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Is this true?

A first-order quasilinear Cauchy problem with smooth data has a global smooth solution

No — it is false.

The counterexample

with u(x, 0) = −x

Characteristics all meet at y = 1: the solution blows up in gradient (a shock).

The kind of mistake this is

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this
PDE

Others that fail the same way

From Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics

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