14 1. Simple Examples of Propagation The Cauchy data, V (0,x) are continuous, piecewise smooth, and singular only at x = 0. Theorem 1.1.4 shows that V is piecewise smooth with singularities only on the characteristics through (0, 0). In addition u = ∂xv 3 (in the sense of distributions) since they both satisfy the same initial value problem. Thus v and u = ∂3v x have singular support only on the characteristics through (0, 0). Interesting things happen if one adds a lower order term. For example, consider the Klein–Gordon equation (1.2.6) utt − uxx + u = 0 . In sharp contrast with equation (1.2.2), there are hardly any undistorted progressing wave solutions. Exercise 1.2.2. Find all solutions of (1.2.6) of the form f(x − ct) and all solutions of the form ei(τt−xξ). Discussion. The solutions ei(τt−xξ) with ξ ∈ R are particularly important since the general solution is a Fourier superposition of these special plane waves. The equation τ = τ(ξ) defining such solutions is called the dispersion relation of (3.1.6). There is an energy conservation law. Denote by S(Rd) the Schwartz space of rapidly decreasing smooth functions. That is, functions such that for all α, β, sup x∈Rd xβ∂αψ(x) x ∞ . Exercise 1.2.3. Prove that if u ∈ C∞(R : S(R)) is a real valued solution of the Klein–Gordon equation, then u2 t + u2 x + u2 dx is independent of t. This quantity is called the energy. Hint. Justify carefully differentiation under the integral sign and integration by parts. If you find weaker hypotheses which suﬃce, that is good. The fundamental solution of the Klein–Gordon equation with initial data (1.2.4), is not as simple as the fundamental solutions of the wave equation. Theorem 1.1.4 implies that the singular support lies on {x = ±t}. The proof is as for the wave equation except that the zeroth order term in the equation for V is replaced by ⎛ ⎝0 1 0 −1 1 −1⎠ 0 0 −1 ⎞ V .

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