BLOW-UP TIME OF SOLUTIONS FOR
SOME NONLINEAR PARABOLIC EQUATIONS
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


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DOI: 10.12732/ijam.v29i1.4

Volume: 29
Issue: 1
Year: 2016