Evolution equations governed by quasilinear operators satisfying Carathéodory’s conditions
Volume 571 / 2022
Abstract
The aim of this paper is to study the abstract quasilinear evolution equation $u’(t)=A(t,u(t))u(t)$ under a strong measurability condition with respect to $t$ on a family $\{A(t,w)\}$ of linear operators instead of a strong continuity condition used in previous researches. Our strategy is that we introduce several types of solutions and approximate solutions to a ‘linearized’ problem of the nonautonomous form $u’(t)=A(t,v(t))u(t)$, where $v$ is a given solution to the original problem in some sense, and establish the local well-posedness of strong solutions under a commutator condition on $A(t,w)$ by using a theorem on convergence of approximate solutions of the linearized problem. We also give a criterion for the continuation of local strong solutions and use it to obtain a global well-posedness theorem, which applies to solving the Cauchy problem for an abstract inhomogeneous Kirchhoff equation with linear dissipation. Finally, we establish an analogue of the Neveu–Trotter–Kato approximation theorem for the abstract quasilinear evolution equation.
