The influence of damping and source terms on solutions of nonlinear wave equations

We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces. One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions

[1]  J. Serrin,et al.  Existence for a nonlinear wave equation with damping and source terms , 2003, Differential and Integral Equations.

[2]  Howard A. Levine,et al.  Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations , 1974 .

[3]  V. Barbu,et al.  BLOW-UP OF GENERALIZED SOLUTIONS TO WAVE EQUATIONS WITH NONLINEAR DEGENERATE DAMPING AND SOURCE TERMS , 2007 .

[4]  V. Barbu,et al.  Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms , 2005 .

[5]  Enzo Vitillaro,et al.  Global existence for the wave equation with nonlinear boundary damping and source terms , 2002 .

[6]  Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions , 2001 .

[7]  Glenn F. Webb,et al.  Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation , 1980, Canadian Journal of Mathematics.

[8]  Grozdena Todorova,et al.  Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms , 2000 .

[9]  M. Willem.  Minimax Theorems , 1997 .

[10]  Viorel Barbu,et al.  Differential equations in Banach spaces , 1976 .

[11]  Vladimir Georgiev,et al.  Existence of a Solution of the Wave Equation with Nonlinear Damping and Source Terms , 1994 .

[12]  R. Temam.  Navier-Stokes Equations , 1977 .

[13]  J. Lions,et al.  Non homogeneous boundary value problems for second order hyperbolic operators , 1986 .

[14]  D. Sattinger,et al.  Saddle points and instability of nonlinear hyperbolic equations , 1975 .

[15]  P. Grisvard,et al.  Caractérisation de quelques espaces d'interpolation , 1967 .

[16]  I. Segal.  Non-Linear Semi-Groups , 1963 .

[17]  V. Barbu,et al.  On nonlinear wave equations with degenerate damping and source terms , 2005 .

[18]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[19]  J. Serrin,et al.  Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy , 1998 .

[20]  M. Reed.  Abstract non-linear wave equations , 1976 .

[21]  P. Grisvard.  Équations différentielles abstraites , 1969 .

[22]  Howard A. Levine,et al.  Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation , 1998 .

[23]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[24]  John M. Ball,et al.  REMARKS ON BLOW-UP AND NONEXISTENCE THEOREMS FOR NONLINEAR EVOLUTION EQUATIONS , 1977 .

[25]  M. Rammaha,et al.  Global existence and non-existence theorems for nonlinear wave equations , 2002 .

[26]  Jacques-Louis Lions,et al.  Some non-linear evolution equations , 1965 .

[27]  Daniel Tataru,et al.  ON THE REGULARITY OF BOUNDARY TRACES FOR THE WAVE EQUATION , 1998 .

[28]  R. Glassey.  Blow-up theorems for nonlinear wave equations , 1973 .

[29]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[30]  I. Lasiecka,et al.  Global solvability and uniform decays of solutions to quaslinear equation with nonlinear boundary dissipation , 1999 .

[31]  J. Lagnese.  Boundary Stabilization of Thin Plates , 1987 .

[32]  Howard A. Levine,et al.  Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation , 1997 .

[33]  I. Lasiecka,et al.  Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping , 2008 .

[34]  S. Krein,et al.  Linear Differential Equations in Banach Space , 1972 .

[35]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[36]  Irena Lasiecka,et al.  Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction , 2007 .

[37]  Marcelo M. Cavalcanti,et al.  On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source , 2009 .

[38]  Fritz John,et al.  Nonlinear wave equations, formation of singularities , 1990 .

[39]  Semilinear wave equations , 1992, math/9201268.

[40]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .

[41]  Grozdena Todorova.  Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms , 2000 .

[42]  Keith M. Agre,et al.  Systems of nonlinear wave equations with damping and source terms , 2006, Differential and Integral Equations.

[43]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[44]  I. Lasiecka,et al.  Hadamard Well-posedness of Weak Solutions in Nonlinear Dynamic Elasticity-full von Karman Systems , 2002 .

[45]  D. Ang,et al.  Mixed problem for some semi-linear wave equation with a nonhomogeneous condition , 1988 .

[46]  Mohammad A. Rammaha.  ON NONLINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS , 2002 .

[47]  Howard A. Levine,et al.  Global Existence and Nonexistence Theorems for Quasilinear Evolution Equations of Formally Parabolic Type , 1998 .

[48]  D. G. Figueiredo,et al.  Elliptic equations in R2 with nonlinearities in the critical growth range , 1995 .

[49]  D. A. Kearns.  On Solutions of Linear Differential Equation , 1960 .

[50]  M. Rammaha,et al.  Global existence and nonexistence for nonlinear wave equations with damping and source terms , 2002 .

[51]  Irena Lasiecka,et al.  Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping , 1993, Differential and Integral Equations.

[52]  A. Haraux.  Nonlinear evolution equations: Global behavior of solutions , 1981 .