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.

Show me the PDF file (305 KB), TEX file for this article.

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

Return to the EJDE web page