On the existence of strong solutions to autonomous differential equations with minimal regularity

Abstract

For proving the existence and uniqueness of strong solutions to (1) d Y d t = F ( Y ) , Y ( 0 ) = C , the most quoted condition seen in elementary differential equations texts is that F ( Y ) and its first derivative be continuous. One wonders about the existence of a minimal regularity condition which allows unique strong solutions. In this note, a bizarre example is seen where F ( Y ) is not differentiable at an equilibrium solution; yet unique non-global strong solutions exist at each point, whereas global non-unique weak solutions are allowed. A characterizing theorem is obtained.