On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition

Abstract

We study the following initial-boundary value problem: 1 $$ \left \{ \textstyle\begin{array}{l} u_{t}-(\mu+\alpha\frac{\partial}{\partial t}) ( \frac{\partial ^{2}u}{\partial x^{2}}+\frac{1}{x}\frac{\partial u}{\partial x} ) +f(u)=f_{1}(x,t),\quad 1 0, u_{x}(1,t)=h_{1}u(1,t)+g_{1}(t),\qquad u(R,t)=g_{R}(t), u(x,0)=\tilde{u}_{0}(x),\end{array}\displaystyle \right . $$ where $\mu>0$ , $\alpha>0$ , $h_{1}\geq0$ , $R>1$ are given constants and f, $f_{1}$ , $g_{1}$ , $g_{R}$ , $\tilde{u}_{0}$ are given functions. First, we use the Galerkin and compactness method to prove the existence of a unique weak solution $u(t)$ of Problem (1) on $(0,T)$ , for every $T>0$ . Next, we study the asymptotic behavior of the solution $u(t)$ as $t\rightarrow+\infty$ . Finally, we prove the existence and uniqueness of a weak solution of Problem (1)1,2 associated with a ‘ $(N+1)$ -points condition in time’ case, 2 $$ u(x,0)=\sum_{i=1}^{N}\eta_{i}u(x,T_{i}), $$ where $(T_{i},\eta_{i})$ , $i=1,\ldots,N$ , are given constants satisfying $$ 0< T_{1}< T_{2}< \cdots< T_{N-1}< T_{N}\equiv T, \qquad \sum_{i=1}^{N} \vert \eta_{i}\vert \leq1. $$