On the critical regularity of nonlinearities for semilinear classical wave equations

In this paper, we consider the Cauchy problem for semilinear classical wave equations \(\begin{aligned} u_{tt}-\Delta u=|u|^{p_S(n)}\mu (|u|) \end{aligned}\) $$\begin{array}{c}\hfill u_{\mathit{tt}}-\mathrm{\Delta }u={|u|}^{p_S(n)}\mu (|u|)\end{array}$$ with the Strauss exponent \(p_S(n)\) $p_S(n)$ and a modulus of continuity \(\mu =\mu (\tau ),\) $\mu =\mu (\tau ),$ which provides an additional regularity of nonlinearities in \(u=0\) $u=0$ comparing with the power nonlinearity \(|u|^{p_S(n)}.\) ${|u|}^{p_S(n)}.$ We obtain a sharp condition on \(\mu \) $\mu$ as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted \(L^{\infty }_tL^{\infty }_r\) $L_t^{\infty }{L}_r^{\infty }$ estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply a conjecture for the critical regularity of source nonlinearities for semilinear classical wave equations. We verify this conjecture in the 3d case.