TY - JOUR

T1 - Smooth solutions to a class of quasilinear wave equations

AU - Hsu, Cheng Hsiung

AU - Lin, Song Sun

AU - Makino, Tetu

N1 - Funding Information:
∗Corresponding author. E-mail addresses: [email protected] (C.-H. Hsu), [email protected] (S.-S. Lin), [email protected] (T. Makino). 1The work of C.-H. Hsu was partially supported by the National Science Council of Taiwan and National Center for Theoretical Sciences, Mathematical Division, Taiwan. 2The work of S.-S. Lin was partially supported by the National Science Council of Taiwan and National Center for Theoretical Sciences, Mathematical Division, Taiwan. 3T. Makino was financially supported by the National Center for Theoretical Sciences, Mathematical Division, Taiwan, and Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan.

PY - 2006/5/15

Y1 - 2006/5/15

N2 - This article investigates the existence/nonexistence of smooth solutions of nonlinear vibration equations which arise from the one-dimensional motion of polytropic gas without external forces contained in a finite interval. For any fixed arbitrarily long time, we show that there are smooth small amplitude solutions of the nonlinear equations for which the periodic solutions of the linearized equation are the first-order approximations. On the other hand, when the nonlinearity is strictly convex or concave, there exists no time-periodic solutions which are twice continuously differentiable. An example of possible singularities which occur at the second derivatives is illustrated. We also give another kind of exact solutions with singularity such that shocks occur after a finite time. Furthermore, we get an estimate of the life span of smooth solutions to the initial-boundary value problem.

AB - This article investigates the existence/nonexistence of smooth solutions of nonlinear vibration equations which arise from the one-dimensional motion of polytropic gas without external forces contained in a finite interval. For any fixed arbitrarily long time, we show that there are smooth small amplitude solutions of the nonlinear equations for which the periodic solutions of the linearized equation are the first-order approximations. On the other hand, when the nonlinearity is strictly convex or concave, there exists no time-periodic solutions which are twice continuously differentiable. An example of possible singularities which occur at the second derivatives is illustrated. We also give another kind of exact solutions with singularity such that shocks occur after a finite time. Furthermore, we get an estimate of the life span of smooth solutions to the initial-boundary value problem.

UR - http://www.scopus.com/inward/record.url?scp=33645996303&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2005.06.007

DO - 10.1016/j.jde.2005.06.007

M3 - 期刊論文

AN - SCOPUS:33645996303

SN - 0022-0396

VL - 224

SP - 229

EP - 257

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -