Asymptotic Behavior of Solutions of Evolution Equations

Publisher Summary This chapter presents the methods of investigation of the asymptotic behavior of solutions of evolution equations, endowed with a dissipative mechanism, based on the study of the structure of the ω-limit set of trajectories of the evolution operator generated by the equation. The dissipative mechanism usually manifests itself by the presence of a Liapunov functional, which is constant on ω -limit sets; the central idea of the approach presented in the chapter is to use this information in conjunction with properties of ω -limit sets such as invariance and minimality. The chapter discusses two examples of wave equations with weak damping for which the scheme set up by Hale applies. In particular, this requires that the Liapunov functional be continuous on phase space. The chapter explores the case of a hyperbolic conservation law that generates a semigroup on space. It also presents a survey of various applications and extensions of these ideas that may serve as a guide to those interested in learning more about the method.

[1]  M. Slemrod,et al.  Asymptotic behavior of nonlinear contraction semigroups , 1973 .

[2]  Jeffrey Rauch,et al.  Decay of solutions to nondissipative hyperbolic systems on compact manifolds , 1975 .

[3]  E. F. Infante,et al.  A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type. , 1974 .

[4]  Constantine M. Dafermos,et al.  Contraction semigroups and trend to equilibrium in continuum mechanics , 1976 .

[5]  M. Slemrod.  The linear stabilization problem in Hilbert space , 1972 .

[6]  Constantine M. Dafermos,et al.  Applications of the invariance principle for compact processes. I. Asymptotically dynamical systems , 1971 .

[7]  J. Ball.  Stability Theory for an Extensible Beam , 1973 .

[8]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[9]  Michael G. Crandall,et al.  The semigroup approach to first order quasilinear equations in several space variables , 1972 .

[10]  L. A. Peletier,et al.  Global attraction for the one-dimensional heat equation with nonlinear time-dependent boundary conditions , 1977 .

[11]  J. Ralston.  Solutions of the wave equation with localized energy , 1969 .

[12]  G. Webb.  A Bifurcation Problem for a Nonlinear Hyperbolic Partial Differential Equation , 1979 .

[13]  Nobuhisa Iwasaki.  Local Decay of Solutions for Symmetric Hyperbolic Systems with Dissipative and Coercive Boundary Conditions in Exterior Domains , 1969 .

[14]  Jeffrey Rauch,et al.  Qualitative behavior of dissipative wave equations on bounded domains , 1976 .

[15]  M. Slemrod.  A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space , 1974 .

[16]  Marshall Slemrod,et al.  A hereditary partial differential equation with applications in the theory of simple fluids , 1976 .

[17]  Measurability and Continuity Conditions for Evolutionary Processes , 1976 .

[18]  J. Walker.  Some results on Liapunov functions and generated dynamical systems , 1978 .

[19]  George R. Sell,et al.  NONAUTONOMOUS DIFFERENTIAL EQUATIONS AND TOPOLOGICAL DYNAMICS. I. THE BASIC THEORY , 1967 .

[20]  Constantine M. Dafermos,et al.  Applications of the invariance principle for compact processes II. Asymptotic behavior of solutions of a hyperbolic conservation law , 1972 .

[21]  John M. Ball,et al.  Feedback stabilization of distributed semilinear control systems , 1979 .

[22]  Jack K. Hale,et al.  Dynamical systems and stability , 1969 .

[23]  J. Ball.  Measurability and continuity conditions for nonlinear evolutionary processes , 1976 .

[24]  L. A. Peletier,et al.  Stabilization of concentration profiles in catalyst particles , 1976 .

[25]  Barbara Keyfitz Quinn,et al.  Solutions with shocks: An example of an L1-contractive semigroup† , 1971 .

[26]  James Alfred Walker,et al.  On the application of Liapunov's direct method to linear dynamical systems , 1976 .

[27]  Zvi Artstein,et al.  The limiting equations of nonautonomous ordinary differential equations , 1977 .

[28]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[29]  C. Dafermos.  ALMOST PERIODIC PROCESSES AND ALMOST PERIODIC SOLUTIONS OF EVOLUTION EQUATIONS , 1977 .

[30]  Constantine M. Dafermos,et al.  Uniform processes and semicontinuous Liapunov functionals , 1972 .

[31]  M. Slemrod.  Asymptotic behavior of periodic dynamical systems on banach spaces , 1970 .

[32]  Constantine M. Dafermos,et al.  An invariance principle for compact processes , 1971 .

[33]  J. P. Lasalle.  THE EXTENT OF ASYMPTOTIC STABILITY. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Marshall Slemrod,et al.  Asymptotic behavior of a class of abstract dynamical systems , 1970 .

[35]  James Alfred Walker,et al.  Some results on the precompactness of orbits of dynamical systems , 1975 .

[36]  Richard K. Miller.  Asymptotic behavior of solutions of nonlinear differential equations , 1965 .

[37]  C. Dafermos.  Asymptotic stability in viscoelasticity , 1970 .