Construction of a Multisoliton Blowup Solution to the Semilinear Wave Equation in One Space Dimension

Source abstract

Abstract We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blowup solution with a characteristic point, we refine the blowup behavior first derived by Merle and Zaag. We also refine the geometry of the blowup set near a characteristic point and show that, except for perhaps one exceptional situation, it is never symmetric with respect to the characteristic point. Then, we show that all blowup modalities predicted by those authors do occur. More precisely, given any integer k ≥ 2 and $\zeta _0 \in {\cal R}$ , we construct a blowup solution with a characteristic point a such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs whose centers (in the hyperbolic geometry) have ζ 0 as a center of mass for all times. © 2013 Wiley Periodicals, Inc.