= 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 $c\ge0$, the function
$$
u_c(x)=\begin{cases}0,&x\le c,\\{}[3(x-c)/4]^{4/3},&x\ge c\end{cases}
$$
is 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 \b[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>.
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