91Ó°ÊÓ

Use Duhamel's principle to find the solution of the nonhomogeneous wave equation for three space dimensions \(u_{t t}-c^{2} \Delta u=f(x, t)\) with initial conditions \(u(x, 0)=0=u_{t}(x, 0)\). What regularity in \(f(x, t)\) is required for the solution \(u\) to be \(C^{2}\) ?

The partial differential equation \(u_{t t}=c^{2} \Delta u-q(x) u\) arises in the study of wave propagation in a nonhomogeneous elastic medium: \(q(x)\) is nonnegative and proportional to the coefficient of elasticity at \(x\). (a) Define an appropriate notion of energy for solutions. (b) Verify the corresponding energy inequality. (c) Use the energy method to prove that solutions are uniquely determined by their Cauchy data.

Solve the initial/boundary value problem $$ \left\\{\begin{aligned} u_{t t}-u_{x x} &=1 & \text { for } & 00 \\ u(x, 0) &=0, & u_{l}(x, 0)=0 & \text { for } & 0