hi
Is it possible to prove that the heat equation H_t=H_xx -inf < x < inf, 0<t<T is positive on all of its domain if the terminal condition H(x,T) =(I(e^-x))''> 0 ?
Also given is that I(x) < z + e^-z for some z.
We have been given a report where they said that this result is a result of the comparison principle but i can't find any form of the comparison principle for this type of domain (most of the results i have read is when x is bounded).
The question is from the bottom of page 12 in http://www.ma.utexas.edu/users/zariphop/pdfs/TZ-59.pdf .
Help would be much appreciated.
Is it possible to prove that the heat equation H_t=H_xx -inf < x < inf, 0<t<T is positive on all of its domain if the terminal condition H(x,T) =(I(e^-x))''> 0 ?
Also given is that I(x) < z + e^-z for some z.
We have been given a report where they said that this result is a result of the comparison principle but i can't find any form of the comparison principle for this type of domain (most of the results i have read is when x is bounded).
The question is from the bottom of page 12 in http://www.ma.utexas.edu/users/zariphop/pdfs/TZ-59.pdf .
Help would be much appreciated.