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April  2015, 35(4): 1469-1478. doi: 10.3934/dcds.2015.35.1469

Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid

2. 

Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante

Received  March 2013 Revised  October 2013 Published  November 2014

In this paper we prove decay estimates for solutions to a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions, that is, $$u_t (x,t)=\int_{\Omega\cup\Omega_0} J(x-y) |u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\, dy$$ for $(x,t)\in \Omega\times \mathbb{R}^+$ and $u(x,t)=0$ in $ \Omega_0\times \mathbb{R}^+$. The proof of these estimates is based on bounds for the associated first eigenvalue.
Citation: Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469
References:
[1]

F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations,, J. Evol. Eqns., 8 (2008), 189. doi: 10.1007/s00028-007-0377-9. Google Scholar

[2]

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., 90 (2008), 201. doi: 10.1016/j.matpur.2008.04.003. Google Scholar

[3]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Nonlocal Diffusion Problems,, Vol. 165. Amer. Math. Soc. Mathematical Surveys and Monographs 2010., (2010). doi: 10.1090/surv/165. Google Scholar

[4]

P. Bates, X. Chen and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions,, Calc. Var., 24 (2005), 261. doi: 10.1007/s00526-005-0308-y. Google Scholar

[5]

P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rat. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[6]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar

[7]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion,, J. Diff. Eqns., 234 (2007), 360. doi: 10.1016/j.jde.2006.12.002. Google Scholar

[8]

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. Rat. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8. Google Scholar

[9]

J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh, 137 (2007), 727. doi: 10.1017/S0308210504000721. Google Scholar

[10]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws,, Math. Mod. Meth. Appl. Sci., 23 (2013), 493. doi: 10.1142/S0218202512500546. Google Scholar

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in nonlinear analysis, (2003), 153. Google Scholar

[12]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009), 21. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[13]

V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar

[14]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pures Appl., 92 (2009), 163. doi: 10.1016/j.matpur.2009.04.009. Google Scholar

[15]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Amer. Math. Soc. Transl., 1950 (1950), 199. Google Scholar

[16]

M. L. Parks, R. B. Lehoucq, S. Plimpton and S. Silling, Implementing peridynamics within a molecular dynamics code,, Computer Physics Comm., 179 (2008), 777. doi: 10.1016/j.cpc.2008.06.011. Google Scholar

[17]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces,, J. Mech. Phys. Solids, 48 (2000), 175. doi: 10.1016/S0022-5096(99)00029-0. Google Scholar

[18]

S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory,, J. Elasticity, 93 (2008), 13. doi: 10.1007/s10659-008-9163-3. Google Scholar

show all references

References:
[1]

F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations,, J. Evol. Eqns., 8 (2008), 189. doi: 10.1007/s00028-007-0377-9. Google Scholar

[2]

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., 90 (2008), 201. doi: 10.1016/j.matpur.2008.04.003. Google Scholar

[3]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Nonlocal Diffusion Problems,, Vol. 165. Amer. Math. Soc. Mathematical Surveys and Monographs 2010., (2010). doi: 10.1090/surv/165. Google Scholar

[4]

P. Bates, X. Chen and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions,, Calc. Var., 24 (2005), 261. doi: 10.1007/s00526-005-0308-y. Google Scholar

[5]

P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rat. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[6]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar

[7]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion,, J. Diff. Eqns., 234 (2007), 360. doi: 10.1016/j.jde.2006.12.002. Google Scholar

[8]

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. Rat. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8. Google Scholar

[9]

J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh, 137 (2007), 727. doi: 10.1017/S0308210504000721. Google Scholar

[10]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws,, Math. Mod. Meth. Appl. Sci., 23 (2013), 493. doi: 10.1142/S0218202512500546. Google Scholar

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in nonlinear analysis, (2003), 153. Google Scholar

[12]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009), 21. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[13]

V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar

[14]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pures Appl., 92 (2009), 163. doi: 10.1016/j.matpur.2009.04.009. Google Scholar

[15]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Amer. Math. Soc. Transl., 1950 (1950), 199. Google Scholar

[16]

M. L. Parks, R. B. Lehoucq, S. Plimpton and S. Silling, Implementing peridynamics within a molecular dynamics code,, Computer Physics Comm., 179 (2008), 777. doi: 10.1016/j.cpc.2008.06.011. Google Scholar

[17]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces,, J. Mech. Phys. Solids, 48 (2000), 175. doi: 10.1016/S0022-5096(99)00029-0. Google Scholar

[18]

S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory,, J. Elasticity, 93 (2008), 13. doi: 10.1007/s10659-008-9163-3. Google Scholar

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