Parabolic and Hyperbolic Equations - Key Properties

Parabolic Properties: Maximum principles, smoothing, irreversibility, monotone energy decay, infinite speed but exponential decay with distance.

Hyperbolic Properties: Finite propagation speed, energy conservation, reversibility, domain of dependence/influence, characteristic surfaces.

Energy Methods: For parabolic $u_t = k\Delta u$ : $\frac{d}{dt}\int u^2 = -2k\int|\nabla u|^2 \leq 0$ (monotone decay). For hyperbolic $u_{tt} = c^2\Delta u$ : $E(t) = \frac{1}{2}\int(u_t^2 + c^2|\nabla u|^2)$ is conserved.

DefinitionRegularity and Smoothing

Parabolic: Solutions are $C^\infty$ for $t > 0$ regardless of initial data regularity (immediate smoothing).

Hyperbolic: Solutions preserve regularity of initial data but no smoothing. Singularities propagate along characteristics.

Propagation Properties: Parabolic disturbances spread instantaneously with Gaussian-like decay. Hyperbolic disturbances travel at finite speed $c$ , confined to light cones $|x - x_0| \leq c(t - t_0)$ .

Remark

Mixed-type equations (parabolic in some regions, hyperbolic in others) arise in transonic flow and phase transitions, combining features of both types with complex behavior near type-change boundaries.