- #1

Shackleford

- 1,656

- 2

and from $$ \Delta_x \quad \text{to} \quad \Delta_y$$.

\begin{gather*}

\begin{split}

u_t(x,t) - \Delta u(x,t) & = \int_{0}^{t} \int_{\mathbb{R}^n}^{} \Phi(y,s) [(\frac{\partial}{\partial t}-\Delta_x)f(x-y,t-s)] \; dyds \\

& + \int_{\mathbb{R}^n}^{} \Phi(y,t) f(x-y,0) \; dy \\

& = \int_{\varepsilon}^{t} \int_{\mathbb{R}^n}^{} \Phi(y,s) [(-\frac{\partial}{\partial s}-\Delta_y)f(x-y,t-s)] \; dyds \\

& + \int_{0}^{\varepsilon} \int_{\mathbb{R}^n}^{} \Phi(y,s) [(-\frac{\partial}{\partial s}-\Delta_y)f(x-y,t-s)] \; dyds \\

& + \int_{\mathbb{R}^n}^{} \Phi(y,t) f(x-y,0) \; dy

\end{split}

\end{gather*}