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Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

Part of: Functional-differential and differential-difference equations Groups and semigroups of linear operators, their generalizations and applications General theory of linear operators Differential equations in abstract spaces

Published online by Cambridge University Press:  15 October 2018

Shangquan Bu  and
Gang Cai
Shangquan Bu
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, China Email: sbu@math.tsinghua.edu.cn
Gang Cai
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China Email: caigang-aaaa@163.com
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Abstract

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In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

Keywords

C𝛼 -well-posednessdegenerate differential equationĊ𝛼 -Fourier multiplierHölder continuous function space

MSC classification

Primary: 34G10: Linear equations
Secondary: 47D06: One-parameter semigroups and linear evolution equations 47A10: Spectrum, resolvent 34K30: Equations in abstract spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 715 - 726
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was supported by the NSF of China (Grants No. 11571194, 11731010, 11771063), the Natural Science Foundation of Chongqing (cstc2017jcyjAX0006, cstc2016jcyjA0116), Science and Technology Project of Chongqing Education Committee (Grants No. KJ1703041, KJZDM201800501, KJ16003162016), the University Young Core Teacher Foundation of Chongqing (020603011714), Talent Project of Chongqing Normal University (Grant No. 02030307-00024). Gang Cai is corresponding author.

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