Nonlocal higher order evolution equations

References

• Fife , P . 2003 . “ Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions ” . In Trends in Nonlinear Anal. , Berlin : Springer .
   Google Scholar
• Bates , P , Chen , X and Chmaj , A . 2005 . Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions . Calc. Var. , 24 : 261 – 281 .
   Google Scholar
• Bates , P , Fife , P , Ren , X and Wang , X . 1997 . Travelling waves in a convolution model for phase transitions . Arch. Rat. Mech. Anal. , 138 : 105 – 136 .
   Web of Science ®Google Scholar
• Coville , J . 2006 . On uniqueness and monotonicity of solutions on non-local reaction diffusion equations . Ann. Mat. Pura Appl. , 185 : 461 – 485 .
   Web of Science ®Google Scholar
• Coville , J . 2007 . Maximum principles, sliding techniques and applications to nonlocal equations . Electronic J. Diff. Eqns. , 2007 ( 68 ) : 1 – 23 .
   Google Scholar
• Coville , J , Dávila , J and Martínez , S . 2008 . Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity . SIAM J. Math. Anal. , 39 ( 5 ) : 1693 – 1709 .
   Google Scholar
• Coville , J and Dupaigne , L . 2005 . Propagation speed of travelling fronts in nonlocal reaction diffusion equations . Nonlinear Anal. , 60 : 797 – 819 .
   Web of Science ®Google Scholar
• Coville , J and Dupaigne , L . 2007 . On a nonlocal equation arising in population dynamics . Proc. Roy. Soc. Edinburgh , 137 : 1 – 29 .
   Google Scholar
• Zhang , L . 2004 . Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks . J. Diff. Eqns. , 197 ( 1 ) : 162 – 196 .
   Web of Science ®Google Scholar
• Bates , P and Chmaj , A . 1999 . An integrodifferential model for phase transitions: Stationary solutions in higher dimensions . J. Stat. Phys. , 95 : 1119 – 1139 .
   Web of Science ®Google Scholar
• Bates , P and Chmaj , A . 1999 . A discrete convolution model for phase transitions . Arch. Rat. Mech. Anal. , 150 : 281 – 305 .
   Web of Science ®Google Scholar
• Carrillo , C and Fife , P . 2005 . Spatial effects in discrete generation population models . J. Math. Biol. , 50 ( 2 ) : 161 – 188 .
   PubMed Web of Science ®Google Scholar
• Chen , X . 1997 . Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations . Adv. Diff. Eqns. , 2 : 125 – 160 .
   Google Scholar
• Fife , P and Wang , X . 1998 . A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions . Adv. Diff. Eqns. , 3 ( 1 ) : 85 – 110 .
   Google Scholar
• Silvestre , L . 2006 . Hölder estimates for solutions of integro differential equations like the fractional laplace . Indiana Univ. Math. J. , 55 ( 3 ) : 1155 – 1174 .
   Web of Science ®Google Scholar
• Wang , X . 2002 . Metastability and stability of patterns in a convolution model for phase transitions . J. Diff. Eqns. , 183 : 434 – 461 .
   Web of Science ®Google Scholar
• Buades , A , Coll , B and Morel , J-M . 2005 . On image denoising methods . SIAM Multiscale Model. Simulation , 4 ( 2 ) : 490 – 530 .
   Google Scholar
• Kindermann , S , Osher , S and Jones , P . 2005 . Deblurring and denoising of images by nonlocal functionals . SIAM Multiscale Model. Simulation , 4 ( 4 ) : 1091 – 1115 .
   Google Scholar
• Gilboa , G and Osher , S . 2006 . Nonlocal linear image regularization and supervised segmentation , California : UCLA CAM Rep. No. 06-47 .
   Google Scholar
• Fife , P . 2000 . Models for phase separation and their mathematics . Electronic J. Diff. Eqns. , 2000 ( 48 ) : 1 – 26 .
   Google Scholar
• Bertozzi , A , Esedoglu , S and Gillette , A . 2007 . Inpainting of binary images using the Cahn–Hilliard equation . IEEE Trans. Image Proc. , 16 ( 1 ) : 285 – 291 .
