Long time behaviour of the integral of the solution to the heat equation

I am interested at longtime behaviour of the solution to the one space dimension heat equation. That is, the solution to the equation $$u_{t} = \frac{1}{2}u_{xx},$$with initial condition $u(0,x)$ which is not necessarly integrable. By the solution I mean the unique one under the suitable growth condition. This solution can be represented by:

$$u(t,x) = \int_{\mathbb{R}}p_{t}(x-y)u(0,y)\mathrm{d}y,$$where $p_{t}(\cdot)$ is the heat kernel.

Let $B \subset \mathbb{R}$ be a compact set. Can we say anything about the following limt:

$$\lim_{t \to \infty}\int_{B}u(t,x)\mathrm{d}x.$$

Does it converge? To what value? Do we know anything about the convergence of its positve part os opposed to its negative part?

Thanks!