Abstract
In this paper, we obtain bounds for the decay rate in the L r (ℝd)-norm for the solutions of a nonlocal and nonlinear evolution equation, namely,
. We consider a kernel of the form K(x, y) = ψ(y−a(x)) + ψ(x−a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form
. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ℝd:
Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1/p1,p as p→∞.
Similar content being viewed by others
References
F. Andreu, J. M. Mazon, J. D. Rossi, and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ. 8 (2008), 189–215.
F. Andreu, J. M. Mazon, J. D. Rossi, and J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl. (9) 90 (2008), 201–227.
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo. The limit as p → ∞ in a nonlocal p-Laplacian evolution equation. A nonlocal approximation of a model for sandpiles, Calc. Var. Partial Differential Equations 35 (2009), 279–316.
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, A nonlocal p-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal. 40 (2009), 1815–1851.
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, Nonlocal Diffusion Problems, Amer. Math. Soc., Providence RI; Real Sociedad Mathemática Española, Madrid, 2010.
G. Bachman and L. Narici, Functional Analysis, Dover, New York, 2000.
P. Bates, X. Chen, and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var. Partial Differential Equations 24 (2005), 261–281.
P. Bates, P. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997), 105–136.
E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. 86 (2006), 271–291.
A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: existence and stability, J. Differential Equations 155 (1999), 17–43.
C. Cortázar, J. Coville, M. Elgueta, and S. Martínez, A non local inhomogeneous dispersal process, J. Differential Equations 241 (2007), 332–358.
C. Cortázar, M. Elgueta, and J. D. Rossi, A nonlocal diffusion equation whose solutions develop a free boundary, Ann. Henri Poincaré 6 (2005), 269–281.
C. Cortázar, M. Elgueta, J. D. Rossi, and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations 234 (2007), 360–390.
C. Cortázar, M. Elgueta, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal. 187 (2008), 137–156.
J. Coville, On uniqueness and monotonicity of solutions on non-local reaction diffusion equations, Ann. Mat. Pura Appl. (4) 185 (2006), 461–485.
J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), 2921–2953.
J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sec. A 137 (2007), 1–29.
Q. Du, M. Gunzburger, R. Lehoucq, and K. Zhou, A nonlocal vector calculus, non-local volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl, Sci. 23 (2013), 493–540.
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer-Verlag, Berlin, 2003, pp. 153–191
J. García-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal. 71 (2009), 6116–6121.
J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations 246 (2009), 21–38.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
V. Hutson, S. Martínez, K. Mischaikow, and G. T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), 483–517.
L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal. 251 (2007), 399–437.
L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl. (9) 92 (2009), 163–187.
L. I. Ignat, J. D. Rossi, and A. San Antolin, Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space, J. Differential Equations 252 (2012), 6429–6447.
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. 10 (1962), 199–325.
M. L. Parks, R. B. Lehoucq, S. Plimpton, and S. Silling, Implementing peridynamics within a molecular dynamics code, Comput. Physics Comm. 179 (2008), 777–783.
W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 (2010), 747–795.
S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), 175–209.
S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elasticity 93 (2008), 13–37.
Author information
Authors and Affiliations
Corresponding author
Additional information
L. I. Ignat is partially supported by grants PN II-RU-TE 4/2010 and PCCE-55/2008 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, MTM2011-29306-C02-00, MICINN, Spain and ERC Advanced Grant FP7-246775 NUMERIWAVES.
D. Pinasco is partially supported by grants ANPCyT PICT 2011-0738 and CONICET PIP 0624.
J. D. Rossi and A. San Antolin are partially supported by the grant MTM2011-27998 MICINN MICINN, Spain.
Rights and permissions
About this article
Cite this article
Ignat, L.I., Pinasco, D., Rossi, J.D. et al. Decay estimates for nonlinear nonlocal diffusion problems in the whole space. JAMA 122, 375–401 (2014). https://doi.org/10.1007/s11854-014-0011-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-014-0011-z