partial-differential-equations-by-evans · Problem 18

Nonlinear First-Order PDE

Mathematics — Calculus · wave-equation-decay-estimate

Problem

Let

u

solve

\begin{cases} u_{tt} - \Delta u = 0 & \text{in } \mathbb{R}^3 \times (0,\infty) \\ u = g,\, u_t = h & \text{on } \mathbb{R}^3 \times \{t=0\}, \end{cases}

where

g, h

are smooth and have compact support. Show there exists a constant

C

such that

|u(x,t)| \leq C/t \quad (x \in \mathbb{R}^3,\, t > 0).