On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles

Abstract

Abstract. We establish the Strauss conjecture concerning small-data global existence for nonlinear wave equations, in the setting of exterior domains to compact obstacles, for space dimensions n = 3 and 4. The obstacle is assumed to be nontrapping, and the solution is assumed to satisfy either Dirichlet or Neumann conditions along the boundary of the obstacle. The key step in the proof is establishing certain "abstract Strichartz estimates" for the linear wave equation on exterior domains.

Global existence for systems of quasilinear wave equations in (1 + 4)-dimensions

“…The first ones are local energy estimates for the solution without a derivative and are obtained by appropriately dividing through by a derivative and using a variant of a Sobolev embedding. These are collected from [2], [7], and [25]. The second class of estimates apply to small, timedependent perturbations of ✷ and, as stated, are from [24] but are heavily influenced by the preceding works [30], [23].…”

Stable soliton resolution for equivariant wave maps exterior to a ball

“…The case when potential V = 0 is proved in [9]. The case with V as in (2.6) can be proved by adapting the argument in [15, Proposition 5.1], which is performed for = 1 to dimension d = 2 + 3.…”

Finite time blowup of solutions to semilinear wave equation in an exterior domain

“…The significant difference to the initial value problem is the effect of reflection at the boundary and the lack of symmetry such as scale-invariance, rotation-invariance and so on. For the existence of global-in-time solutions to (1.1) has been discussed in Du-Metcalfe-Sogge-Zhou [2] and Hidano-Metcalfe-Smith-Sogge-Zhou [3] when p S (N ) < p < N +3 N −1 and N = 3, 4. The sharp lower bounds for lifespan of solutions are shown in Yu [17] 2 < p < p S (3) with N = 3; Zhou-Han [19] proved sharp upper bounds in the case 1 < p < p S (N ) and N ≥ 3.…”