LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC KIRCHHOFF SYSTEMS

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.

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

[2]  Salim A. Messaoudi,et al.  A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation , 2007, Appl. Math. Lett..

[3]  Giuseppina Autuori,et al.  Kirchhoff systems with dynamic boundary conditions , 2010 .

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

[5]  J. Serrin,et al.  Asymptotic stability for nonautonomous dissipative wave systems , 1996 .

[6]  R. Monneau.  Justification of the Nonlinear Kirchhoff-Love Theory of Plates as the Application of a New Singular Inverse Method , 2003 .

[7]  Patrick Joly,et al.  An efficient numerical method for the resolution of the Kirchhoff‐Love dynamic plate equation , 2005 .

[8]  Enzo Vitillaro,et al.  Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation , 1999 .

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

[10]  L. Tsai,et al.  Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation , 2006 .

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

[12]  Yong Zhou,et al.  Global nonexistence for a semilinear Petrovsky equation , 2009 .

[13]  A. Boccuto,et al.  Asymptotic stability for abstract evolution equations and applications to partial differential systems , 1998 .

[14]  Patrick Martinez,et al.  EXISTENCE AND DECAY RATE ESTIMATES FOR THE WAVE EQUATION WITH NONLINEAR BOUNDARY DAMPING AND SOURCE TERM , 2004 .

[15]  Enzo Vitillaro.  Some new results on global nonexistence and blow-up for evolution problems with positive initial energy , 2000 .

[16]  Philippe G. Ciarlet,et al.  A Justi cation of a Nolinear Model in Plate Theory , 1979 .

[17]  Ganesh C. Gorain,et al.  Exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation , 2006, Appl. Math. Comput..

[18]  P. Pucci,et al.  Asymptotic stability for nonlinear Kirchhoff systems , 2009 .

[19]  Giuseppina Autuori,et al.  Global Nonexistence for Nonlinear Kirchhoff Systems , 2010 .

[20]  J. A. Soriano,et al.  GLOBAL EXISTENCE AND ASYMPTOTIC STABILITY FOR THE NONLINEAR AND GENERALIZED DAMPED EXTENSIBLE PLATE EQUATION , 2004 .

[21]  J. Serrin,et al.  Local asymptotic stability for dissipative wave systems , 1998 .

[22]  Local asymptotic stability for polyharmonic Kirchhoff systems† , 2011 .

[23]  Solange Kouémou Patcheu.  On a Global Solution and Asymptotic Behaviour for the Generalized Damped Extensible Beam Equation , 1997 .

[24]  K. Ono.  Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation , 1997 .

[25]  Shun-Tang Wu,et al.  On A system of nonlinear wave equations of kirchhoff type with a strong dissipation , 2007 .

[26]  K. Ono.  Global existence, asymptotic behaviour, and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space , 2000 .

[27]  Monique Dauge,et al.  Koiter Estimate Revisited , 2010 .

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

[29]  N. Alaa,et al.  Nonexistence of Global Solutions to a Class of Nonlinear Differential Inequalities and Application to Hyperbolic–Elliptic Problems , 2003 .

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

[31]  Shun-Tang Wu,et al.  EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS FOR A NONLINEAR WAVE EQUATION , 2009 .

[32]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[33]  P. Pucci,et al.  Kirchhoff systems with nonlinear source and boundary damping terms , 2010 .

[34]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[35]  M. Guedda,et al.  Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions , 2002 .