Electron. J. Differential Equations, Vol. 2018 (2018), No. 04, pp. 1-20.

Existence and asymptotic behavior of solutions of the Dirichlet problem for a nonlinear pseudoparabolic equation

Le Thi Phuong Ngoc, Dao Thi Hai Yen, Nguyen Thanh Long

Abstract:
This article concerns the initial-boundary value problem for nonlinear pseudo-parabolic equation
$$\displaylines{ u_{t}-u_{xxt}-(1+\mu (u_{x}))u_{xx}+(1+\sigma (u_{x}))u=f(x,t),\quad 0<x<1,\; 0<t<T, \cr u(0,t)=u(1,t)=0, \cr u(x,0)=\tilde{u}_0(x), }$$
where f, $\tilde{u}_0$, $\mu $, $\sigma $ are given functions. Using the Faedo-Galerkin method and the compactness method, we prove that there exists a unique weak solution u such that $u\in L^{\infty }(0,T;H_0^1\cap H^2)$, $u'\in L^2(0,T;H_0^1)$ and $\| u\| _{L^{\infty }(Q_{T})}\leq \max \{\|\tilde{u}_0\| _{L^{\infty }(\Omega )}$, $\| f\|_{L^{\infty }(Q_{T})}\}$. Also we prove that the problem has a unique global solution with $H^1$-norm decaying exponentially as $t\to +\infty $. Finally, we establish the existence and uniqueness of a weak solution of the problem associated with a periodic condition.

Submitted January 27, 2017. Published January 4, 2018.
Math Subject Classifications: 34B60, 35K55, 35Q72, 80A30.
Key Words: Nonlinear pseudoparabolic equation; asymptotic behavior; exponential decay; periodic weak solution; Faedo-Galerkin method.

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Le Thi Phuong Ngoc
University of Khanh Hoa
01 Nguyen Chanh Str., Nha Trang City, Vietnam
email: ngoc1966@gmail.com
Dao Thi Hai Yen
Department of Natural Science
Phu Yen University
18 Tran Phu Str., Ward 7, Tuy Hoa City, Vietnam
email: haiyennbh@yahoo.com
Nguyen Thanh Long
Department of Mathematics and Computer Science
VNUHCM - University of Science
227 Nguyen Van Cu Str., Dist. 5, HoChiMinh City, Vietnam
email: longnt2@gmail.com

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