= Waiting-time solutions of a sublinear Bernoulli equation
{title2=$y'=ay+q(x)y^p,\quad0<p<1$}
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 $x>x_0$, define
$$
y(x)=\left[(1-p)e^{(1-p)ax}\int_{x_0}^xq(s)e^{-(1-p)as}\,ds\right]^{1/(1-p)},
$$
and 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.
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