Math 6410 • ODE

Wave Equation. Let \(f(x,t)\in C^\infty (\mathbb {R}^3\times \mathbb {R})\) and \(R>0\) be fixed. Suppose that the support of \(f\) is contained in the cylinder \[ \operatorname {spt} f\subset \{(x,t)\in \mathbb {R}^3\times \mathbb {R}:\lvert x\rvert \le R\text { and }t\ge 0\}. \] Show that there exists a solution \(u(x,t)\) to the inhomogeneous wave equation \((\spadesuit )\). Show that the solution satisfies \(u(x,t)=0\) whenever \(\lvert x\rvert \ge ct+R\). \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}=c^2\Delta u+f(x,t), & \text {if }x\in \mathbb {R}^3\text { and }t\ge 0,\\ u(x,0)=0,\\ \dfrac {\partial u}{\partial t}(x,0)=0, & \text {for }x\in \mathbb {R}^3. \end {cases} \qquad (\spadesuit ) \]