Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“A first-order quasilinear Cauchy problem with smooth data has a global smooth solution” — false

Counterexample: u uₓ + u_y = 0 with u(x, 0) = −x

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

PDE

#2 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“A PDE has a single type throughout its domain” — false

Counterexample: The Tricomi equation y uₓₓ + u_yy = 0

Elliptic in the upper half-plane, hyperbolic in the lower, parabolic on the axis.

PDE

#3 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“All the classical equations satisfy a maximum principle” — false

Counterexample locked — unlock with Notes + PYQ

PDE