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- Initial value problem; Nonuniqueness problem; Differential equations; Scalar case
We consider an initial value problem (1.1) [see pdf for notation] where [see pdf for notation]. Several uniqueness results weaker than a Lipschitz condition are known, see [1,4]. However, results concerning with nonuniqueness criteria are rare. For the case n = 1, a nonuniqueness result was given in [5,6], see also . Very recently the general case was also investigated in . In this paper, we consider nonuniqueness problem from a very general point of view. Our first nonuniqueness result deals with the scalar case which extends the results of [5,6]. It also shows that when the conditions of general uniqueness theorem are violated there results nonuniqueness. We then investigate the general case which demands somewhat different methods, since the techniques employed in the scalar case are not extendable to cover the general situation. Furthermore, our results deal with the case when f is singular at t = 0. That is, f(t,x) blows up in some sense as t -> 0+ and f(0,x) is not defined, see .