# American Institute of Mathematical Sciences

March  2017, 22(2): 339-368. doi: 10.3934/dcdsb.2017016

## Analysis of a nonlocal-in-time parabolic equation

 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

* Corresponding author: Zhi Zhou (zhizhou0125@gmail.com)

Received  March 2016 Revised  July 2016 Published  December 2016

Fund Project: This research is supported in part by the AFOSR MURI center for Material Failure Prediction through peridynamics and the ARO MURI Grant W911NF-15-1-0562.

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. Moreover, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter δ approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings.

Citation: Qiang Du, Jiang Yang, Zhi Zhou. Analysis of a nonlocal-in-time parabolic equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016
##### References:

show all references

##### References:
Plot of $\|u_h^\delta-u_h^0\|_{L^2(0,T;L^2(\Omega))}$ for example (a), (b) and (c), with $\tau=10^{-4}$, $h=10^{-3}$
$\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (a) with $\delta=1$, $h=1/N$, $\tau=10^{-5}$, $T=1$
 $\gamma$ $\text{norm}\backslash N$ $10$ $20$ $40$ $80$ $160$ rate $-0.8$ $L^2$ 4.58e-4 1.15e-4 2.87e-5 7.18e-6 1.79e-6 $\approx$ 2.00 (2.00) $H^1$ 9.42e-3 4.71e-3 2.35e-3 1.18e-3 5.53e-4 $\approx$ 1.05 (1.00) $-0.2$ $L^2$ 2.99e-4 7.51e-5 1.88e-5 4.69e-6 1.17e-6 $\approx$ 2.00 (2.00) $H^1$ 5.57e-3 2.78e-3 1.39e-3 6.94e-4 3.26e-4 $\approx$ 1.05 (1.00) $0.2$ $L^2$ 2.54e-4 6.36e-5 1.59e-5 3.97e-6 9.90e-7 $\approx$ 2.00 (2.00) $H^1$ 4.59e-3 2.29e-3 1.14e-3 5.72e-4 2.69e-4 $\approx$ 1.07 (1.00) $0.8$ $L^2$ 2.22e-4 5.58e-5 1.40e-5 3.49e-6 8.69e-7 $\approx$ 2.00 (2.00) $H^1$ 3.98e-3 1.99e-3 9.94e-4 4.97e-4 2.34e-4 $\approx$ 1.06 (1.00)
 $\gamma$ $\text{norm}\backslash N$ $10$ $20$ $40$ $80$ $160$ rate $-0.8$ $L^2$ 4.58e-4 1.15e-4 2.87e-5 7.18e-6 1.79e-6 $\approx$ 2.00 (2.00) $H^1$ 9.42e-3 4.71e-3 2.35e-3 1.18e-3 5.53e-4 $\approx$ 1.05 (1.00) $-0.2$ $L^2$ 2.99e-4 7.51e-5 1.88e-5 4.69e-6 1.17e-6 $\approx$ 2.00 (2.00) $H^1$ 5.57e-3 2.78e-3 1.39e-3 6.94e-4 3.26e-4 $\approx$ 1.05 (1.00) $0.2$ $L^2$ 2.54e-4 6.36e-5 1.59e-5 3.97e-6 9.90e-7 $\approx$ 2.00 (2.00) $H^1$ 4.59e-3 2.29e-3 1.14e-3 5.72e-4 2.69e-4 $\approx$ 1.07 (1.00) $0.8$ $L^2$ 2.22e-4 5.58e-5 1.40e-5 3.49e-6 8.69e-7 $\approx$ 2.00 (2.00) $H^1$ 3.98e-3 1.99e-3 9.94e-4 4.97e-4 2.34e-4 $\approx$ 1.06 (1.00)
$\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=0.5$
 $\gamma$ $\text{norm}\backslash m$ $11$ $21$ $41$ $81$ $161$ $321$ rate $-0.8$ $L^2$ 2.40e-3 8.04e-4 2.72e-4 9.29e-5 3.21e-5 1.10e-5 $\approx$ 1.55 $H^1$ 7.40e-2 5.10e-2 3.54e-2 2.45e-2 1.66e-2 1.09e-2 $\approx$ 0.55 $-0.2$ $L^2$ 6.04e-3 2.24e-3 8.08e-4 2.88e-4 1.02e-4 3.52e-5 $\approx$ 1.48 $H^1$ 2.22e-1 1.59e-9 1.12e-9 7.86e-2 5.38e-2 3.53e-2 $\approx$ 0.53 $0.2$ $L^2$ 7.67e-3 2.88e-3 1.05e-3 3.77e-4 1.33e-4 4.63e-5 $\approx$ 1.47 $H^1$ 2.88e-1 2.07e-1 1.47e-1 1.03e-1 7.08e-2 4.66e-2 $\approx$ 0.53 $0.8$ $L^2$ 9.35e-3 3.56e-3 1.30e-3 4.70e-4 1.66e-4 5.78e-5 $\approx$ 1.47 $H^1$ 3.57e-1 2.58e-1 1.84e-1 1.29e-1 8.50e-2 5.81e-2 $\approx$ 0.52
