BLOW-UP TIME OF SOLUTIONS FOR
SOME NONLINEAR PARABOLIC EQUATIONS
Diabate Nabongo1, N'Guessan Koffi2, Toure Kidjegbo Augustin3
1,2Alassane Ouattara University of Bouake
UFR SED
Bouake, P.O.Box V18 Bouake, IVORY COST
3National Polytechnic Institute Houphouet Boigny
Department of Mathematics
Yamoussoukro - P.O.Box 1093, IVORY COST
Abstract. In this paper, we consider the following initial-boundary value problem

\begin{displaymath}%\\ \left\{% \begin{array}{ll} \hbox{$u_{t}=\varepsilon \... ...x) \ \ \ \mbox{in} \quad\Omega$,} %\\ \end{array}% \right. \end{displaymath}

where $b\in C^{1}(\mathbb{R}_{+})$, $b(t)\geq b_{0}>0$, $b^{'}(t)\geq0$ for $t\geq 0$, $\varepsilon$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial\Omega$, $f(s)$ is positive, nondecreasing, convex function for positive values of $s$ and $\int^{\infty}\frac{ds}{f(s)}<\infty$. We show that if $\varepsilon$ is small enough, the solution $u$ of the above problem blows up in a finite time and its blow-up time tends to the one of the solution of the following differential equation

\begin{displaymath}%\\ \left\{% \begin{array}{ll} \hbox{$\alpha^{'}(t)=b(t)f... ...} \\ [4pt] \hbox{$\alpha(0)=M$,} %\\ \end{array}% \right. \end{displaymath}

as $\varepsilon$ goes to zero, where $M=\sup_{x\in \Omega}u_{0}(x)$.

Finally, we give some numerical results to illustrate our analysis.

AMS Subject Classification: 35B40, 35B50, 35K60, 65M06
Key Words and Phrases: blow-up, nonlinear parabolic equation, numerical blow-up time, finite difference, convergence


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DOI: 10.12732/ijam.v29i1.4

Volume: 29
Issue: 1
Year: 2016