I have a question about a property of the solutions to the heat equation. Let $u(t,x)$ be a solution to the (one-space dimension) heat equation $$u_t = u_{xx}.$$ Is it true that $\int_{\mathbb{R}}u(t,x) \mathrm{d}x$ is constant with respect to $t$? That is, I want to verify (and find a reference) for the following equation:
$$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}}u(t,x) \mathrm{d}x = 0.$$ Where can I find a reference for this fact? In addition, do we have to assume $u(0,\cdot) \in L^1(\mathbb{R})$ for the above to hold? Or is it true even if the initial condition is not integrable? Many thanks!
Edit: I understand why conservation of mass holds under some conditions specified in an answer below. I still looking for a reference for this statement (I only found it in some private lecture notes, but not in any books or papers). I also want to know whether it is true even if the initial condition satisfies $\int_{\mathbb{R}}u(0,x)\mathrm{d}x = \infty$.