Maximum principle for hyperbolic PDEs

Question

I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof regarding that. It would be really helpful if I get a help or resource regarding the same.

Answer 1

Fundamentally, the classical maximum principles for second order elliptic PDEs are based on the simple facts that

1. The Hessian of a function at a local maximum is positive semidefinite.
2. The full contraction (Hilbert-Schmidt product) between two (symmetric) positive semidefinite matrices is non-negative.

In point 2, one of the PSD matrices is the Hessian, and the other is the leading order coefficients of your second order PDE.

For hyperbolic PDEs you cannot invoke point 2, as the symbol of the equation is required to be indefinite. This is what people usually mean when they say that there is no maximum principle for hyperbolic PDEs: that the basic heuristic scheme can't even be started. This certainly does not rule out individual equations/solutions exhibiting something that looks a bit like a maximum principle.

(For an example of a fake maximum principle: the kernel for the linear wave equation in dimension 1, 2, and 3 are signed. This means that if you solve $\Box u = f$ and $\Box v = g$, both with vanishing data and with $f \geq g$ pointwise, you will find the solutions satisfy also $u \geq v$ pointwise. This looks sort of like a maximum principle, but is true for completely different reasons.)