Nonlinear Parabolic Equation on Manifolds

Gladson Antunes, Ivo F. Lopez, Maria Darci G. da Silva, Luiz Adauto Medeiros, Angela Biazutti


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).

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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