Followed by the topic where we discuss a pointwise estimate of solution to a class of heat equations. We are now interested in the case of wave equations.
Consider the wave equation in
together with the following initial data
We assume . We will prove the following estimate
for some constant depending only on the given data.
Proof. The assumption on implies the existence of two positive constants and such that
From the Poisson formula (but is known as Kirchhoff formula) in 3D, the solution to the problem is given as the following
It is easy to verify that: the area of the intersection is less than or equals to the area of . Therefore, we obtain