English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study the following initial-boundary value problem: 1{ut−(μ+α∂∂t)(∂2u∂x2+1x∂u∂x)+f(u)=f1(x,t),10,ux(1,t)=h1u(1,t)+g1(t),u(R,t)=gR(t),u(x,0)=u˜0(x),$$ \left \{ \textstyle\begin{array}{l} u_{t}-(\mu+\alpha\frac{\partial}{\partial t}) ( \frac{\partial ^{2}u}{\partial x^{2}}+\frac{1}{x}\frac{\partial u}{\partial x} ) +f(u)=f_{1}(x,t),\quad 1< x< R, t>0, \\ u_{x}(1,t)=h_{1}u(1,t)+g_{1}(t),\qquad u(R,t)=g_{R}(t), \\ u(x,0)=\tilde{u}_{0}(x),\end{array}\displaystyle \right . $$ where μ>0$\mu>0$, α>0$\alpha>0$, h1≥0$h_{1}\geq0$, R>1$R>1$ are given constants and f, f1$f_{1}$, g1$g_{1}$, gR$g_{R}$, u˜0$\tilde{u}_{0}$ are given functions. First, we use the Galerkin and compactness method to prove the existence of a unique weak solution u(t)$u(t)$ of Problem (1) on (0,T)$(0,T)$, for every T>0$T>0$. Next, we study the asymptotic behavior of the solution u(t)$u(t)$ as t→+∞$t\rightarrow+\infty$. Finally, we prove the existence and uniqueness of a weak solution of Problem (1)1,2 associated with a ‘(N+1)$(N+1)$-points condition in time’ case, 2u(x,0)=∑i=1Nηiu(x,Ti),$$ u(x,0)=\sum_{i=1}^{N}\eta_{i}u(x,T_{i}), $$ where (Ti,ηi)$(T_{i},\eta_{i})$, i=1,…,N$i=1,\ldots,N$, are given constants satisfying 0

On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition