This is maybe a trivial question but i need some clarification to make it clearer in my mind. Consider the fundamental solution of the equation $ \partial_{t} u - \partial^{2}_{xx}u=0$ given by the so called green function $H(t,x)=(2 \pi t)^{\frac{-d}{2}} e^{\frac{- \vert x \vert ^{2}}{2t}}$ for $t>0$ and $ x \in R $ , $H(t,x)=0 $ if $t < 0, x \in R $.

The integral: $ \int_{0}^{t} \int_{R} H(t-s,x-y) dsdy$ can be well defined ?

For me H is not L1(R^{2}) so i don't see how this integral can be defined.

Maybe it's really trivial question but it confuses me. Thank you for any element of answer.

New contributor
nour is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • the integral over all $x$ is a Gaussian integral which gives unity. – Carlo Beenakker Dec 6 at 23:06
  • so we can consider the L1 norm on R^{2} of H ? – nour Dec 7 at 1:33
  • you are looking for a solution to the diffusion equation, which is given by the Green function; this solution is integrable, and by particle number conservation is equal to its value at $t=0$, hence it integrates to unity. – Carlo Beenakker Dec 7 at 9:50

Your Answer

nour is a new contributor. Be nice, and check out our Code of Conduct.

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.