Verifying inhomogeneous solution of heat equation.

Question

Here is my question. How (3.21) is induced? I tried several times but I could not get the answer.

Answer 1

A straightforward change of variables shows \begin{align*}w(x,\varepsilon):&=\int_{\mathbb R^n}K(y,\varepsilon)F(x-y,t-\varepsilon)\ dy\\ &=\int_{\mathbb R^n}K(x-z,\varepsilon)F(z,t-\varepsilon)\ dz\\ &=u(x,t;t-\varepsilon) \end{align*} which we know solves (3.16) with $s=t-\varepsilon$. Since all functions involved are continuous, letting $\varepsilon\to0$ and looking at the initial condition verifies the result.