Abstract

In this work we investigate the existence and the uniqueness of solution for a nonlinear differential equation of parabolic type on the lateral boundary $\Sigma$ of a cylinder $Q$, cf. (1). An important part of our study is to transform this initial value problem into another one whose differential operator equation is of the type
\[
u_{t}+a\left({\displaystyle\int_{\Gamma}}udx\right)  \mathcal{A}%
u-\Delta_{\Gamma}u+u^{2k+1}=f \,\, \text{on} \,\, \Sigma,
\]
cf. (9), where $k$ is a positive integer. The operator $\mathcal{A}$ acts in Sobolev spaces on $\Gamma$, boundary of $\Omega$. The initial value problem (9) will be studied in Section $4$. Thus, we obtain  the existence and the uniqueness of weak solution for (9).

This work is licensed under a Creative Commons Attribution 4.0 License.