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Blowup Solutions to the Semilinear Wave Equation with a Stylized Pyramid as a Blowup Surface
Author(s) -
Merle Frank,
Zaag Hatem
Publication year - 2018
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21756
Subject(s) - surface (topology) , mathematics , mathematical analysis , space (punctuation) , differentiable function , stylized fact , subsequence , pyramid (geometry) , point (geometry) , similarity (geometry) , nonlinear system , pure mathematics , geometry , physics , image (mathematics) , bounded function , quantum mechanics , philosophy , linguistics , artificial intelligence , computer science , economics , macroeconomics
Abstract We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite‐time blowup solution with an isolated characteristic blowup point at the origin and a blowup surface that is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in ℝ 2 . The blowup surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one‐dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two‐dimensional stationary solution, whose existence is a by‐product of the proof. At the origin, it behaves like the sum of four solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blowup solution with a characteristic point in higher dimensions, showing a really two‐dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of noncharacteristic points where the blowup surface is nondifferentiable. © 2018 Wiley Periodicals, Inc.