Let $$ u_{t}(x,t) = \Delta u(x,t) \space \space \space \text{ for } \space t \ge 0 \space \space\space\space , \space x \in \mathbb{R}^{n} $$
and $$ u \rightarrow 0 \space \text{ as } \space ||x|| \rightarrow \infty $$
Show $$I(t) = \int_{\mathbb{R}^{n}} {u(x,t) } dx = constant $$
What i have attempted so far is to apply Lipschitz and the Divergence Theorem on a finite sphere and then try and show the limit comes out as zero. This works up until the limit part, then i get a bit stuck. Maybe i'm missing something obvious.
thanks for any hints/advice you can provide!
EDIT: very grateful for the solutions you provided
For given initial conditions $u(x,0)$, you can write the solution as a convolution against the fundamental solution as
$$ u(x,t) = \int_{\mathbb{R}^n}\frac{1}{(4\pi t)^{n/2}}e^{-||x-x'||^2/4t}u(x',0)dx' $$
You can verify that such a representation matches your boundary condition at infinity. Then integrating over x:
$$ \int_{\mathbb{R}^n} u(x,t)dx = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{1}{(4\pi t)^{n/2}}e^{-||x-x'||^2/4t}u(x',0)dx'dx $$
If the initial condition is well behaved enough, we can switch the order of integrations: $$ \int_{\mathbb{R}^n} u(x,t)dx = \int_{\mathbb{R}^n}u(x',0)\left(\int_{\mathbb{R}^n}\frac{1}{(4\pi t)^{n/2}}e^{-||x-x'||^2/4t}dx\right) dx' $$ The inner integral is unity, leaving $$ \int_{\mathbb{R}^n} u(x,t)dx = \int_{\mathbb{R}^n}u(x',0)dx' $$ The heat equation "smears out" the initial condition in time, but conserves it's total integral as time goes on. All that's left is to decide what is regular enough to justify switching the integration order.