# American Institute of Mathematical Sciences

June  2019, 8(2): 359-395. doi: 10.3934/eect.2019019

## 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

Received  February 2018 Revised  July 2018 Published  June 2019 Early access  March 2019

Fund Project: 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).

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.

Citation: Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019
##### References:
 [1] C. O. Alves, M. M. Cavalcanti, V. N. D. Cavalcanti, M. A. Rammaha and D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 583-608.  doi: 10.3934/dcdss.2009.2.583. [2] D. D. Ang and A. P. N. Dinh, Mixed problem for some semilinear wave equation with a nonhomogeneous condition, Nonlinear Analysis, 12 (1988), 581-592.  doi: 10.1016/0362-546X(88)90016-8. [3] M. M. Cavalcanti, V. N. Domingos, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term, Nonlinear Analysis, 38 (1999), 281-294.  doi: 10.1016/S0362-546X(98)00195-3. [4] M. M. Cavalcanti, V. N. Domingos and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Applied Mathematics and Computation, 150 (2004), 439-465.  doi: 10.1016/S0096-3003(03)00284-4. [5] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, The University of Chicago Press, 1988. [6] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, Journal of Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051. [7] Y. Guo, Systems of Nonlinear Wave Equations with Damping and Supercritical Sources, Ph.D thesis, University of Nebraska-Lincoln, 2012. [8] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Archive for Rational Mechanics and Analysis, 100 (1988), 191-206.  doi: 10.1007/BF00282203. [9] J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1989. doi: 10.1137/1.9781611970821. [10] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, 1st edition, Volume I: Ordinary Differential Equations, Academic Press , 1969. [11] J. L. Lions, Quelques Méthodes de Ré Solution Des Problèmes Aux Limites Nonlinéaires, Dunod; Gauthier Villars, Paris, 1969. [12] N. T. Long and L. T. P. Ngoc, On a nonlinear wave equation with boundary conditions of two-point type, Journal of Mathematical Analysis and Applications, 385 (2012), 1070-1093.  doi: 10.1016/j.jmaa.2011.07.034. [13] L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On anonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Analysis, 70 (2009), 3943-3965.  doi: 10.1016/j.na.2008.08.004. [14] L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences, 37 (2014), 464-487.  doi: 10.1002/mma.2803. [15] L. T. P. Ngoc, C. H. Hoa and N. T. Long, Existence, blow-up, and exponential decay estimates for a system of semilinear wave equations associated with the helical flows of Maxwell fluid, Mathematical Methods in the Applied Sciences, 39 (2016), 2334-2357.  doi: 10.1002/mma.3643. [16] M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic Journal of Differential Equations, 38 (2002), 1-17. [17] L. X. Truong, L. T. P. Ngoc, A. P. N. Dinh and N. T. Long, Existence, blow-up and exponential decay estimates for a nonlinear wave equations with nonlinear boundary conditions of two-point type, Nonlinear Analysis, 74 (2011), 6933-6949.  doi: 10.1016/j.na.2011.07.015. [18] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasgow Mathematical Journal, 44 (2002), 375-395.  doi: 10.1017/S0017089502030045. [19] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Communications in Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.

