Nonlinear stability of explicit self-similar solutions for the timelike extremal hypersurfaces in $${\mathbb {R}}^{1+3}$
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Calculus of Variations
Nonlinear stability of explicit self-similar solutions for the timelike extremal hypersurfaces in R1+3 Weiping Yan1 Received: 5 December 2018 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime R1+3 . We find that there are two explicit lightlike selfsimilar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime R1+3 , the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincaré can’t be used) in solving the difference equation by construction of a Newton’s polygon when we carry out the analysis of spectrum for the linear operator. Mathematics Subject Classification 37L15 · 35A10 · 35L05 · 35A21
Contents 1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear mode unstable of lightlike self-similar solutions . . . . . . . . . . . . . 2.1 Two explicit lightlike self-similar solutions . . . . . . . . . . . . . . . . . 2.2 Mode unstable of self-similar solutions u ± T (t, r ) . . . . . . . . . . . . . . 3 Well-posedness of linearized time evolution . . . . . . . . . . . . . . . . . . . 3.1 The C0 -semigroup of linearized operator at the initial approximation step 3.2 The spectrum of linearized operator at the initial approximation step . . . 3.3 Decay in time of the general approximation solution . . . . . . . . . . . . 4 Nonlinear stability of explicit lightlike self-similar solutions . . . . . . . . . . 4.1 The approximation scheme . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convergence of the approximation scheme . . . . . . . . . . . . . . . . .
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Communicated by A.Chang.
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Weiping Yan [email protected] School of Mathematics, Xiamen University, Xiamen 361000, People’s Republic of China 0123456789().: V,-vol
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W. Yan
4.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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