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

2019524 16:30 
2019524 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 $\ln ({T_\varepsilon}) \approx \varepsilon^{q_c(q_c1)}$ in $n=3$, and improve the lower bound of $T_\varepsilon$ from $\exp ({c\varepsilon^{(q_c1)}})$ to $\exp ({c\varepsilon^{(q_c1)^2/2}})$ 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 3dimensional space with $q<2$. We obtain that this system admits a global solution above a $pq$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.