Long-time behavior of nonlinear integro-differential evolution equations

Abstract We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space.

[1]  Jan Prüss,et al.  Global existence for a semilinear parabolic Volterra equation , 1992 .

[2]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[3]  E. Feireisl,et al.  Long-time stabilization of solutions to a phase-field model with memory , 2001 .

[4]  R. Chill,et al.  Convergence to steady states in asymptotically autonomous semilinear evolution equations , 2003 .

[5]  G. Simonett,et al.  Quasilinear evolutionary equations and continuous interpolation spaces , 2004 .

[6]  R. C. MacCamy,et al.  An integro-differential equation with application in heat flow , 1977 .

[7]  Vicente Vergara.  Convergence to steady states of solutions to nonlinear integral evolution equations , 2011 .

[8]  Olof J. Staffans,et al.  Volterra Integral and Functional Equations , 1990 .

[9]  Jace W. Nunziato,et al.  On heat conduction in materials with memory , 1971 .

[10]  Ralph Chill,et al.  On the Łojasiewicz–Simon gradient inequality , 2003 .

[11]  J. Prüss,et al.  Well-posedness and long-time behaviour for the non-isothermalCahn-Hilliard equation with memory , 2009 .

[12]  Convergence to steady states of solutions of semilinear evolutionary integral equations , 2005 .

[13]  G. Gripenberg.  Volterra integro-differential equations with accretive nonlinearity , 1985 .

[14]  S. Huang,et al.  Convergence in gradient-like systems which are asymptotically autonomous and analytic , 2001 .

[15]  Rico Zacher,et al.  Maximal regularity of type Lp for abstract parabolic Volterra equations , 2005 .

[16]  M. Gurtin,et al.  A general theory of heat conduction with finite wave speeds , 1968 .

[17]  Rico Zacher,et al.  Lyapunov functions and convergence to steady state for differential equations of fractional order , 2008 .

[18]  Herbert Amann,et al.  Operator‐Valued Fourier Multipliers, Vector ‐ Valued Besov Spaces, and Applications , 1997 .

[19]  Jan Prüss,et al.  Asymptotic behaviour of a semilinear viscoelastic beam model , 2001 .