Parabolic PDE whose solution is a product of lower dimensional solutions?

Question

Consider the partial differential equation:

$$ \bigg(\sum_{i=1}^n s_i\bigg) \Delta \Phi=nu \Phi_u $$

for $\Phi(u,s_1,s_2,\cdot\cdot\cdot,s_n).$

This is a linear parabolic partial differential equation reminiscent of the heat equation on $\Bbb R^n.$

I would like a solution that is a product of one dimensional solutions: $ \Phi(u,s_1,s_2,\cdot\cdot\cdot,s_n)=\Phi(u,s_1)\cdot \Phi(u,s_2)\cdot\cdot\cdot \Phi(u,s_n).$

Is this possible?

I'm not dealing with any boundary conditions.

Answer 1

There is a possibility. Let's try the ansatz $$ \Phi(u,s_1,s_2,\cdot\cdot\cdot,s_n)=e^{\phi(u)\sum_{i=1}^ns_i} =\prod_{i=1}^ne^{\phi(u)s_i}. \tag{1} $$ Substituting $(1)$ in the PDE, we obtain an ODE for $\phi(u)$: $$ \bigg(\sum_{i=1}^n s_i\bigg) n\phi^2(u) e^{\phi(u)\sum_{i=1}^ns_i} =nu \phi'(u)\bigg(\sum_{i=1}^n s_i\bigg)e^{\phi(u)\sum_{i=1}^ns_i} $$ $$ \Rightarrow u\phi'(u)=\phi^2(u). \tag{2} $$ The solution to $(2)$ is $$ \phi(u)=-\frac{1}{C+\ln|u|}, \tag{3} $$ where $C$ is an arbitrary constant. We conclude that yes, the PDE has a solution that is a product of one-dimensional solutions.

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P.S. - In addition to the solution presented above, there are some trivial solutions satisfying the condition $\Phi(u,s_1,\ldots,s_n)=\prod_{i=1}^n\Phi(u,s_i)$: (a) $\Phi=0$, (b) $\Phi=1$, and (c) if $n$ is odd, $\Phi=-1$.