Global Nonexistence for Nonlinear Kirchhoff Systems

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, ut), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped.

[1]  P. Hästö.  On the density of continuous functions in variable exponent Sobolev space , 2007 .

[2]  Vicentiu D. Rădulescu,et al.  Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent , 2008 .

[3]  A. E. Hamidi,et al.  Sharp Sobolev Asymptotics for Critical Anisotropic Equations , 2009 .

[4]  J. Serrin,et al.  Some remarks on the global nonexistence problem for nonautonomous abstract evolution equations , 1997 .

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

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

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

[8]  S. Antontsev,et al.  Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions , 2006 .

[9]  Qihu Zhang,et al.  Eigenvalues of p(x)-Laplacian Dirichlet problem , 2005 .

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

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

[12]  Jiří Rákosník,et al.  On spaces $L^{p(x)}$ and $W^{k, p(x)}$ , 1991 .

[13]  M. Dreher,et al.  The Kirchhoff equation for the p-Laplacian , 2006 .

[14]  Giuseppe Mingione,et al.  Gradient estimates for the p (x)-Laplacean system , 2005 .

[15]  Giuseppina Autuori,et al.  Asymptotic stability for anisotropic Kirchhoff systems , 2009 .

[16]  Giovany M. Figueiredo,et al.  On an elliptic equation of p-Kirchhoff type via variational methods , 2006, Bulletin of the Australian Mathematical Society.

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

[18]  Xianling Fan,et al.  On the Spaces Lp(x)(Ω) and Wm, p(x)(Ω) , 2001 .

[19]  M. Dreher,et al.  The Wave Equation for the p-Laplacian , 2007 .

[20]  Jiří Rákosník,et al.  Sobolev embeddings with variable exponent , 2000 .

[21]  Peter Hästö,et al.  The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values , 2006 .

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

[23]  Zhikov.  On Lavrentiev's Phenomenon. , 1995 .

[24]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[25]  P. Hästö,et al.  Sobolev inequalities with variable exponent attaining the values $1$ and $n$ , 2008 .