Caltech/UCLA Joint Analysis Seminar

We give two new examples of nonlinear wave equations Box u = F(u) (with u vector-valued, and F some smooth nonlinearity) that exhibit blowup in finite time. Firstly, in three dimensions, we give an example of finite time blowup with F "defocusing" (the gradient of some non-negative potential) and also growing at any supercritical rate p > 5; somewhat surprisingly, the construction requires the Nash embedding theorem (which in turn requires the dimension of the range of u to be large - at least 40).  Secondly, in the case when F has all derivatives bounded, we give an example of blowup in eleven and higher dimensions, almost complementing positive results in nine and fewer dimensions due to Pecher, von Wahl, Brenner, and others.

The location is MS6221 UCLA.