 $\gamma$ $\text{norm}\backslash m$ $11$ $21$ $41$ $81$ $161$ $321$ rate $-0.8$ $L^2$ 2.40e-3 8.04e-4 2.72e-4 9.29e-5 3.21e-5 1.10e-5 $\approx$ 1.55 $H^1$ 7.40e-2 5.10e-2 3.54e-2 2.45e-2 1.66e-2 1.09e-2 $\approx$ 0.55 $-0.2$ $L^2$ 6.04e-3 2.24e-3 8.08e-4 2.88e-4 1.02e-4 3.52e-5 $\approx$ 1.48 $H^1$ 2.22e-1 1.59e-9 1.12e-9 7.86e-2 5.38e-2 3.53e-2 $\approx$ 0.53 $0.2$ $L^2$ 7.67e-3 2.88e-3 1.05e-3 3.77e-4 1.33e-4 4.63e-5 $\approx$ 1.47 $H^1$ 2.88e-1 2.07e-1 1.47e-1 1.03e-1 7.08e-2 4.66e-2 $\approx$ 0.53 $0.8$ $L^2$ 9.35e-3 3.56e-3 1.30e-3 4.70e-4 1.66e-4 5.78e-5 $\approx$ 1.47 $H^1$ 3.57e-1 2.58e-1 1.84e-1 1.29e-1 8.50e-2 5.81e-2 $\approx$ 0.52
$\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=1.2$
 $\gamma$ $\text{norm}\backslash m$ $11$ $21$ $41$ $81$ $161$ $321$ rate $-0.8$ $L^2$ 1.36e-4 3.75e-5 9.86e-6 2.53e-6 6.37e-7 1.58e-7 $\approx$ 1.95 $H^1$ 1.56e-3 8.09e-4 4.08e-4 2.02e-4 9.77e-5 4.47e-5 $\approx$ 1.03 $-0.2$ $L^2$ 4.11e-4 1.13e-4 2.98e-5 7.63e-6 1.93e-6 4.77e-7 $\approx$ 1.95 $H^1$ 5.98e-3 3.10e-3 1.57e-3 7.78e-4 3.76e-4 1.72e-4 $\approx$ 1.02 $0.2$ $L^2$ 5.22e-4 1.44e-4 3.78e-5 9.68e-6 2.44e-6 6.06e-7 $\approx$ 1.95 $H^1$ 8.01e-3 4.15e-3 2.10e-3 1.04e-4 5.03e-4 2.30e-4 $\approx$ 1.02 $0.8$ $L^2$ 6.32e-4 1.74e-4 4.58e-5 1.17e-6 2.96e-6 7.34e-7 $\approx$ 1.95 $H^1$ 1.01e-2 5.24e-3 2.65e-3 1.31e-4 6.34e-4 2.90e-4 $\approx$ 1.02
 $\gamma$ $\text{norm}\backslash m$ $11$ $21$ $41$ $81$ $161$ $321$ rate $-0.8$ $L^2$ 1.36e-4 3.75e-5 9.86e-6 2.53e-6 6.37e-7 1.58e-7 $\approx$ 1.95 $H^1$ 1.56e-3 8.09e-4 4.08e-4 2.02e-4 9.77e-5 4.47e-5 $\approx$ 1.03 $-0.2$ $L^2$ 4.11e-4 1.13e-4 2.98e-5 7.63e-6 1.93e-6 4.77e-7 $\approx$ 1.95 $H^1$ 5.98e-3 3.10e-3 1.57e-3 7.78e-4 3.76e-4 1.72e-4 $\approx$ 1.02 $0.2$ $L^2$ 5.22e-4 1.44e-4 3.78e-5 9.68e-6 2.44e-6 6.06e-7 $\approx$ 1.95 $H^1$ 8.01e-3 4.15e-3 2.10e-3 1.04e-4 5.03e-4 2.30e-4 $\approx$ 1.02 $0.8$ $L^2$ 6.32e-4 1.74e-4 4.58e-5 1.17e-6 2.96e-6 7.34e-7 $\approx$ 1.95 $H^1$ 1.01e-2 5.24e-3 2.65e-3 1.31e-4 6.34e-4 2.90e-4 $\approx$ 1.02
$\| (u-u_h)(t) \|_{\dot H^{p}\left( \Omega \right)}$ for example (b) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $t=0.5$
 $\gamma$ $\text{norm}\backslash m$ $10$ $20$ $40$ $80$ $160$ $320$ rate $-0.8$ $L^2$ 2.88e-4 7.21e-5 1.80e-5 4.51e-6 1.12e-6 2.77e-7 $\approx$ 2.00 $H^1$ 7.88e-3 3.95e-3 1.97e-3 9.87e-4 4.64e-4 2.10e-4 $\approx$ 1.05 $0.8$ $L^2$ 6.27e-5 1.57e-5 3.94e-6 9.85e-7 2.45e-7 6.07e-8 $\approx$ 2.00 $H^1$ 1.43e-3 7.12e-4 3.56e-4 1.78e-4 8.37e-5 3.78e-5 $\approx$ 1.05
 $\gamma$ $\text{norm}\backslash m$ $10$ $20$ $40$ $80$ $160$ $320$ rate $-0.8$ $L^2$ 2.88e-4 7.21e-5 1.80e-5 4.51e-6 1.12e-6 2.77e-7 $\approx$ 2.00 $H^1$ 7.88e-3 3.95e-3 1.97e-3 9.87e-4 4.64e-4 2.10e-4 $\approx$ 1.05 $0.8$ $L^2$ 6.27e-5 1.57e-5 3.94e-6 9.85e-7 2.45e-7 6.07e-8 $\approx$ 2.00 $H^1$ 1.43e-3 7.12e-4 3.56e-4 1.78e-4 8.37e-5 3.78e-5 $\approx$ 1.05
$\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (c) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $T=1$
