Semigroup Forum ( IF 0.7 ) Pub Date : 2021-03-02 , DOI: 10.1007/s00233-021-10165-2 Vu Trong Luong , Nguyen Van Minh
In this paper we present a simple spectral theory of polynomially bounded functions on the half line, and then apply it to study the asymptotic behavior of solutions of fractional differential equations of the form \(D^{\alpha }_Cu(t)=Au(t)+f(t)\), where \(D^{\alpha }_Cu(t)\) is the derivative of the function u in Caputo’s sense, A is generally an unbounded closed operator, f is polynomially bounded. Our main result claims that if u is a mild solution of the Cauchy problem such that \(\lim _{h\downarrow 0} \sup _{t\ge 0} \Vert u(t+h)-u(t)\Vert /(1+t)^n=0\), and \(\sup _{t\ge 0} \Vert u(t)\Vert /(1+t)^n <\infty \), then, \(\lim _{t\rightarrow \infty } u(t)/(1+t)^n =0\) provided that the spectral set \(\Sigma (A,\alpha )\cap i{\mathbb {R}}\) is countable, where \(\Sigma (A,\alpha )={\mathbb {C}}\backslash \rho (A,\alpha ), \rho (A,\alpha )\) is defined to be the set of complex numbers \(\xi \) such that \(\lambda ^{\alpha -1} (\lambda ^\alpha -A)^{-1}\) is analytic in a neighborhood of \(\xi \), and u satisfies some ergodic conditions with zero means. The obtained result extends known results on strong stability of solutions to fractional equations.
中文翻译:
多项式有界解的简单谱理论及其在微分方程中的应用
在本文中,我们提出了半线上的多项式有界函数的简单谱理论,然后将其用于研究形式为\(D ^ {\ alpha} _Cu(t)= Au的分数阶微分方程解的渐近行为。(t)+ f(t)\),其中\(D ^ {\ alpha __Cu(t)\)是卡普托意义上的函数u的导数,A通常是无界的封闭算子,f是多项式有界的。我们的主要结果认为,如果u是柯西问题的温和解,则\(\ lim _ {h \ downarrow 0} \ sup _ {t \ ge 0} \ Vert u(t + h)-u(t) \ Vert /(1 + t)^ n = 0 \)和\(\ sup _ {t \ ge 0} \ Vert u(t)\ Vert /(1 + t)^ n <\ infty \),然后,\(\ lim _ {t \ rightarrow \ infty} u(t)/(1 + t)^ n = 0 \)假设频谱集\(\ Sigma(A,\ alpha)\ cap i {\ mathbb { R}} \)是可数,其中\(\西格玛(A,\阿尔法)= {\ mathbb {C}} \反斜杠\ RHO(A,\阿尔法),\ RHO(A,\阿尔法)\)被定义成为复数\(\ xi \)的集合,使得\(\ lambda ^ {\ alpha -1}(\ lambda ^ \ alpha -A)^ {-1} \)在\( \ xi \),并且u以零均值满足某些遍历条件。所得结果扩展了分数阶方程解的强稳定性的已知结果。