A class of dissipative nonautonomous nonlocal second-order evolution equations

References

• Alali B, Lipton R. Multiscale dynamics of heterogeneous media in the peridynamic formulation. J. Elast. 2012;106:71–103.
   Web of Science ®Google Scholar
• Bourdin B, Francfort G, Marigo J-J. The variational approach to fracture. J. Elast. 2008;91:5–148.
   Web of Science ®Google Scholar
• Bourdin B, Larsen C, Richardson C. A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 2011;168:133–143.
   Web of Science ®Google Scholar
• Emmrich E, Puhst D. Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function. Commun. Math. Sci. 2013;11:1039–1049.
   Web of Science ®Google Scholar
• Erbay HA, Erkip A, Muslu GM. The Cauchy problem for the one dimensional nonlinear peridynamic model. J. Differ. Equ. 2012;252:4392–4409.
   Web of Science ®Google Scholar
• Lipton R. Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 2014;117:21–50.
   Web of Science ®Google Scholar
• Lipton R. Cohesive dynamics and brittle fracture. J. Elast. 2016;124:143–191. doi:10.1007/s10659-015-9564-z
   Web of Science ®Google Scholar
• Duruk N, Erbay HA, Erkip A. Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity. Nonlinearity. 2010;23:107–118.
   Web of Science ®Google Scholar
• Duruk N, Erbay HA, Erkip A. Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations. J. Differ. Equ. 2011;250:1448–1459.
   Web of Science ®Google Scholar
• Erbay HA, Erbay S, Erkip A. The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials. Nonlinearity. 2011;24:1347–1359.
   Web of Science ®Google Scholar
• Andreu F, Mazón JM, Rossi JD, et al. A nonlocal p--Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. 2008;90:201–227.
   Web of Science ®Google Scholar
• Barros SRM, Pereira AL, Possani C, et al. Spatially periodic equilibria for a non local evolution equation. Discrete Contin. Dyn. Syst. 2003;9:937–948.
   Web of Science ®Google Scholar
• Bellettini G, de Masi A, Presutti E. Energy levels of a non local functional. J. Math. Phys. 2005;46:1–31.
   Web of Science ®Google Scholar
• Bezerra FDM, Pereira AL, da Silva SH. Existence and continuity of global attractors and of nonhomogeneous equilibria for a class of evolution equation with non local terms. J. Math. Anal. Appl. 2012;396:590–600.
   Web of Science ®Google Scholar
• da Silva SH. Properties of an equation for neural fields a bounded domain. Electron. J. Differ. Equ. 2012;42:1–9.
   Google Scholar
• Grasselli M, Petzeltová H, Schimperna G. A nonlocal phase-field system with inertial term. Quart. Appl. Math. 2007;65:451–469.
   Web of Science ®Google Scholar
• French DA. Identification of a free energy functional in an integro-differential equation model for neuronal network activity. Appl. Math. Lett. 2004;17:1047–1051.
   Web of Science ®Google Scholar
• Masi A, Orland E, Presutti E, et al. Stability of the interface in a model of phase separation. Proc. R. Soc. Edinburgh Sect. A. 1994;124:1013–1022.
   Web of Science ®Google Scholar
• Pereira AL. Global attractor and nonhomogeneous equilibria for a non local evolution equation in an unbounded domain. J. Differ. Equ. 2006;226:352–372.
   Web of Science ®Google Scholar
• Pereira AL, da Silva SH. Existence of global attractors and gradient property for a class of non local evolution equations. São Paulo J. Math. Sci. 2008;2:1–20.
   Google Scholar
• Hopfield JJ. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. U.S.A. 1984;81:3088–3092.
   PubMed Web of Science ®Google Scholar
• Carvalho AN, Langa JA, Robinson JC. Attractors for infinite-dimensional non-autonomous dynamical systems. New York (NY):Springer-Verlag; 2012. (Applied Mathematical Sciences; vol. 182).
   Google Scholar
• Carbone VL, Nascimento MJD, Schiabel-Silva K, et al. Pullback attractors for a singularly nonautonomous plate equation. Electron. J. Differ. Equ. 2011;77:1–13.
   Google Scholar
• Carvalho AN, Cholewa JW. Attractors for strongly damped wave equations with critical nonlinearities. Pacific J. Math. 2002;207:287–310.
   Web of Science ®Google Scholar
• Carvalho AN, Cholewa JW, Dlotko T. Damped wave equations with fast growing dissipative nonlinearities. Discrete Contin. Dyn. Syst. 2009;24:1147–1165.
   Web of Science ®Google Scholar
• Cholewa JW, Dlotko T. Global attractors in abstract parabolic problems. Cambridge: Cambridge University Press; 2000.
   Google Scholar
• Arrieta JM, Carvalho AN, Hale JK. A damped hyperbolic equation with critical exponent. Comm. Partial Differ. Equ. 1992;17:841–866.
   Web of Science ®Google Scholar
• Babin AV, Vishik MI. Attractors of evolution equations. Amsterdam: North-Holland Publishing Co.; 1992.
   Google Scholar