Waiting-time solutions of a sublinear Bernoulli equation ($y^{\prime }=ay+q(x)y^p,0<p<1$)

Let q be continuous and strictly positive, with a constant a. A nonnegative solution of a differential equation may stay zero until any prescribed point $x_0$ and become positive afterwards. For $x>x_0$, defineand set $y=0$ before $x_0$. The Bernoulli power substitution verifies the equation after $x_0$. Continuity of q gives $y=O((x-x_0{)}^{1/(1-p)})$, so the joining derivative is zero and the equation also holds at the join. These different waiting times give nonunique solutions from zero. The right side lacks Lipschitz continuity as a function of y near zero, so the uniqueness conclusion of the Picard-Lindelöf theorem does not apply there.