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

Shangquan Bu¹

¹ Department of Mathematical Science University of Tsinghua Beijing 100084, China

Studia Mathematica, Tome 214 (2013) no. 1, pp. 1-16

Résumé

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 ($P_2$): $\frac {d}{dt}(Mu')(t) =Au(t)+f(t)$ $(0\leq t\leq 2\pi )$ with periodic boundary conditions $ u(0)=u(2\pi )$, $(Mu')(0)=(Mu')(2\pi )$, in Lebesgue–Bochner spaces $L^p({\mathbb T}, X)$, periodic Besov spaces $B_{p,q}^s({\mathbb T}, X)$ and periodic Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb T}, X)$, where $A$ and $M$ are closed operators in a Banach space $X$ satisfying $D(A)\subset D(M)$. Our results generalize the previous results of W. Arendt and S. Q. Bu when $M= I_X$.

Détail

DOI : 10.4064/sm214-1-1

Keywords: using known results operator valued fourier multipliers vector valued function spaces necessary sufficient conditions well posedness second order degenerate equations frac leq leq periodic boundary conditions lebesgue bochner spaces mathbb periodic besov spaces mathbb periodic triebel lizorkin spaces mathbb where closed operators banach space satisfying subset results generalize previous results arendt