Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

Vo Anh Khoa; Le Thi Phuong Ngoc; Nguyen Thanh Long

doi:10.3934/eect.2019019

Article Contents

2019, Volume 8, Issue 2: 359-395. Doi: 10.3934/eect.2019019

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Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

1.
Institute for Numerical and Applied Mathematics, University of Goettingen, Lotzestraẞe 16-18, 37083 Goettingen, Germany
2.
Department of Mathematics and Computer Science, VNUHCM-University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
3.
Faculty of Sciences, Hasselt University, Campus Diepenbeek, Agoralaan Building D, BE3590 Diepenbeek, Belgium
4.
University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam

^* Corresponding author: Nguyen Thanh Long
^* Corresponding author: Nguyen Thanh Long

Received: January 31, 2018

Revised: June 30, 2018

Early access: March 2019

Published: May 2019

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant no. B2017-18-04. The work of the first author was partly supported by a postdoctoral fellowship of the Research Foundation-Flanders (FWO).

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Abstract

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.

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Mathematics Subject Classification: 35L05, 35L15, 35L20, 35L55, 35L70.

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Figure 1. Exact solutions.

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Figure 2. Approximate solutions.

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Table 1. Numerical results at nodes $ \left( \frac{4}{5} , t_{n}\right) $ for $ n\in\left\{10, 20, 30\right\}. $

$ n $	$ u_{ex}\left( \frac{4}{5}, t_{n}\right) $	$ u\left( \frac{4}{5} , t_{n}\right) $	$ \left\vert u_{ex}\left( \frac{4}{5}, t_{n}\right) -u\left( \frac{4}{5}, t_{n}\right) \right\vert $
$ {\small 10} $	$ {\small 1.54436330E-03} $	$ {\small 2.91855517E-03} $	$ {\small 1.37419186E-03} $
$ {\small 20} $	$ {\small 2.82860006E-05} $	$ {\small 7.20712002E-05} $	$ {\small 4.37851996E-05} $
$ {\small 30} $	$ {\small 5.18076174E-07} $	$ {\small 1.77972692E-06} $	$ {\small 1.26165074E-06} $

$n$	$v_{ex}\left( \frac{4}{5}, t_{n}\right) $	$v\left( \frac{4}{5}% , t_{n}\right) $	$\left\vert v_{ex}\left( \frac{4}{5}, t_{n}\right) -v\left( \frac{4}{5}, t_{n}\right) \right\vert $
${\small 10}$	${\small 3.86090827E-04}$	${\small 7.29514168E-04}$	${\small 3.43423340E-04}$
${\small 20}$	${\small 7.07150017E-06}$	${\small 1.80147701E-05}$	${\small 1.09432699E-05}$
${\small 30}$	${\small 1.29519043E-07}$	${\small 6.22799676E-06}$	${\small 4.93280633E-07}$

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Table 2. Numerical results for the $ l_{\infty } $ norm error $ \mathcal{E}_{N, K} $

$ K $	$ N $	$ \mathcal{E}_{N, K}\left( u\right) $	$ \mathcal{E}_{N, K}\left( v\right) $
$ {\small 50} $	$ {\small 50} $	$ {\small 6.68545424E-03} $	$ {\small 6.68150701E-03} $
$ {\small 100} $	$ {\small 100} $	$ {\small 3.59475057E-03} $	$ {\small 3.59201931E-03} $
$ {\small 150} $	$ {\small 150} $	$ {\small 2.45841870E-03} $	$ {\small 2.45632948E-03} $
$ {\small 200} $	$ {\small 200} $	$ {\small 1.86793338E-03} $	$ {\small 1.86628504E-03} $

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