# Does multiplication commute with taking of fundamental solution (heat equation)

Let $\Phi(t,x)$ be a heat function, $$\Phi(t,x) = \frac{1}{\sqrt{4 \pi t}} \exp\left(-\frac{x^2}{4t}\right).$$ Then $(\partial_{t} - \partial_{xx})\Phi(t,x) = \delta(t)\delta(x)$. Furthermore, $$(\partial_{t} - \partial_{xx} - \partial_{yy})[\Phi(t,x)\Phi(t,y)] = \delta(t)\delta(x)\delta(y).$$ In other words, fundamental solution of heat equation on product of two lines is equal to product of fundamental solutions of heat equations on the line (taking fundamental solution and taking product here commute in some sense). Let $\Psi(t,z,z_{0})$ be a solution of system where $z \in [0,L]$ $$(\partial_{t}-\partial_{zz})\Psi = \delta(t)\delta(z-z_0), \\ \Psi(0,z,z_0) = 0, \\ + \text{ homogenous boundary conditions of mixed type}$$ Is it true that $$(\partial_{t} - \partial_{xx} -\partial_{zz})[\Phi(t,x)\Psi(t,z,z_0)] = \delta(t)\delta(x)\delta(z-z_0)$$ In other words, is it true that product of fundamental solutions of heat equation on the line and on the segment is a fundamental solution of heat equation on line $\times$ segment?

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