While searching for information about properties of parabolic problems, I stumbled upon a publication titled "Study of Nonlinear Parabolic Problems".
This made me wonder why the writers specifically mentioned the word "nonlinear" since I believed that parabolic problems were inherently nonlinear. This led me to question the exact definition of a parabolic problem or equation.
Is it possible to have a linear parabolic problem?
If so, what is the definition of a linear parabolic equation?
Check "Handbook of Differential Equations" 3rd edition Daniel Zwillinger.
"6. Classification of Partial Differential Equations"
Page 37 :
Page 38 :
(Point 1) In general , $A,B,C$ may or may not be Constants.
When these are Constants , then Discriminant $D$ is also Constant. Discriminant $D$ having Zero value indicates Parabolic PDE.
Eg :
$(A,B,C)=(0.1,2,10)$ : $D=0$
$(A,B,C)=(x^2,2xy,y^2)$ : $D=0$
(Point 2) When $u$ is a Solution & $v$ is a Solution , we could try $u+v$ , which will also be a Solution when $\Psi$ is linear.
Otherwise , we can not add the Solutions , due to nonlinear $\Psi$
Hence we can have Parabolic linear PDE & Parabolic nonlinear PDE & $n^{th}$ Degree PDE& $k^{th}$ Order PDE & ETC.
Classification is almost unlimited.