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

Abstract

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity |u| p in exterior domain in R N (N ≥ 3). Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in R N . At least in the case N = 4, their estimates are sharp in view of the work by Zha-Zhou [21]. The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa-Yordanov [15]. Mathematics Subject Classification (2010): Primary: 35L05, 35L20, 35B44.

Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds

“…(p,q)-curve . However, we do not need to assume that the initial data f and g are positive in the pointwise sense assumed in the previous results and we only need the assumption I[g] = R N g(x)dx > 0 (see also [16,31] for the classical case m = 0).…”

Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

“…Remark 1.7. When the spatial dimension is not greater than 4, all of the previous blow-up results and lifespan estimates for exterior problem with critical power heavily rely on the assumption that the obstacle is a ball (see [14], [16], [31]), under which they can construct some special test functions explicitly. In contrast, our results hold for very general obstacle in 2-D.…”

“…Zhou-Han [47] Lai-Zhou [14] Hidano et al [9] 4 Zhou-Han [47] Sobajima-Wakasa [31] Du et al [3], reproved by Hidano et al [9] ≥ 5 Zhou-Han [47] Lai-Zhou [15], reproved by Sobajima-Wakasa [31] p = 2, Metcalfe-Sogge [25], reproved by Wang [40] Just like the Cauchy problem, it is meaningful to study the lifespan for the blowup exponent. We expect the same estimate as that of the Cauchy problem, regardless of the boundary obstacle, at least when the obstacle is nontrapping.…”

“…L : ? U : Zhou-Han [47] L : Zha-Zhou [43] reproved by Wang [40] U :Sobajima-Wakasa [31] ≥ 5 L : ? U : Zhou-Han [47] L : ?…”

“…U : Zhou-Han [47] L : ? U : Lai-Zhou [15], reproved by Sobajima-Wakasa [31] We also have to mention the generalization of problem (1.8) from Euclidean space to other manifold, such as asymptotically Euclidean manifolds (see [26], [32], [38] and references therein), and black hole spacetime (see [1], [19], [21], [24] and references therein). One can find a detailed description of such kind of generalization in a recent survey paper [39].…”