TY - JOUR
T1 - Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients
AU - Han , Wei
JO - Journal of Partial Differential Equations
VL - 2
SP - 138
EP - 150
PY - 2013
DA - 2013/06
SN - 26
DO - http://doi.org/10.4208/jpde.v26.n2.4
UR - https://global-sci.org/intro/article_detail/jpde/5158.html
KW - Semilinear wave equation
KW - critical exponent
KW - initial-boundary value problem
KW - blow up
KW - lifespan
AB - <p>This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x &gt; 0, t &gt; 0, n=1,$ 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 $1 &lt; p &lt; +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ 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.</p>