In this mini-course, we will examine the global existence vs blow up for solutions to the power type semilinear wave equations with small data ($(\partial_t^2-\Delta) u=|u|^p$, $p>1$), particularly for spatial dimension three. The problem is also known as the Strauss conjecture. In the process, we will present various analysis tools to handle this problem. Many of these tools have been proven to be useful for similar problems in broader context, e.g., the analog of the Strauss conjecture on the Schwarzschild and Kerr black hole backgrounds.

 

Lecture I. Semilinear wave equations: heuristics and what we expect;

Lecture II. Blow up: ODE argument, Yordanov-Zhang method of test functions;

Lecture III. Global existence using Klainerman-Sobolev estimates, Strichartz estimates;

Lecture IV. Modern proof: (integrated) local energy estimates, Weighted Strichartz estimates;

Lecture V. More robust proof of weighted Strichartz estimates, long time existence;

Lecture VI. History and discussion.