= Solution
When the velocity depends on $u$, different initial values travel at different speeds. The resulting <characteristic crossing> can destroy a classical solution in finite time, even though the initial function is smooth and bounded. For the <Inviscid Burgers equation>, take $u_0(a)=-\tanh a$. Before crossing,
$$
X(t,a)=a-t\tanh a,\qquad u(t,X(t,a))=-\tanh a,
$$
and
$$
u_x(t,X(t,a))=\frac{-\operatorname{sech}^2a}{1-t\operatorname{sech}^2a}.
$$
At $a=0$ this tends to $-\infty$ as $t\uparrow1$, while $|u|\le1$. Thus $\boxed{\text{bounded states can undergo finite-time gradient blow-up and shock formation}.}$ The <bounded-derivative Burgers flux> agrees with this equation on the whole state range, if a global bounded-speed flux is desired. After crossing, arbitrary <weak solutions> need not be unique; an <entropy solution> condition is needed to select the admissible continuation.
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