The Threshold between Effective and Noneffective Damping for Semilinear Waves

In this paper we study the global existence of small data solutions to the Cauchy problem utt −△u+ μ 1 + t ut = f(t, u) , u(0, x) = u0(x) , ut(0, x) = u1(x) , where μ ≥ 2. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. We extend our results to a model with polynomial speed of propagation, and to a model with an exponential speed of propagation and a constant damping ν ut.

[1]  K. Nishihara,et al.  Decay property of solutions for damped wave equations with space–time dependent damping term☆ , 2011 .

[2]  M. D’Abbicco,et al.  Long time asymptotics for 2 by 2 hyperbolic systems , 2011 .

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

[4]  Kenji Nishihara,et al.  Decay Properties for the Damped Wave Equation with Space Dependent Potential and Absorbed Semilinear Term , 2010 .

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

[6]  R. Ikehata,et al.  Decay estimates of solutions for dissipative wave equations in R^N with lower power nonlinearities , 2004 .

[7]  Masahito Ohta,et al.  Critical exponents for semilinear dissipative wave equations in RN , 2002 .

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

[9]  J. Wirth.  Asymptotic properties of solutions to wave equations with time-dependent dissipation , 2004 .

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

[11]  Blow-up of the energy at infinity for 2 by 2 systems , 2012 .

[12]  Xinfu Li.  Critical exponent for semilinear wave equation with critical potential , 2013 .

[13]  Taeko Yamazaki,et al.  Asymptotic behavior for abstract wave equations with decaying dissipation , 2006, Advances in Differential Equations.

[14]  Kenji Nishihara,et al.  Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping , 2011 .

[15]  M. D’Abbicco,et al.  Hyperbolic-like estimates for higher order equations , 2012 .

[16]  Hiroshi Fujita,et al.  On the blowing up of solutions of the Cauchy problem for $u_1+\Delta u+u^{1+\alpha} , 1966 .

[17]  M. D’Abbicco,et al.  A class of dissipative wave equations with time-dependent speed and damping , 2013 .

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

[19]  J. Wirth,et al.  Generalised energy conservation law for wave equations with variable propagation speed , 2008, 0802.0409.

[20]  Taeko Yamazaki,et al.  Diffusion phenomenon for abstract wave equations with decaying dissipation , 2007 .

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

[22]  Kosuke Ono,et al.  Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations , 1993 .

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

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