## Asymptotic behavior for numerical solutions of a semilinear parabolic equation with a nonlinear boundary condition

Nabongo Diabate and Théodore K. Boni

### Abstract

This paper concerns the study of the numerical approximation for the following initial-boundary value problem, \begin{eqnarray*} &&u_t=u_{xx}-au^{p},\quad 0 < x < 1,\quad t>0, \\ &&u_x(0,t)=0,\quad u_x(1,t)+bu^{q}(1,t)=0,\quad t>0,\\ &&u(x,0)=u_{0}(x)\geq0,\quad 0\leq x\leq 1, \end{eqnarray*} where $a>0$, $b>0$ and $p>q>1$. We show that the solution of a semidiscrete form of the initial value problem above goes to zero as $t$ approaches infinity and give its asymptotic behavior. We provide some numerical experiments that illustrate our analysis.

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### Key words

semidiscretizations, semilinear parabolic equation, asymptotic behavior, convergence

### AMS subject classifications

35B40, 35B50, 35K60, 65M06

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