Does this PDE admit steady state solutions?

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SUMMARY

The discussion centers on the existence of steady state solutions for a linear partial differential equation (PDE) represented as L f(x,y,t) = 0, where L is a second-order linear operator dependent on spatial variables x, y, and their partial derivatives, but not on time derivatives. The key conclusion is that by setting all time derivative terms to zero, one can analyze the equation for time-independent solutions f(x,y). This approach is essential for determining the conditions under which steady state solutions exist.

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mbp
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Hello to everyone. I am new with this forum and I am asking help with PDE.

I have a linear PDE:
L f(x,y,t) = 0
where L is a second order linear operator depending on x, y, their partial derivatives, and t, but not on derivatives with respect to t. The question is: does this PDE in general admit steady state (time independent) solutions f(x,y)?

Thanks in advance.
 
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Set all derivative terms w.r.t. t equal to zero and see if you can solve it.
 

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