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##### References:
 [1] C. O. Alves, M. M. Cavalcanti, V. N. D. Cavalcanti, M. A. Rammaha and D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 583-608.  doi: 10.3934/dcdss.2009.2.583. [2] D. D. Ang and A. P. N. Dinh, Mixed problem for some semilinear wave equation with a nonhomogeneous condition, Nonlinear Analysis, 12 (1988), 581-592.  doi: 10.1016/0362-546X(88)90016-8. [3] M. M. Cavalcanti, V. N. Domingos, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term, Nonlinear Analysis, 38 (1999), 281-294.  doi: 10.1016/S0362-546X(98)00195-3. [4] M. M. Cavalcanti, V. N. Domingos and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Applied Mathematics and Computation, 150 (2004), 439-465.  doi: 10.1016/S0096-3003(03)00284-4. [5] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, The University of Chicago Press, 1988. [6] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, Journal of Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051. [7] Y. Guo, Systems of Nonlinear Wave Equations with Damping and Supercritical Sources, Ph.D thesis, University of Nebraska-Lincoln, 2012. [8] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Archive for Rational Mechanics and Analysis, 100 (1988), 191-206.  doi: 10.1007/BF00282203. [9] J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1989. doi: 10.1137/1.9781611970821. [10] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, 1st edition, Volume I: Ordinary Differential Equations, Academic Press , 1969. [11] J. L. Lions, Quelques Méthodes de Ré Solution Des Problèmes Aux Limites Nonlinéaires, Dunod; Gauthier Villars, Paris, 1969. [12] N. T. Long and L. T. P. Ngoc, On a nonlinear wave equation with boundary conditions of two-point type, Journal of Mathematical Analysis and Applications, 385 (2012), 1070-1093.  doi: 10.1016/j.jmaa.2011.07.034. [13] L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On anonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Analysis, 70 (2009), 3943-3965.  doi: 10.1016/j.na.2008.08.004. [14] L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences, 37 (2014), 464-487.  doi: 10.1002/mma.2803. [15] L. T. P. Ngoc, C. H. Hoa and N. T. Long, Existence, blow-up, and exponential decay estimates for a system of semilinear wave equations associated with the helical flows of Maxwell fluid, Mathematical Methods in the Applied Sciences, 39 (2016), 2334-2357.  doi: 10.1002/mma.3643. [16] M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic Journal of Differential Equations, 38 (2002), 1-17. [17] L. X. Truong, L. T. P. Ngoc, A. P. N. Dinh and N. T. Long, Existence, blow-up and exponential decay estimates for a nonlinear wave equations with nonlinear boundary conditions of two-point type, Nonlinear Analysis, 74 (2011), 6933-6949.  doi: 10.1016/j.na.2011.07.015. [18] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasgow Mathematical Journal, 44 (2002), 375-395.  doi: 10.1017/S0017089502030045. [19] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Communications in Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.
Exact solutions.
Approximate solutions.
Numerical results at nodes $\left( \frac{4}{5} , t_{n}\right)$ for $n\in\left\{10, 20, 30\right\}.$
 $n$ $u_{ex}\left( \frac{4}{5}, t_{n}\right)$ $u\left( \frac{4}{5} , t_{n}\right)$ $\left\vert u_{ex}\left( \frac{4}{5}, t_{n}\right) -u\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 1.54436330E-03}$ ${\small 2.91855517E-03}$ ${\small 1.37419186E-03}$ ${\small 20}$ ${\small 2.82860006E-05}$ ${\small 7.20712002E-05}$ ${\small 4.37851996E-05}$ ${\small 30}$ ${\small 5.18076174E-07}$ ${\small 1.77972692E-06}$ ${\small 1.26165074E-06}$ $n$ $v_{ex}\left( \frac{4}{5}, t_{n}\right)$ $v\left( \frac{4}{5}% , t_{n}\right)$ $\left\vert v_{ex}\left( \frac{4}{5}, t_{n}\right) -v\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 3.86090827E-04}$ ${\small 7.29514168E-04}$ ${\small 3.43423340E-04}$ ${\small 20}$ ${\small 7.07150017E-06}$ ${\small 1.80147701E-05}$ ${\small 1.09432699E-05}$ ${\small 30}$ ${\small 1.29519043E-07}$ ${\small 6.22799676E-06}$ ${\small 4.93280633E-07}$
 $n$ $u_{ex}\left( \frac{4}{5}, t_{n}\right)$ $u\left( \frac{4}{5} , t_{n}\right)$ $\left\vert u_{ex}\left( \frac{4}{5}, t_{n}\right) -u\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 1.54436330E-03}$ ${\small 2.91855517E-03}$ ${\small 1.37419186E-03}$ ${\small 20}$ ${\small 2.82860006E-05}$ ${\small 7.20712002E-05}$ ${\small 4.37851996E-05}$ ${\small 30}$ ${\small 5.18076174E-07}$ ${\small 1.77972692E-06}$ ${\small 1.26165074E-06}$ $n$ $v_{ex}\left( \frac{4}{5}, t_{n}\right)$ $v\left( \frac{4}{5}% , t_{n}\right)$ $\left\vert v_{ex}\left( \frac{4}{5}, t_{n}\right) -v\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 3.86090827E-04}$ ${\small 7.29514168E-04}$ ${\small 3.43423340E-04}$ ${\small 20}$ ${\small 7.07150017E-06}$ ${\small 1.80147701E-05}$ ${\small 1.09432699E-05}$ ${\small 30}$ ${\small 1.29519043E-07}$ ${\small 6.22799676E-06}$ ${\small 4.93280633E-07}$
Numerical results for the $l_{\infty }$ norm error $\mathcal{E}_{N, K}$
 $K$ $N$ $\mathcal{E}_{N, K}\left( u\right)$ $\mathcal{E}_{N, K}\left( v\right)$ ${\small 50}$ ${\small 50}$ ${\small 6.68545424E-03}$ ${\small 6.68150701E-03}$ ${\small 100}$ ${\small 100}$ ${\small 3.59475057E-03}$ ${\small 3.59201931E-03}$ ${\small 150}$ ${\small 150}$ ${\small 2.45841870E-03}$ ${\small 2.45632948E-03}$ ${\small 200}$ ${\small 200}$ ${\small 1.86793338E-03}$ ${\small 1.86628504E-03}$
 $K$ $N$ $\mathcal{E}_{N, K}\left( u\right)$ $\mathcal{E}_{N, K}\left( v\right)$ ${\small 50}$ ${\small 50}$ ${\small 6.68545424E-03}$ ${\small 6.68150701E-03}$ ${\small 100}$ ${\small 100}$ ${\small 3.59475057E-03}$ ${\small 3.59201931E-03}$ ${\small 150}$ ${\small 150}$ ${\small 2.45841870E-03}$ ${\small 2.45632948E-03}$ ${\small 200}$ ${\small 200}$ ${\small 1.86793338E-03}$ ${\small 1.86628504E-03}$
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