Authors: ANASS LAMAIZI, ABDELLAH ZEROUALI, OMAR CHAKRONE, BELHADJ KARIM

Abstract: This paper is concerned with the existence and blow-up of solutions to the following linear parabolic equation: $ ~~ u_t - \Delta u +u = 0 \quad \text{ in } \Omega \times (0,T) $, under nonlinear boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^n $, $n \geq 1$, with smooth boundary. We obtain a threshold result for the global existence of solutions, next we shall prove the existence time $T$ of solution is finite when the initial energy satisfies certain condition.

Keywords: Parabolic problem, heat equation, nonlinear boundary condition, blow-up

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