 184
|
nr 2
103-119
EN
We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u''(t) + αu'(t) + d/dt (∫^{t}_{-∞} b(t-s)u(s)ds) = Au(t) - ∫^{t}_{-∞} a(t-s)Au(s)ds + f(t)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u'(0) = u'(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions are given for the problem (P) to have maximal regularity on $L^{p}(𝕋,X)$, periodic Besov spaces $B_{p,q}^{s}(𝕋,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^{s}(𝕋,X)$
3
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Fourier multipliers for Hölder continuous functions and maximal regularity

51%
Arendt W., Batty C., Bu S.
Studia Mathematica
|
2004
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tom 160
|
nr 1
23-51
EN
Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^{p}$-situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.
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