# Maximum principle and inferring existence of solution

If I have a nonlinear parabolic PDE on domain $I \times [0,T]$ $$u_t = Lu$$ $$u|_{t=0} > 0$$ and I want to show existence to it.

Can I say this: if we assume there exists a $u$ that's $C^1$ in time and $C^2$ in space that satisfies this PDE then this $u > 0$ by the maximum principle. This implies strict parabolicity of the PDE (since the coefficients of $u_{xx}$ depend on $u$ as it's nonlinear), and I can use standard results of non-linear strictly parabolic PDEs to show that the solution does indeed exist and that it is $C^{2,1}$ since I used the maximum principle.

Is that right? It seems a bit circular.

Thanks!

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I am not sure your reasoning is correct. Existence, for nonlinear PDE's, should be proved first. You are actually thinking of a nonlinear equation as if it were linear, but this is generally useless, for existence. It is usefuls in regularity theory, on the contrary. –  Siminore Jun 27 '12 at 9:34
@Siminore thanks –  soup Jun 28 '12 at 18:05
If your operator $L$ is general nonlinear there is very little you can say. However, $L$ usually denotes a linear operator. In both the previous if your operator $L$ does not include time then you can use existence for those positive solutions. I think you should reconsider your problem from the begin. While there is surely existence you are hardly going to find unicity, thus the problem is no well-possed. If you have a physical motivation you should look into boundary conditions. –  D... Jan 6 at 2:07
Thanks @D... ... –  soup Jan 9 at 18:27