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

• * Corresponding author: Nguyen Thanh Long
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.

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

 Citation: • • Figure 1.  Exact solutions.

Figure 2.  Approximate solutions.

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

 $n$ $u_{ex}\left( \frac{4}{5}, t_{n}\right)$ $u\left( \frac{4}{5} , t_{n}\right)$ $\left\vert u_{ex}\left( \frac{4}{5}, t_{n}\right) -u\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 1.54436330E-03}$ ${\small 2.91855517E-03}$ ${\small 1.37419186E-03}$ ${\small 20}$ ${\small 2.82860006E-05}$ ${\small 7.20712002E-05}$ ${\small 4.37851996E-05}$ ${\small 30}$ ${\small 5.18076174E-07}$ ${\small 1.77972692E-06}$ ${\small 1.26165074E-06}$ $n$ $v_{ex}\left( \frac{4}{5}, t_{n}\right)$ $v\left( \frac{4}{5}% , t_{n}\right)$ $\left\vert v_{ex}\left( \frac{4}{5}, t_{n}\right) -v\left( \frac{4}{5}, t_{n}\right) \right\vert$ ${\small 10}$ ${\small 3.86090827E-04}$ ${\small 7.29514168E-04}$ ${\small 3.43423340E-04}$ ${\small 20}$ ${\small 7.07150017E-06}$ ${\small 1.80147701E-05}$ ${\small 1.09432699E-05}$ ${\small 30}$ ${\small 1.29519043E-07}$ ${\small 6.22799676E-06}$ ${\small 4.93280633E-07}$

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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