American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 1105-1114. doi: 10.3934/proc.2015.1105

Blow-up of solutions to semilinear wave equations with non-zero initial data

Kyouhei Wakasa 1,

1. 

Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

Received  September 2014 Revised  December 2014 Published  November 2015

In this paper, we are concerned with the initial value problem for $u_{tt}-\Delta u=|u_t|^p$ in $\mathbb{R}^n\times[0,\infty)$ with the initial data $u(x,0)=f(x)$, $u_t(x,0)=g(x)$, where $(f,g)$ are slowly decaying.
    H. Takamura [13] obtained the blow-up result for the case where $f\equiv0$ and $g\not\equiv0$. Our purpose in this paper is to show the blow-up result for the case where the both initial data do not vanish identically.
Keywords: semilinear wave equations, high dimensions, point-wise estimates., Initial value problem, blow-up.
Mathematics Subject Classification: Primary: 35L70; Secondary: 35B44, 35E1.
Citation: Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105
References:
[1]

R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math., 73 (1991), 153-162.

[2]

R. T. Glassey, MathReview to "Global behavior of solutions to nonlinear wave equations in three space dimensions" of Sideris, Comm. Partial Differential Equations (1983).

[3]

K. Hidano, Initial value problem of semilinear wave equations in three space dimensions, Nonlinear Anal., 26 (1996), 941-970.

[4]

K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995), 1273-1305.

[5]

K. Hidano, C. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, Journal de Math ématiques Pures et Appliqu ées (9), 98 (2012), 518-541.

[6]

F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.

[7]

H. Kubo, Blow-up of solutions to semilinear wave equations with initial data of slow decay in low space dimensions, Differential and Integral Equations., 7 (1994), 315-321.

[8]

K. Masuda, Blow-up solutions for quasi-linear wave equations in two space dimensions, Lecture Notes in Num. Appl. Anal., 6 (1983), 87-91.

[9]

M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.

[10]

M. A. Rammaha, H. Takamura, H. Uesaka and K. Wakasa, Blow-Up of Positive Solutions to Wave Equations in High Space Dimensions,, to appear in Differential and Integral Equations, (). 

[11]

J. Schaeffer, Finite-time blow-up for $u_{t t}-\Delta u=H(u_r,u_t)$ in two space dimensions, Comm. Partial Differential Equations, 11 (1986), 513-543.

[12]

T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations, 8, (1983) 1219-1323.

[13]

H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661.

[14]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equations with small data, Tsukuba Math. J., 22 (1998), 198-211.

[15]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chin. Ann. Math. Ser.B, 22 (2001), 275-280.

show all references

References:
[1]

R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math., 73 (1991), 153-162.

[2]

R. T. Glassey, MathReview to "Global behavior of solutions to nonlinear wave equations in three space dimensions" of Sideris, Comm. Partial Differential Equations (1983).

[3]

K. Hidano, Initial value problem of semilinear wave equations in three space dimensions, Nonlinear Anal., 26 (1996), 941-970.

[4]

K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995), 1273-1305.

[5]

K. Hidano, C. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, Journal de Math ématiques Pures et Appliqu ées (9), 98 (2012), 518-541.

[6]

F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.

[7]

H. Kubo, Blow-up of solutions to semilinear wave equations with initial data of slow decay in low space dimensions, Differential and Integral Equations., 7 (1994), 315-321.

[8]

K. Masuda, Blow-up solutions for quasi-linear wave equations in two space dimensions, Lecture Notes in Num. Appl. Anal., 6 (1983), 87-91.

[9]

M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.

[10]

M. A. Rammaha, H. Takamura, H. Uesaka and K. Wakasa, Blow-Up of Positive Solutions to Wave Equations in High Space Dimensions,, to appear in Differential and Integral Equations, (). 

[11]

J. Schaeffer, Finite-time blow-up for $u_{t t}-\Delta u=H(u_r,u_t)$ in two space dimensions, Comm. Partial Differential Equations, 11 (1986), 513-543.

[12]

T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations, 8, (1983) 1219-1323.

[13]

H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661.

[14]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equations with small data, Tsukuba Math. J., 22 (1998), 198-211.

[15]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chin. Ann. Math. Ser.B, 22 (2001), 275-280.

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