 Solution

At zero the right side is continuous but fails every local Lipschitz condition in its second variable, since ${\delta }^{1/4}/\delta ={\delta }^{-3/4}\to \infty$. For every $c\ge 0$, the functionis continuously differentiable and satisfies the differential equation, including at the joining point where both one-sided derivatives are zero. Each has $u_c(0)=0$. The identically zero function is another possibility. Thus solutions exist but are not unique, even locally: choosing $c=0$ and taking the zero function already gives two different solutions on every neighborhood of zero. These are instances of waiting-time solutions of a sublinear Bernoulli equation.