On the critical regularity of nonlinearities for semilinear classical wave equations

Abstract

AbstractIn this paper, we consider the Cauchy problem for semilinear classical wave equations $$\begin{aligned} u_{tt}-\Delta u=|u|^{p_S(n)}\mu (|u|) \end{aligned}$$ u tt - Δ u = | u | p S ( n ) μ ( | u | ) with the Strauss exponent $$p_S(n)$$ p S ( n ) and a modulus of continuity $$\mu =\mu (\tau ),$$ μ = μ ( τ ) , 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 $$\mu $$ μ 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 ∞ L r ∞ 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.