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| Existence of Weak Solutions for a Kind of Parabolic Steklov Problems |
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| KeyWord:Parabolic problem, global existence, blow-up, asymptotic behavior |
| Author Name | Affiliation | | Anass Lamaizi | Department of Mathematics, Mohammed First University, Oujda, 60000, Morocco | | Abdellah Zerouali | Department of Mathematics, Regional Centre of Trades Education and Training, Oujda, 60000, Morocco | | Omar Chakrone | Department of Mathematics, Mohammed First University, Oujda, 60000, Morocco | | Belhadj Karim | Department of Mathematics, School of Education and Training, Oujda, 60000, Morocco |
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| Abstract: |
| Our focus in this study revolves around investigating the following parabolic problem
$$
\begin{cases}
u_{t}-\Delta u +u=0 \quad ~ \text{ in } ~ \Omega ,~ t>0 , \ \frac{\partial u}{\partial \nu }= g(u) \quad \quad \quad \quad ~~ \text{ on } \partial \Omega ,~ t>0 , \u(x, 0)=u_{0}(x) \quad ~~~ \text{ in } ~ \Omega .
\end{cases}
$$
By using the Galerkin approximation and a family of potential wells, we obtain the existence of global solution and finite time blow-up under some suitable conditions. On the other hand, the results for asymptotic behavior of certain solutions with positive initial energy are also given. |
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