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

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  • 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.
    Mathematics Subject Classification: 35B40, 35P15, 45K05.


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  • [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), 189-215.doi: 10.1007/s00028-007-0377-9.


    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), 201-227.doi: 10.1016/j.matpur.2008.04.003.


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


    P. Bates, X. Chen and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var., 24 (2005), 261-281.doi: 10.1007/s00526-005-0308-y.


    P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal., 138 (1997), 105-136.doi: 10.1007/s002050050037.


    E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.doi: 10.1016/j.matpur.2006.04.005.


    C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Diff. Eqns., 234 (2007), 360-390.doi: 10.1016/j.jde.2006.12.002.


    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), 137-156.doi: 10.1007/s00205-007-0062-8.


    J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics, Proc. Roy. Soc. Edinburgh, 137 (2007), 727-755.doi: 10.1017/S0308210504000721.


    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), 493-540.doi: 10.1142/S0218202512500546.


    P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, pp. 153-191, Springer-Verlag, Berlin, 2003.


    J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.doi: 10.1016/j.jde.2008.04.015.


    V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.doi: 10.1007/s00285-003-0210-1.


    L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92 (2009), 163-187.doi: 10.1016/j.matpur.2009.04.009.


    M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 1950 (1950), 199-325.


    M. L. Parks, R. B. Lehoucq, S. Plimpton and S. Silling, Implementing peridynamics within a molecular dynamics code, Computer Physics Comm., 179 (2008), 777-783.doi: 10.1016/j.cpc.2008.06.011.


    S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), 175-209.doi: 10.1016/S0022-5096(99)00029-0.


    S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elasticity, 93 (2008), 13-37.doi: 10.1007/s10659-008-9163-3.

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