On the critical regularity of nonlinearities for semilinear classical wave equations

Abstract

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

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