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I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:

What is an example of a non-separable linear PDE?

Preferably a simple homogeneous one.

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Maybe a non-homogeneous modification of a homogeneous equation suffices? – J. M. Nov 19 '10 at 12:07
Yes, perhaps, but are there homogeneous ones? – Jonas Teuwen Nov 19 '10 at 12:22
Most variable coefficient PDEs cannot be separated. Separability is very closely tied to symmetries of the coefficients, so as long as you cannot choose a coordinate system in which the coefficients are independent of one (or several) of the variables, you cannot make it separable. – Willie Wong Nov 19 '10 at 16:15
On the other hand, to use a $C_0$ semigroup to solve an evolutionary PDE presupposes time independence of the coefficients. So at least the time-variable is separable. If you want to break that you'd have to use a $C_0$ semi-groupoid instead... – Willie Wong Nov 19 '10 at 16:17
up vote 3 down vote accepted

See my comments above on some general ideas. For specific examples, perhaps a good starting point would be L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," Phys. Rev. 74, 87-89 (1948). So just take an arbitrary potential function $V(x)$ that is not in Eisenhart's list, and consider the Schroedinger equation

$$ [-i \partial_t + \triangle + V(x)]\psi(t,x) = 0$$

This is Linear, homogeneous, and aside from the trivial separability of the $t$ variable, satisfies your requirements.

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Yes, a nice one. – Jonas Teuwen Nov 19 '10 at 19:43

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