Long-Time Existence for Small Amplitude Semilinear Wave Equations

We prove that a certain class of semilinear wave equations has global solutions if the initial data is small. These existence results rely on mixed-norm angular-radial space-time estimates. More specifically, we consider power nonlinearities, $\square u=F_{p}(u)$, where $F_{p}(u)\sim |u|^{p}$ and $\square =\partial _{t}^{2}-\Delta $ denotes the D'Alembertian on ${\Bbb R}_{+}^{1+n}$. In 1979, John investigated this equation for small initial data in dimension n = 3. He proved that if p > 1 + √2 then there is a global solution whereas for p < 1 + √2, the solution may blow up. Shortly afterwards Strauss conjectured that a similar result should hold in n dimensions, and the critical power $p_{c}$ above which global solutions exist should be the positive root of $(n-1)p_{c}^{2}-(n+1)p_{c}-2=0$. Sideris proved that for all n the solutions can blow up if $pp_{c}=2$. Here we prove that for n ≤ 8 one has global solutions if $p>p_{c}$. Furthermore, we proved that for all n, one has global solutions if $p>p_{c}$ and the initial data are spherically symmetric. In the radial case we use $L^{p}$ estimates, derived from the explicit form of the fundamental solution using inequalities from classical analysis. In the nonradial case we write the fundamental solution in polar coordinates and use Fourier integral and special function estimates to handle the angular part.