• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.105-116
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• 2013

In this paper, we consider the positive solution to a Cauchy problem in $\mathbb{B}^N$ of the fast diffusive equation: ${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$, with nontrivial, nonnegative initial data. Here $\frac{2N+m}{N+m+1}$ < $p$ < 2, $q$ > 1 and 0 < $m{\leq}n$ < $qm+N(q-1)$. We prove that $q_c=p-1{\frac{p+n}{N+m}}$ is the critical Fujita exponent. That is, if 1 < $q{\leq}q_c$, then every positive solution blows up in finite time, but for $q$ > $q_c$, there exist both global and non-global solutions to the problem.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.55-66
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• 2014

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation $u_t=div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+a{\int}_{\Omega}u^q(y,t)dy$, 1 < p < 2, in a bounded domain ${\Omega}{\subset}R^N$ with $N{\geq}1$. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in ${\Omega}$ for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of $a{\mu}$, where ${\mu}{\int}_{\Omega}{\phi}^{p-1}(x)dx$ and ${\phi}$ is the unique positive solution of the elliptic problem -div(${\mid}{\nabla}{\phi}{\mid}^{p-2}{\nabla}{\phi}$) = 1, $x{\in}{\Omega}$; ${\phi}(x)$=0, $x{\in}{\partial}{\Omega}$. This is a main difference between equations with local and nonlocal sources.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.4
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• pp.201-210
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• 2010

We propose a new robust and accurate method for the numerical solution of medical image segmentation. The modified Allen-Cahn equation is used to model the boundaries of the image regions. Its numerical algorithm is based on operator splitting techniques. In the first step of the splitting scheme, we implicitly solve the heat equation with the variable diffusive coefficient and a source term. Then, in the second step, using a closed-form solution for the nonlinear equation, we get an analytic solution. We overcome the time step constraint associated with most numerical implementations of geometric active contours. We demonstrate performance of the proposed image segmentation algorithm on several artificial as well as real image examples.