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I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of page and number of theorem ?

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For weakly parabolic systems, see e.g. by D'Ancona and Spagnolo. The paper also gives citations to the strongly parabolic case. – Willie Wong Jun 19 '12 at 12:34
Dear Willie , I saw it, but I couldn't found Short time existence on nonlinear parabolic PDE , can you give another reference with details, number of theorems and number of pages? – baba ab dad Jun 19 '12 at 13:10
up vote 12 down vote accepted

This is kind of meta-theorem. It has several version, and if someone was willing to write one theorem containing all the situations, it would be unreadible.

The fully non-linear case (example $\partial_t u=\det(\nabla^2u)$ with $u(t=0,\cdot)$ convex), is really involved. In the quasilinear case (example $\partial_t u={\rm div}(|\nabla u|^{p-1}\nabla u)$, it is still complicated, but several books treat it extensively.

The semi-linear case $\partial_tu+Lu=f(u,\nabla u)$, where $L$ is an elliptic linear operator (like the Laplacian $-\Delta$), has a rather simple philosophy. One follows the guidelines of the Cauchy-Lipschitz theory for ODEs. If $a\in X$ is the initial data, one rewrites the Cauchy problem as an integral equation $$u(t)=e^{-tL}a+\int_0^t e^{(s-t)L}f(u(s),\nabla u(s))ds=:Nu(t).$$ When $T>0$ is small enough and the Banach space $X$ is appropriate, one proves that $N$ is a contraction in some ball $B(a;r)$ of $C(0,T;X)$. Then Picard's theorem tells that there is a unique fixed point $u$ ; this is the local solution.

By appropriate, I mean that the operators $S_t:=e^{-tL}$ form a strongly continuous semi-group over $X$. In particular $S_t\in{\mathcal L}(X)$ and $$\|S_t\|_{{\mathcal L}(X)}\le Ce^{\omega t},\qquad\forall t>0$$ for suitable constants $C$ and $\omega$. In practice, we may deal with a scale of Banach spaces $X_s$, like the Sobolev spaces $H^s$, and we have $$\|S_t\|_{{\mathcal L}(X_s,X_r)}\le Ct^{\alpha(s-r)}e^{\omega t},\qquad\forall t>0$$ for some $\alpha>0$, which is related to the order of $L$.

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Cher Denis Serre!, merci pour votre réponse belle – baba ab dad Jun 19 '12 at 14:12
Dear Denis Serre, can you please name some good references. – Sajjad Lakzian Jul 20 '12 at 0:29
If you know how to do the quasilinear case but for a system and not just a scalar equation, then you can do the fully nonlinear case using "prolongation" which means adding new unknown functions and setting them equal to the spatial first and second derivatives of the original unknown function. – Deane Yang Jul 20 '12 at 1:07
Dear Sajjad, for an appropriate reference please see… anyway if you need the PDF file of this book, you can send email me this book is very comprehensive about parabolic pde – baba ab dad Jul 20 '12 at 20:16
I guess the quasilinear case can be done using some sort of parabolic regularization. – Fan Zheng Apr 12 at 18:15

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