Ngô Quốc Anh

February 26, 2010

Heat equations: The first pointwise estimates

Filed under: Các Bài Tập Nhỏ, PDEs — Ngô Quốc Anh @ 17:08

I am going to discuss a series of results involving pointwise estimates of heat and wave equations. Let us start with the following  simple problem

Consider the Cauchy problem for the heat operator in $\mathbb R$

$\displaystyle u_t = u_{xx} \quad x\in \mathbb R, t>0$

with the initial condition

$\displaystyle u(x,0)=f(x), \quad x \in \mathbb R$.

We assume $f$ is bounded, continuous, and satisfies

$\displaystyle\int_{ - \infty }^{ + \infty } {{{\left| {f(x)} \right|}^2}dx} < \infty$.

We will show that there exists a constant $C$ such that

$\displaystyle \left| {u(x,t)} \right| \leqslant \frac{C}{{{t^{\frac{1}{4}}}}}$

for all $x \in \mathbb R$ and $t>0$.

Proof. The solution to the problem is given by the Poisson formula

$\displaystyle u(x,t) = \frac{1}{{\sqrt {2\pi t} }}\int_{ - \infty }^{ + \infty } {f(y){e^{ - \frac{{{{(x - y)}^2}}}{{4t}}}}dy}$.

By the Cauchy inequality, we have

$\displaystyle\left| {u(x,t)} \right| \leqslant \frac{1}{{\sqrt {2\pi t} }}{\left( {\int_{ - \infty }^{ + \infty } {{{\left| {f(y)} \right|}^2}dy} } \right)^{\frac{1}{2}}}{\left( {\int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(x - y)}^2}}}{{2t}}}}dy} } \right)^{\frac{1}{2}}}$.

Set $y=x+\sqrt{2t}\eta$ in the second integral of the above inequality. Then we get immediately

$\displaystyle\int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(x - y)}^2}}}{{2t}}}}dy} = \int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(\sqrt {2t} \eta )}^2}}}{{2t}}}}\sqrt {2t} d\eta } = \sqrt {2t} \int_{ - \infty }^{ + \infty } {{e^{ - {\eta ^2}}}d\eta }$.

Thus

$\displaystyle\left| {u(x,t)} \right| \leqslant \frac{1}{{\sqrt {2\pi } {t^{\frac{1}{2}}}}}{\left( {\int_{ - \infty }^{ + \infty } {{{\left| {f(y)} \right|}^2}dy} } \right)^{\frac{1}{2}}}{\left( {\int_{ - \infty }^{ + \infty } {{e^{ - {\eta ^2}}}d\eta } } \right)^{\frac{1}{2}}}{\left( {2t} \right)^{\frac{1}{4}}} = \frac{C}{{{t^{\frac{1}{4}}}}}$

for some constant $C$.

Note that the above pointwise estimate shows us that the function $u$ is decreasing with respect to time $t$. This is reasonable because, for example, the heat equation predicts that if a hot body is placed in a box of cold water, the temperature of the body will decrease, and eventually (after infinite time, and subject to no external heat sources) the temperature in the box will equalize.