Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles

“…In this case, under a somewhat more restrictive smallness condition, global existence was established by John [16] for the case where n = 3, then Glassey [12] for n = 2, Zhou [40] for n = 4, Lindblad and Sogge [22] for n ≤ 8 and then Georgiev, Lindblad and Sogge [11] for all n (see also Tataru [37]). For obstacles, when n = 4, ∆ g = ∆, the results in Theorem 1.2 for the Dirichlet-wave equation outside of nontrapping obstacles were obtained in [8] under a somewhat more restrictive smallness assumption.…”

Blow up of solutions to nonlinear wave equation in 2D exterior domains

“…Glassey [4,5] showed the global existence and nonexistence [except the critical case p = p c (2)] in the 2D case. Schaeffer [9] showed the nonexistence in the critical case p = p c (n) (n = 2, 3).…”

“…In 2008, for n = 4, p > 2 = p c (4) and Ω c is a nontrapping obstacle, Du et al [2] proved, among other results, that global solutions exist if the initial values are suitably small and have compact support. Hidano et al [6] showed the existence of global solutions without compactness assumption on the initial values.…”

“…In a recent paper, Smith et al [11] successfully verified Strauss' conjecture for obstacles when n = 2. They proved that if p > p c (2) and Ω c is a nontrapping obstacle, there exist global solutions if the initial data are small and regular.…”

“…Using the method of Yordanov and Zhang [14], they proved that when n ≥ 3, 1 < p < p c (n) and Ω satisfies a certain mild assumption, the solution to This article proves that, on two dimensional exterior domain, the solutions of the initial boundary value problem (1.2) blow up when 1 < p < p c (2). The following assumptions about domain and initial data are needed.…”