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Blow up for a class of quasilinear wave equations in one space dimension
Author(s) -
Martel Yvan
Publication year - 2000
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(20000525)23:8<751::aid-mma138>3.0.co;2-u
Subject(s) - mathematics , gravitational singularity , wave equation , class (philosophy) , dimension (graph theory) , term (time) , conservation law , space (punctuation) , mathematical analysis , perturbation (astronomy) , pure mathematics , physics , quantum mechanics , artificial intelligence , computer science , linguistics , philosophy
For suitable σ and F , we prove that all classical solutions of the quasilinear wave equation $\phi_{tt}-(\sigma(\phi_{x}))_{x}=F(\phi)$ , with initial data of compact support, develop singularities in finite time. The assumptions on σ and F include in particular the model case $\phi_{tt}-\phi_{xx}(1+\phi^{2}_{x})=\varepsilon\phi^{q+1}$ , for q ⩾ 2,and ϵ = ±1. The starting point of the proof is to write the equation under the form of a first order system of two equations, in which F ( ϕ ) appears as a nonlocal term. Then, we present a new idea to control the effect of this perturbation term, and we conclude the proof by using well‐known methods developed for 2 × 2 systems of conservation laws. Copyright © 2000 John Wiley & Sons, Ltd.