Hyperbolic Equations

Abstract

The solutions of hyperbolic equations and inequalities do not exhibit the type of maximum principle that was studied in the preceding chapters. Even in the simplest case of the wave equation in two independent variables*

$$ {{u}_{{xx}}} - {{u}_{{tt}}} = 0, $$

((1))

it is easily seen that the maximum of a nonconstant solution u in a domain D may occur at an interior point. For example, we observe that the function

$$ u = \sin x\sin t $$

satisfies the above equation, and that it attains its maximum in the square 0 < x < π, 0 < t < π, at the center (π/2, π/2).