Ngô Quốc Anh

February 26, 2010

Wave equations: The first pointwise estimates

Filed under: Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 19:15

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 \mathbb R^3

\displaystyle u_{tt}-\Delta u =0, \quad x\in \mathbb R^3, t>0

together with the following initial data

\displaystyle u(x,0)=0, \quad u_t(x,0)=g(x).

We assume g \in C_0^\infty(\mathbb R^3). We will prove the following estimate

\displaystyle |u(x,t)| \leq \frac{C}{t}, \quad t>0

for some constant C depending only on the given data.

Proof. The assumption on g implies the existence of two positive constants R and M such that

\displaystyle {\rm supp}(g) \subset B_R = \{ x \in \mathbb R^3:|x|<R\}

and

\displaystyle |g(x)| \leq M, \quad \forall x \in \mathbb R^3.

From the Poisson formula (but is known as Kirchhoff formula) in 3D, the solution to the problem is given as the following

\displaystyle u(x,t)=\frac{1}{4\pi t} \int_{|y-x|=t} {g(y)dS_y}.

It is easy to verify that: the area of the intersection \{ y \in \mathbb R^3:|y-x|=t\} \cap B_R is less than or equals to the area of \partial B_R. Therefore, we obtain

\displaystyle |u(x,t)| \leq\frac{1}{4\pi t} M 4\pi R^2 = \frac{MR^2}{t}, \quad t>0.

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