I have a question concerning the heat equation:
Formulate the homogeneous Cauchy problem for the heat equation on $\mathbb{R^n}$, and give uniqueness and existence results, including a solution formula, under a boundedness condition. Assume now that the initial function $\phi$ is real-valued, nonnegative, compactly supported and not identically $0$. Show that the solution $f(t,x)$ satisfies
- $f(t,x)>0$ for all $(t,x)\in(0,\infty)\times\mathbb{R^n}$,
- $\lim_{|x|\rightarrow\infty}f(t,x)=0$ for any fixed $t>0$,
- $\lim_{t\rightarrow\infty}f(t,x)=0$ for any fixed $x\in \mathbb{R^n}$.
I have no clue what to do, I do not even know how to formulate since I have never seen it before. I know the form of the solutions of the heat equation without any condition. Furthermore I treated the Dirichlet problem and I know Green's formulas and functions, I have seen the fundamental solutions of $-\Delta$, but I do not know If this helps here. I think I should express the heat equation with a boundedness condition as for the Dirichlet problem. Can somebody tell me how to formulate this problem and how to proceed.
Any help would be very much appreciated. Thank you in advance!
