Gaussian lower bounds on the Dirichlet heat kernel and blowup of nonlinear heat equations with Osgood-type nonlinearities
Vortrag von Prof. Dr. James Robinson
Sprecher eingeladen von: Prof. Dr. Michel Chipot
Datum: 28.11.13 Zeit: 17.15 - 18.45 Raum: Y27H25
Abstract: I will discuss the relationship between the existence of solutions for the ODE $\dot u=f(u)$ and for the PDE $u_t-\Delta u=f(u)$, in two surprising situations.First I will illustrate the phenomenon of ``diffusion-induced blowup" using an example due to Weinberger: a coupled system of ODEs in which all solutions tend to zero but for which there are solutions of the corresponding PDE that blow up in finite time.
Then I will show, based on joint work with Robert Laister and Mikolaj Sierzega, that even in the case of a single equation there are ODEs in which solutions exist for all time, but for which the corresponding PDE has no solution for initial data in $L^p$ if $p<\infty$. The argument requires Gaussian lower bounds on the Dirichlet heat kernel, for which I will give a simple proof.