## 応用解析セミナー

開催情報 木曜日　16:00～17:30　数理科学研究科棟(駒場) 002号室 石毛 和弘

### 2016年10月27日(木)

16:00-17:30   数理科学研究科棟(駒場) 126号室
Fred Weissler 氏 (パリ第13大学)
Sign-changing solutions of the nonlinear heat equation with positive initial value
(ENGLISH)
[ 講演概要 ]
We consider the nonlinear heat equation with a power nonlinear source term on all of N-dimensional space. It is well known that the associated Cauchy problem is locally well-posed in a variety of function spaces, including certain Lebesgue spaces, depending on the power. In other Lebesgue spaces, it can happen that the Cauchy problem is not well-posed. In particular, there exist non-negative initial values for which no local (in time) non-negative solution exists. This can happen also for some homogeneous functions, where the homogeneity is linked to the scaling properties of the equation.

I will discuss recent work, in collaboration with T. Cazenave, F. Dickstein and I. Naumkin. We show that for a certain class of non-negative initial values which, as mentioned above, do not admit local non-negative solutions, there exist in fact local (or global) solutions which change sign. In particular, in the case of non-negative homogeneous initial data which do not admit non-negative solutions, we construct sign-changing self-similar solutions with the given initial data.

https://www.ms.u-tokyo.ac.jp/~miyamoto/Weissler-abstract.pdf