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

with the initial condition

.

We assume is bounded, continuous, and satisfies

.

We will show that there exists a constant such that

for all and .

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

.

By the Cauchy inequality, we have

.

Set in the second integral of the above inequality. Then we get immediately

.

Thus

for some constant .

Note that the above pointwise estimate shows us that the function is decreasing with respect to time . 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.

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