Let us consider the heat equation $\partial_t u(t,x)=\Delta u(t,x)$ on (e.g.) $\mathbb{R}$ with initial condition $u(x,0)=u_0(x)$.
Denoting with $\Gamma(t,x)$ the heat Kernel $\Gamma(t,x)=(4\pi t)^{-1/2}e^{-\frac{|x|^2}{4t}}$ is the fundamental solution and thus $$u(t,x)=\Gamma_t*u_0(x)=\int_{\mathbb{R}}(4\pi t)^{-1/2}e^{-\frac{|x-y|^2}{4t}}u_0(y)dy.$$ I want to investigate the spatial decay (e.g. the beahaviour of the tails) of $u$ at a fixed finite time $T$.
It is knwon (see e.g. the survey from Vazquez, Asymptotic behaviour methods for the Heat Equation. Convergence to the Gaussian) that if $u_0$ is compactly supported probability density then $$\lim_{x\to \infty}\frac{\log u(x,T)}{|x|^2}=-\frac{1}{4T}$$ (proposition 4.2) that is the tails have Gaussian decay.
Moreover even if $u_0$ has Gaussian behaviour than so will $u(T,x)$ since the convolution of two Gaussian is a Gaussian itself.
When the initial data has fatter tails like exponential (again from Vazquez) we observe the same behaviour for the solution, i.e. $$\lim_{x\to \infty}e^{x}u(T,x)=e^T$$ My questions are the following:
- How do tails behave knowind the tails of a general initial data? Is it true that the tails'behaviour at time T is the behaviour of the tails of the slowest decaying function between $\Gamma $ and $u_0$, (under some hypotheses on the minimum speed of decay of $u_0$, I think).
1.1) What happens for example if $u_0$ exibit continuous Poisson-like decay? (that is $u_0\sim e^{-|x|log|x|}$).
1.2) Looking at the dual problem of the previous example how does behave $\Gamma*\phi$ where $\phi$ is like $\phi(x)=e^{|x|log|x|}$
1.3) Another way to look the problem is: if $X$ is a normal random variable and $Y$ is (in this case) Poisson random variable. How does behave in the tails the density of $X+Y$. How does the density of $X+Y$ behaves for a general absolutely continuous random variable $Y$?
- Most importantly: Which are the good references for this problem?