Remarks on some dissipative sine-Gordon equations

References

• Temam R. Infinite dimensional dynamical systems in mechanics and physics. 2nd ed. New York: Springer-Verlag; 1997.
   Google Scholar
• Chueshov I, Lasiecka I. Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, HCDTE lecture notes. Part I. Nonlinear hyperbolic PDEs, dispersive and transport equations, 96 pp., AIMS Ser. Appl. Math., 6, Am Inst Math Sci. (AIMS), Springfield, MO; 2013.
   Google Scholar
• Ghidaglia JM, Temam R. Attractors for damped nonlinear hyperbolic equations. J Math Pures Appl. 1987;66:273–319.
   Web of Science ®Google Scholar
• Ghidaglia JM, Temam R. Dimension of the universal attractor describing the periodically driven Sine-Gordon equations. Transp Theory Stat Phys. 1987;16(2–3):253–265. doi: 10.1080/00411458708204663
   Google Scholar
• Ghidaglia JM, Temam R. Periodic dynamical systems with applications to since-Gordon equations: estimate of the fractal dimension of the universal attractor. Contempory Math AMS. 1989;99:143–164. doi: 10.1090/conm/099/1034498
   Google Scholar
• Hale J. Asymptotic behaviour and dynamics in infinite dimensions. Boston: Pitman; 1985. p. 1–41. (Research Notes in Mathematics; 132).
   Google Scholar
• Haraux A. Two remarks on dissipative hyperbolic problems. In: Brezis H and Lions JL, editors. Nonlinear PDE and their applications. Collège de France seminar. Vol. VII. London: Pitman; 1985.
   Google Scholar
• Raugel G. Global attractors in partial differential equations. In: Handbook of dynamical systems. Vol. 2. Amsterdam: North-Holland; 2002. p. 885–982.
   Google Scholar
• Haraux A, Jendoubi M. Convergence of bounded weak solutions of the wave equations with dissipation and analytic nonlinearity. Cal Var PDE. 1999;9:95–124. doi: 10.1007/s005260050133
   Web of Science ®Google Scholar
• Haraux A, Jendoubi M. The convergence problem for dissipative autonomous systems: classical methods and recent advances. Vol. 7. Cham: Springer, Bilbao: BCAM Basque Center for Applied Mathematics; 2015. (SpringerBriefs in Mathematics. BCAM SpringerBriefs).
   Google Scholar
• Pierre M, Rogeon P. Convergence to equilibrium for a time semi-discrete damped wave equation. J Appl Anal Comput. 2016;6(4):1041–1048.
   Google Scholar
• Goubet O, Ricceri B. Non-existence results for an eigenvalue problem involving Lipschitzian nonlinearities with non-positive primitive. Bull. London Math. Soc. Forthcoming.
   Google Scholar
• Ricceri B. Four Conjectures in Nonlinear Analysis. In: Rassias T. editor. Applications of Nonlinear Analysis. Springer Optimization and Its Applications, vol 134. Cham: Springer; 2018. p. 681–710.
   Google Scholar
• Dupaigne L. Stable solutions of elliptic partial differential equations. 2011. Boca Raton, FL: Chapman and Hall (monographs and surveys in pure and applied mathematics).
   Google Scholar
• Lieb E, Loss M. Analysis. 2nd ed. Providence: American Mathematical Society; 2001. (Graduates studies in Mathematics; 14).
   Google Scholar
• Farina A, Valdinoci E. A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature. Adv Math. 2010;225:2808–2827. doi: 10.1016/j.aim.2010.05.008
   Web of Science ®Google Scholar
• Modica L. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm Pure Appl Math. 1985;38:679–684. doi: 10.1002/cpa.3160380515
   Web of Science ®Google Scholar
• Miranville A, Zelik S. Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Handbook of differential equations: evolutionary equations. Vol. IV. Amsterdam: Elsevier/North-Holland; 2008. p. 103–200. (Handb. Differ. Equ.).
   Google Scholar
• Zelik S. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete Contin Dyn Syst. 2001;7(3):593–641. doi: 10.3934/dcds.2001.7.593
   Web of Science ®Google Scholar
• Abounouh M, Goubet O, Hakim A. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system. Differential Integral Equations. 2003;16(5):573–581.
   Google Scholar
• Bourgain J. Global solutions of nonlinear Schrödinger equations. Vol. 46. AMS colloquium publication. Providence: American Mathematical Society; 1999.
   Google Scholar
• Ghidaglia JM, Temam R. Regularity of the solutions of second order evolution equations and their attractors. Ann Scuola Norm Sup Pisa Cl Sci. 1987;14(3):485–511.
   Google Scholar