Math 220: Properties of Solutions of Second Order Pde

There are a few facts that can be read off from this expression immediately. We consider t > 0 here; t < 0 is similar. First, for t0 > 0, u(x0, t0) depends on the initial data φ just at the two points x0± ct0, while it depends on the values of ψ in the whole interval [x0 − ct0, x0 + ct0]. Thus, we call the interval [x0 − ct0, x0 + ct0] the domain of dependence of (x0, t0): if the initial conditions vanish there, the solution vanishes at (x0, t0). Note that the straight lines x − ct = x0 − ct0 and x+ ct = x0 + ct0 which go through (x0, t0) and (x0 ± ct0, 0) are characteristics. In fact, it is convenient (for reasons that will be more clear when we solve the inhomogeneous wave equation, u = f) to consider the domain of dependence of (x, t) to be the whole region D− x0,t0 = {(x, t) : t ≤ t0, |x− x0| ≤ c(t0 − t)}. This is the backward characteristic triangle from (x0, t0): its sides are the characteristics x− ct = x0 − ct0 and x+ ct = x0 + ct0. With this definition, if the initial data are imposed at t = T instead, where T < t0, then the solution u at (x0, t0) depends on the initial data in the interval [x0 − c(t0 − T ), x0 + c(t0 − T )], which is just the segment in which the line t = T intersects the backward characteristic triangle. Indeed, a change of variables (replacing t by τ = t − T ) shows that the solution of