Introduction to Partial Differential Equations
Exercise Sheet 11
Ross Ogilvie 11th November,
2024
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Consider the fundamental solution of the heat equation given in Definition 4.5.
We want to show that is a distribution. Clearly it is linear. Fix a set and let .
(1 point)
Hence is a continuous linear functional. (2 points)
Finally, we want to show that (in the sense of distributions) .
as , uniformly in . (1 point)
as . (3 points)
(2 points)
Together these integrals show that
for all test functions . Therefore as claimed.
First a corollary to Theorem 4.7:
(2 points)
The above corollary shows how solutions to the heat equation on with such initial conditions behave: they tend to zero as . Physically this is because if then there is a finite amount of total heat, which over time becomes evenly spread across the space.
On open and bounded domains we can have different behaviour, due to the boundary conditions holding the temperature steady. In this question we determine the long time behaviour of solutions to the heat equation on open and bounded sets with on and .We claim as .
Hint. Consider .