Global existence, uniqueness, and continuous dependence for a semilinear initial value problem
The author considers the semilinear initial value problem \[ u'(t)+ Au(t)= f(t, u(t)),\quad t\geq 0,\quad u(0)+ u_0,\tag{IVP} \] in a Banach space \(X\) in which \(f:[0,\infty)\times X\to X\) is continuous in \(t\) and satisfies a Lipschitz-type condition in \(x\), \(A\) is a closed linear operator whose spectrum lies in a certain sector of the complex plane and whose resolvent satisfies \[ \|(z- A)^{-1}\|\leq M|z|^\gamma\tag{1} \] for all \(z\) outside that sector, where \(\gamma\in (-1,0)\). (Compare condition (1) to the inequality \(\|(z- A)^{-1}\|\leq M|z|^{-1}\), which has been used elsewhere in the literature.) Along with some growth conditions, the author proves global existence and uniqueness of a mild solution to (IVP) and also demonstrates that this solution depends continuously on the initial value. Examples are given which consider the application of these ideas to parabolic partial differential equations, fractional differential equations and the Korteweg-de Vries operator. This article is based on the work of \textit{F. Periago} and \textit{B. Straub} [J. Evol. Equ. 2, No. 1, 41-68 (2002; Zbl 1005.47015)].
scientific article; zbMATH DE number 1917783
scientific article