a symmetric hyperbolic system
Some of the deepest well-posedness results in PDE come not from clever formulas but from a single clean inequality: an energy estimate. A symmetric hyperbolic system is a form into which a great many physical systems can be cast precisely so that this energy method works, guaranteeing that the linear initial-value problem has one and only one solution that depends continuously on the data.
Friedrichs' definition: the first-order system A_0 U_t + sum_j A_j U_{x_j} = B U + F is symmetric hyperbolic if the coefficient matrices A_0, A_1, ..., A_n are all real symmetric and A_0 is positive definite. The magic is then a one-line computation. Multiply the equation by U on the left (an inner product) and integrate over space; symmetry of the A_j makes the spatial terms integrate to a boundary term, and one is left with d/dt of (1/2) integral U^T A_0 U bounded by a constant times the same energy. Gronwall's inequality then bounds the energy E(t) by E(0) times e^(Ct). That bound is everything: it forbids two different solutions from the same data, it forbids blow-up of the energy in finite time, and it gives continuous dependence — the three pillars of well-posedness.
Why it matters: a huge range of physical laws — Maxwell's equations, linear elasticity, the linearised Euler and shallow-water equations, and (after symmetrising) many more — are symmetric hyperbolic or can be symmetrised by multiplying by a suitable positive matrix. This is why these theories are deterministic and stable in the right norm. The caveat is that the clean estimate is a LINEAR result: for nonlinear systems it gives only local-in-time existence, because the constants in the energy estimate themselves depend on the solution and can blow up as a shock forms.
Maxwell's equations in vacuum, written for the fields (E, B), take the form U_t + sum A_j U_{x_j} = 0 with constant real symmetric matrices A_j and A_0 = identity. The energy (1/2) integral (|E|^2 + |B|^2) dx is conserved — exactly the symmetric-hyperbolic energy. That conserved energy is why the Cauchy problem for light in vacuum is perfectly well-posed.
Symmetry of the coefficient matrices yields a conserved (or bounded) energy, hence well-posedness.
Symmetric hyperbolic is a sufficient, very convenient structure, not a necessary one — a system can be hyperbolic and well-posed without being symmetric, but symmetrisability is the cleanest route to the energy estimate. Strict hyperbolicity and symmetrisability are different notions; many physical systems are symmetrisable but not strictly hyperbolic.