# Classification of 2D time dependent diffusion equation

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 Mathematics be better suited for this maths question? – Kyle Kanos Feb 24 at 2:44

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 $$t=constant$$.