Let $u(x,t)$ satisfy $$u_{t}=u_{xx}, x \in\mathbb{R}, t >0$$ $$u(x,0)= \begin{cases} 1, \ \ x \in [0,1] \\ 0,\ \ otherwise \end{cases}$$ Then what is the value of $ \lim_{t \to 0^{+}} u(1,t)$ ?
I tried to use variable separable solution method but it didn't help me. Can anyone suggest something?
I think this is most easily used with the heat kernel method: $$ u(y,t) = \int \frac{e^{-\frac{(x-y)^2}{4t}}}{\sqrt{4\pi t}} u(x,0)dx $$ now $\lim\limits_{t\to0^+}\frac{e^{-\frac{(x-y)^2}{4t}}}{\sqrt{4\pi t}} = \delta(x-y)$ which unsurprisingly will yield $\lim\limits_{t\to0^+} u(y,t) = u(y,0)$.