Short time existence on nonlinear parabolic PDE - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:00:34Z http://mathoverflow.net/feeds/question/99994 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99994/short-time-existence-on-nonlinear-parabolic-pde Short time existence on nonlinear parabolic PDE Hassan Jolany 2012-06-19T11:42:01Z 2013-04-13T11:32:05Z <p>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 ?</p> http://mathoverflow.net/questions/99994/short-time-existence-on-nonlinear-parabolic-pde/100003#100003 Answer by Denis Serre for Short time existence on nonlinear parabolic PDE Denis Serre 2012-06-19T13:41:39Z 2012-06-19T13:41:39Z <p>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. </p> <p>The fully non-linear case (example $\partial_t u=\det(\nabla^2u)$ with $u(t=0,\cdot)$ convex, it is really advanced. In the quasilinear case (example $\partial_t u={\rm div}(|\nabla|^{p-1}\nabla u)$, it is still complicated, but several books treat it extensively. </p> <p>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.</p> http://mathoverflow.net/questions/99994/short-time-existence-on-nonlinear-parabolic-pde/102708#102708 Answer by Sajjad Lakzian for Short time existence on nonlinear parabolic PDE Sajjad Lakzian 2012-07-20T00:29:01Z 2012-07-20T00:29:01Z <p>Dear Denis Serre, can you please name some good references. </p>