Maximum principle and initial conditions (easy) Can someone tell me why it's true that for
$$u_t = a(u)u_{xx} + b(u)u_{x} + c(u)$$
$$u|_{t=0} = u_0$$
if $u_0 > 0$ then the solution $u > 0$ too? Do the functions $a$, $b$, and $c$ need to be constrained in some way for this to hold? There is no boundary in this equation and the domain can be something like the unit circle.
I know it's max principle but I really cannot find anything online about max principles and initial conditions. Thanks a lot. 
 A: The maximum and minimum principles for elliptic and parabolic equations belong to the class of what is called a priori estimates. To be more precise, these types of results are of the form

If a solution were to exists for a certain equation, and if the coefficients of the equation were to satisfy certain properties, then the solution must also satisfy certain property. 

One of the nice things about these types of estimates/principles/theorems/lemmas is that one can easily adapt them to nonlinear cases from the linear case results. In your case:


*

*Assume that $u$ is a fixed given function which exists as a solution to your equation. 

*For that $u$, define the coefficient functions $A(x,t) = a(u)$ and $B(x,t) = b(u)$ and $C(x,t) = c(u)$. Then clearly $u$ also solves the linear equation $$ u_t(x,t) = A(x,t)u_{xx} + B(x,t) u_x + C(x,t) $$

*Hence if the functions $A,B,C$ can be assumed to satisfy also the conditions for the classical maximum principle for linear parabolic equations, $u$, now being a solution to the linear equation must also follow the maximum principle. 


That is to say: if the map $u\mapsto a(u)$ and $u\mapsto c(u)$ satisfy the condition that $a(u)$ is uniformly positive and $c(u)$ is non-positive, the classical maximum principle will apply for any classical solution of the quasilinear equation just as well as how it applies to the linear equation. 
Remark: Philosophically this is another instance of the method of freezing coefficients. For qualitative and quantitative results about a (presumed to exist) solution to a quasilinear equation, if one can make certain assumptions on how the coefficients behave, one can treat it just like a solution to an appropriate linear equation. It is usually in the step for proving existence of solutions that the quasilinearity becomes a problem. 
