Counterexample

Is this true?

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

No — it is false.

The counterexample

$u u_{x} + u_y = 0$ with u(x, 0) = −x

Why it fails

Characteristics $x = x_{0}(1 - y)$ all meet at y = 1: the solution blows up in gradient (a shock).

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Keep practicing

Existence vs uniqueness

A theorem giving one was read as giving both.

See real exam options built on this mistake Drill statements like this

Go deeper· Explained in full in First-order PDE: characteristics, Lagrange and Charpit· Asked in CSIR NET June 2025· From Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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