Existence of strong solutions to singular nonlinear evolution equations. (English) Zbl 0592.34042

Freedman, M. A.

Pac. J. Math. 120, 331-344 (1985).

The author is concerned with the following problem: Suppose that the abstract Cauchy problem \(du/dt+A(t)u(t)\ni 0\), \(s<t<T\), \(u(s)=x\) has a strong solution for \(0<s\). Will there also exist a solution for \(s=0 ?\) Sufficient conditions are given for an affirmative answer that still allows A(t) to be singular at \(t=0\). The conditions require A(t) to be m- accretive for each fixed t and regular in t (in a certain sense) as t varies. Examples are given to show that no solution exists for \(s=0\) if the conditions are not met. The conditions are illustrated by some examples of partial differential equations.

Reviewer: G.F.Webb

MSC:

34G20

Nonlinear differential equations in abstract spaces

Keywords:

nonlinear accretive operator; strong solution; first order; differential equation; abstract Cauchy problem; examples of partial differential equations