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I was trying to classify the following PDE:

$$\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}}+\frac{\partial^2{u}}{\partial{y^2}}$$

where $u = u(x,y,t)$. I was originally using the definition of $B^2-4AC$ and found this equation to be elliptic, which is true for the Laplace equation however I was wondering if the dependence on time changes this. I was also wondering if this PDE is inhomogeneous and linear? Thank you!

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  • $\begingroup$ Might Mathematics be better suited for this maths question? $\endgroup$ – Kyle Kanos Feb 24 at 2:44
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homogenous, linear and parabolic.

In a generalization of the 2-dimensional equation, any equation of the form $$ \partial_t y = -L u $$ where $L$ is positive elliptic (such as $-\nabla^2$) is said to be parabolic. It shares with the 2d case the fact that it has well defined solutions with inital value data an a line with $t=constant$.

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  • $\begingroup$ thank you! could you please explain why it is parabolic? $\endgroup$ – Shaun Feb 23 at 18:27
  • $\begingroup$ Added a note to my rather short (because of lack of time) original answer. $\endgroup$ – mike stone Feb 23 at 21:55

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