Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.

[1]  J. Kenigson,et al.  Energy decay estimates for the dissipative wave equation with space–time dependent potential , 2011 .

[2]  Kenji Nishihara,et al.  Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application , 2003 .

[3]  Hendrik J. Kuiper,et al.  LIFE SPAN OF NONNEGATIVE SOLUTIONS TO CERTAIN QUASILINEAR PARABOLIC CAUCHY PROBLEMS , 2003 .

[4]  A. Milani,et al.  On the diffusion phenomenonof quasilinear hyperbolic waves , 2000 .

[5]  B. Yordanov,et al.  Critical Exponent for a Nonlinear Wave Equation with Damping , 2000 .

[6]  Tosio Kato.  Biow‐up of solutions of some nonlinear hyperbolic equations , 1980 .

[7]  M. D’Abbicco,et al.  A Modified Test Function Method for Damped Wave Equations , 2013 .

[8]  B. Yordanov,et al.  Optimal decay rate of the energy for wave equations with critical potential , 2013 .

[9]  Kenji Nishihara,et al.  Critical exponent for the semilinear wave equation with time-dependent damping , 2012 .

[10]  M. Ikeda,et al.  A note on the lifespan of solutions to the semilinear damped wave equation , 2012, 1212.1772.

[11]  H. Fujita.  On the blowing up of solutions fo the Cauchy problem for u_t=Δu+u^ , 1966 .

[12]  Michael Reissig,et al.  Semi-linear wave equations with effective damping , 2012, Chinese Annals of Mathematics, Series B.

[13]  Nakao Hayashi,et al.  Damped wave equation with super critical nonlinearities , 2004, Differential and Integral Equations.

[14]  Yuta Wakasugi.  Critical exponent for the semilinear wave equation with scale invariant damping , 2014 .

[15]  Jens Wirth,et al.  Wave equations with time-dependent dissipation II. Effective dissipation , 2006 .

[16]  Marcello D'Abbicco,et al.  The threshold of effective damping for semilinear wave equations , 2015 .

[17]  Kenji Nishihara,et al.  The Lp–Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media , 2003 .

[18]  B. Yordanov,et al.  Weighted L2-estimates for dissipative wave equations with variable coefficients , 2009 .

[19]  K. Tanizawa,et al.  Global existence of solutions for semilinear damped wave equations in RN with noncompactly supported initial data , 2005 .

[20]  A. Matsumura,et al.  On the Asymptotic Behavior of Solutions of Semi-linear Wave Equations , 1976 .

[21]  K. Mochizuki.  Scattering Theory for Wave Equations with Dissipative Terms , 1976 .

[22]  M. D’Abbicco,et al.  A modified test function method for damped waves , 2012, 1211.0453.

[23]  A. Milani,et al.  On the Diffusion Phenomenon of Quasilinear Hyperbolic Waves , 1980 .

[24]  Ryo Ikehata,et al.  Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential , 2009 .

[25]  Maisa Khader.  Global existence for the dissipative wave equations with space–time dependent potential , 2013 .

[26]  Jens Wirth.  Solution representations for a wave equation with weak dissipation , 2002 .

[27]  Takayoshi Ogawa,et al.  Large time behavior and Lp−Lq estimate of solutions of 2-dimensional nonlinear damped wave equations , 2004 .

[28]  Takashi Narazaki,et al.  L^P-L^Q Estimates for Damped Wave Equations and their Applications to Semi-Linear Problem , 2004 .

[29]  Yuta Wakasugi.  Small data global existence for the semilinear wave equation with space–time dependent damping , 2012, 1202.5379.

[30]  Qi S. Zhang.  A blow-up result for a nonlinear wave equation with damping: The critical case , 2001 .