Blow Up of Solutions to One Dimensional Initial-Boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

摘要

This paper is devoted to studying the following initial-boundary value problem for one-dimensional semilinear wave equations with variable coefficients and with subcritical exponent:utt-(e)x(a(x)(e)xu)=|u|p,x＞0,t＞0,n=1,(0.1)where u =u(x,t) is a real-valued scalar unknown function in [0,+∞) × [0,+∞),here a(x) is a smooth real-valued function of the variable x ∈ (0,+∞).The exponents p satisfies 1＜p＜ +∞ in (0.1).It is well-known that the number pc(1) =+∞ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g.,[1]).We will establish a blowup result for the above initial-boundary value problem,it is proved that there can be no global solutions no matter how small the initial data are,and also we give the lifespan estimate of solutions for above problem.