   PubMed Web of Science ®Google Scholar
• Bertozzi , A , Esedoglu , S and Gillette , A . 2007 . Analysis of a two-scale Cahn–Hilliard model for image inpainting . Multiscale Mod. Simul. , 6 ( 3 ) : 913 – 936 .
   Web of Science ®Google Scholar
• Michelson , D . 1986 . Steady solutions of the Kuramoto–Sivashinsky equation . Physica D , 19 : 89 – 111 .
   Web of Science ®Google Scholar
• Cross , MC and Hohenberg , PC . 1993 . Pattern formation outside of equilibrium . Rev. Mod. Phys. , 65 ( 3 ) : 851 – 1112 .
   Web of Science ®Google Scholar
• Polyanin , AD . 2001 . Handbook of Linear Partial Differential Equations for Engineers and Scientists , London : Chapman and Hall/CRC .
   Google Scholar
• Happel , J and Brenner , H . 1973 . Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Mechanics of Fluids and Transport Processes). , Berlin : Springer .
   Google Scholar
• Den Hartog , JP . 1987 . Advanced Strength of Materials , New York : Dover Publications .
   Google Scholar
• Birkner , M , Lòpez-Mimbelab , JA and Wakolbing , A . 2005 . Comparison results and steady states for the Fujita equation with fractional Laplacian . Ann. Inst. Henri Poincare (C) Non-Linear Anal. , 22 ( 1 ) : 83 – 97 .
   Google Scholar
• Fino , A and Karch , G . Decay of mass for nonlinear equation with fractional Laplacian . Monatsh. Math. (to appear) DOI: 10.1007/s00605-009-0093-3 ,
   Google Scholar
• Chen , W and Holm , S . 2004 . Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency . J. Acoust. Soc. Am. , 115 ( 4 ) : 1424 – 1430 .
   PubMed Web of Science ®Google Scholar
• Han , J . 2004 . The Cauchy problem and steady state solutions for a nonlocal Cahn–Hilliard equation . Electronic J. Diff. Eqns. , 2004 ( 113 ) : 1 – 9 .
   Google Scholar
• Nishiura , Y and Ohnishi , I . 1995 . Some mathematical aspects of the micro-phase separation in diblock copolymers . Physica D , 84 : 31 – 39 .
   Google Scholar
• Ohta , T and Kawasaki , K . 1986 . Equilibrium morphology of block copolymer melts . Macromolecules , 19 : 2621 – 2632 .
   Web of Science ®Google Scholar
• Chasseigne , E , Chaves , M and Rossi , JD . 2006 . Asymptotic behaviour for nonlocal diffusion equations . J. Math. Pures Appl. , 86 : 271 – 291 .
   Web of Science ®Google Scholar
• Ignat , LI and Rossi , JD . 2008 . Refined asymptotic expansions for nonlocal evolution equations . J. Evol. Eqns. , 8 ( 4 ) : 617 – 629 .
   Google Scholar
• Andreu , F , Mazon , JM , Rossi , JD and Toledo , J . 2008 . The Neumann problem for nonlocal nonlinear diffusion equations . J. Evol. Eqns. , 8 ( 1 ) : 189 – 215 .
   Google Scholar
• Cortazar , C , Elgueta , M , Rossi , JD and Wolanski , N . 2007 . Boundary fluxes for non-local diffusion . J. Diff. Eqns. , 234 : 360 – 390 .
   Web of Science ®Google Scholar
• Cortazar , C , Elgueta , M , Rossi , JD and Wolanski , N . 2008 . How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems . Arch. Rat. Mech. Anal. , 187 ( 1 ) : 137 – 156 .
   Google Scholar
• Ignat , LI and Rossi , JD . 2007 . A nonlocal convection–diffusion equation . J. Funct. Anal. , 251 : 399 – 437 .
   Web of Science ®Google Scholar
• Cortázar , C , Coville , J , Elgueta , M and Martínez , S . 2007 . A non local inhomogeneous dispersal process . J. Diff. Eqns. , 241 ( 2 ) : 332 – 358 .
   Google Scholar
• Cortazar , C , Elgueta , M and Rossi , JD . 2005 . A non-local diffusion equation whose solutions develop a free boundary . Ann. Henri Poincaré , 6 ( 2 ) : 269 – 281 .
   Google Scholar