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I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{[-1,1]} 1_{x \in [-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.

In particular I am interested in obtaining non-trivial bounds on $u(t,0)$. Of course the heat kernel gives a trivial bound on $u(t,0)$ but I am struggling to obtain anything stronger.

Perhaps the equation has a closed form solution from which it is easy to read such information off?

Added later: Of course appropriate growth conditions at infinity are assumed to ensure a unique solution.

Correction: The indicator function is a function of the $x$ variable only.

2 added 113 characters in body

I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{[-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.

In particular I am interested in obtaining non-trivial bounds on $u(t,0)$. Of course the heat kernel gives a trivial bound on $u(t,0)$ but I am struggling to obtain anything stronger.

Perhaps the equation has a closed form solution from which it is easy to read such information off?

Added later: Of course appropriate growth conditions at infinity are assumed to ensure a unique solution.

1

# Heat equation bounds

I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{[-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.

In particular I am interested in obtaining non-trivial bounds on $u(t,0)$. Of course the heat kernel gives a trivial bound on $u(t,0)$ but I am struggling to obtain anything stronger.

Perhaps the equation has a closed form solution from which it is easy to read such information off?