Classification of 2D time dependent diffusion equation

Question

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!

Might [math.se] be better suited for this maths question? – Kyle Kanos Feb 24 '19 at 02:44 — Kyle Kanos, Feb 24 '19 at 02:44

mike stone · Accepted Answer · 2019-02-23T21:55:01.103

3

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 $t=constant$.

edited Feb 23 '19 at 21:55

answered Feb 23 '19 at 18:24

mike stone

52,996

thank you! could you please explain why it is parabolic? – Shaun Feb 23 '19 at 18:27
Added a note to my rather short (because of lack of time) original answer. – mike stone Feb 23 '19 at 21:55

Classification of 2D time dependent diffusion equation

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