Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals (Q6626999)

In this paper, the author establishes the well-posedness of the nonautonomous abstract Cauchy problem \N\[\dot{u}(t)=A(t)u(t), t\in [s,b), u(s)=s \] \Nwith \(a\leq s<b\) in a general Banach space \(X\) under some dissipativity conditions expressed by a metric-like functional.\NHis result extends R.H. Martin's result (Zbl 0249.47065) on the generation of an evolution operator in a class of uniformly convex Banach spaces and gives an affirmative answer to K\(\overline{o}\)mura's conjecture (Zbl.0163.38302) hat an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition. Moreover, this paper also provides two examples as applications, one is effectively solving degenerate wave equations, the other is well-posedness of nonautonomous evolution equations of Kirchhoff type.