 $\gamma$ $\text{norm}\backslash m$ $21$ $41$ $81$ $161$ $321$ rate $-0.5$ $L^2$ 4.67e-3 1.73e-3 6.23e-4 2.21e-4 7.68e-5 $\approx$ 1.47 ($1.50$) $H^1$ 3.41e-1 2.43e-1 1.71e-1 1.17e-1 7.71e-2 $\approx$ 0.52 ($0.50$) $0.5$ $L^2$ 4.68e-3 1.73e-3 6.23e-4 2.21e-4 7.68e-5 $\approx$ 1.47 ($1.50$) $H^1$ 3.41e-1 2.43e-1 1.71e-1 1.17e-1 7.72e-2 $\approx$ 0.52 ($0.50$)
 $\gamma$ $\text{norm}\backslash m$ $21$ $41$ $81$ $161$ $321$ rate $-0.5$ $L^2$ 4.67e-3 1.73e-3 6.23e-4 2.21e-4 7.68e-5 $\approx$ 1.47 ($1.50$) $H^1$ 3.41e-1 2.43e-1 1.71e-1 1.17e-1 7.71e-2 $\approx$ 0.52 ($0.50$) $0.5$ $L^2$ 4.68e-3 1.73e-3 6.23e-4 2.21e-4 7.68e-5 $\approx$ 1.47 ($1.50$) $H^1$ 3.41e-1 2.43e-1 1.71e-1 1.17e-1 7.72e-2 $\approx$ 0.52 ($0.50$)
$\| u-u_h \|_{L^2(0,T;\dot H^{p}\left( \Omega \right))}$ for example (c) with $\delta=1$, $h=1/m$, $\tau=10^{-4}$, $T=1$
 $\gamma$ $\text{norm}\backslash m$ $10$ $20$ $40$ $80$ $160$ $320$ rate $-0.5$ $L^2$ 2.51e-4 6.29e-5 1.57e-5 3.93e-6 9.79e-7 2.42e-7 $\approx$ 2.00 $H^1$ 9.33e-3 4.68e-3 2.34e-3 1.17e-3 5.51e-4 2.49e-4 $\approx$ 1.07 $0.5$ $L^2$ 2.06e-4 5.18e-5 1.29e-5 3.23e-6 8.06e-7 1.99e-7 $\approx$ 2.02 $H^1$ 7.12e-3 3.57e-3 1.79e-3 8.93e-4 4.20e-4 1.90e-4 $\approx$ 1.08
 $\gamma$ $\text{norm}\backslash m$ $10$ $20$ $40$ $80$ $160$ $320$ rate $-0.5$ $L^2$ 2.51e-4 6.29e-5 1.57e-5 3.93e-6 9.79e-7 2.42e-7 $\approx$ 2.00 $H^1$ 9.33e-3 4.68e-3 2.34e-3 1.17e-3 5.51e-4 2.49e-4 $\approx$ 1.07 $0.5$ $L^2$ 2.06e-4 5.18e-5 1.29e-5 3.23e-6 8.06e-7 1.99e-7 $\approx$ 2.02 $H^1$ 7.12e-3 3.57e-3 1.79e-3 8.93e-4 4.20e-4 1.90e-4 $\approx$ 1.08
 [1] Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 [2] Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803 [3] Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205 [4] Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927 [5] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [6] Li-Xia Liu, Sanyang Liu, Chun-Feng Wang. Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property. Journal of Industrial and Management Optimization, 2011, 7 (1) : 53-66. doi: 10.3934/jimo.2011.7.53 [7] Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197 [8] Aiting Le, Chenyin Qian. Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022057 [9] Alberto Maspero, Beat Schaad. One smoothing property of the scattering map of the KdV on $\mathbb{R}$. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1493-1537. doi: 10.3934/dcds.2016.36.1493 [10] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 [11] Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657 [12] Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076 [13] Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265 [14] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 [15] Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279 [16] Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 [17] Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156 [18] Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 [19] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [20] Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134

2020 Impact Factor: 1.327