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Volume 26, Issue 2
Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

Wei Han

DOI: 10.4208/jpde.v26.n2.4

J. Part. Diff. Eq., 26 (2013), pp. 138-150.

Published online: 2013-06

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  • Abstract

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ 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 $1 < p < +∞$ 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.

  • Keywords

Semilinear wave equation critical exponent initial-boundary value problem blow up lifespan

  • AMS Subject Headings

35L05, 35L70, 35L15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

sh_hanweiwei1@126.com (Wei Han)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-26-138, author = {Han , Wei}, title = {Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients}, journal = {Journal of Partial Differential Equations}, year = {2013}, volume = {26}, number = {2}, pages = {138--150}, abstract = {

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ 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 $1 < p < +∞$ 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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v26.n2.4}, url = {http://global-sci.org/intro/article_detail/jpde/5158.html} }
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 -

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ 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 $1 < p < +∞$ 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.

Han , Wei. (2013). Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients. Journal of Partial Differential Equations. 26 (2). 138-150. doi:10.4208/jpde.v26.n2.4
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