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 & Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016
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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
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