PDE Seminar (2019/5/24): Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Date
2019-5-24 16:30 - 2019-5-24 18:00
Place
Faculty of Science Building #3, Room 309
Speaker/Organizer
Wei Dai (Hokkaido University)
 
Firstly, we study the equation $\square u = |u|^{q_c}+ |\partial u|^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p \geq q_c$. We obtain the optimal estimate of the lifespan $\ln ({T_\varepsilon}) \approx \varepsilon^{-q_c(q_c-1)}$ in $n=3$, and improve the lower bound of $T_\varepsilon$ from $\exp ({c\varepsilon^{-(q_c-1)}})$ to $\exp ({c\varepsilon^{-(q_c-1)^2/2}})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u = |v|^q$, $\square v = |\partial_t u|^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.