• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1161-1178
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• 2018

In this paper, we consider the Cauchy problem for a class of nonlinear sixth-order wave equation. The global existence and the finite time blow-up for the problem are proved by the potential well method at both low and critical initial energy levels. Furthermore, we present some sufficient conditions on initial data such that the weak solution exists globally at supercritical initial energy level by introducing a new stable set.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1697-1710
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• 2014

The initial-boundary value problem for a class of nonlinear higher-order wave equations system with a damping and source terms in bounded domain is studied. We prove the existence of global solutions. Meanwhile, under the condition of the positive initial energy, it is showed that the solutions blow up in the finite time and the lifespan estimate of solutions is also given.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.529-540
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• 2008

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system, $${u_t}-{\triangle}_{m,p}u=u^{{\alpha}_1}\;{\int}_{\Omega}\;{\upsilon}^{{\beta}_1}\;(x,\;t)dx,\;{\upsilon}_t-{\triangle}_{n,p}{\upsilon}={\upsilon}^{{\alpha}_2}\;{\int}_{\Omega}\;u^{{\beta}_2}\;(x,{\;}t)dx,$$ with homogeneous Dirichlet boundary condition. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depends on the initial data and the relations of the parameters in the system.

• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.547-554
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• 2014

In this paper, we consider the existence and uniqueness of global weak solution for a sixth-order classical surface-diffusion equation in one spatial dimension. Moreover, the regularity and blow-up of solutions are also studied.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1119-1132
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• 2009

In this paper, we consider the quasilinear elliptic system $\\div(|{\nabla}u|^{p-2}{\nabla}u)=u(a_1u^{m1}+b_1(x)u^m+{\delta}_1v^n),\;\\div(|{\nabla}_v|^{q-2}{\nabla}v)=v(a_2v^{r1}+b_2(x)v^r+{\delta}_2u^s)$, in $\Omega$ where m > $m_1$ > p-2, r > $r_1$ > q-, p, q $\geq$ 2, and ${\Omega}{\subset}R^N$ is a smooth bounded domain. By constructing certain super and subsolutions, we show the existence of positive blow-up solutions and give a global estimate.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.201-208
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• 2015

This paper reviews recent mathematical progresses made on the study of the initial-value problem for nonlinear Dirac equations in one space dimension. We also prove the global existence of solutions to some nonlinear Dirac equations and propose a model problem (3.6).

• Journal of the Korean Mathematical Society
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• v.48 no.3
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• pp.513-527
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• 2011

The paper deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical Fujita curve is conjectured with the aid of some new results.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.37-56
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• 2013

This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $$\{u_t=(\mid(u^{m_1})_x{\mid}^{{p_1}^{-1}}(u^{m_1})_x)_x+u^{l_{11}}{{\int_0}^a}v^{l_{12}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T),\\{v_t=(\mid(v^{m_2})_x{\mid}^{{p_2}^{-1}}(v^{m_2})_x)_x+v^{l_{22}}{{\int_0}^a}u^{l_{21}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T)}$$ with nonlinear boundary conditions $u_x{\mid}{_{x=0}}=0$, $u_x{\mid}{_{x=a}}=u^{q_{11}}u^{q_{12}}{\mid}{_{x=a}}$, $v_x{\mid}{_{x=0}}=0$, $v_x|{_{x=a}}=u^{q21}v^{q22}|{_{x=a}}$ and the initial data ($u_0$, $v_0$), where $m_1$, $m_2{\geq}1$, $p_1$, $p_2$ > 1, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$ > 0. Under appropriate hypotheses, the authors establish local theory of the solutions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1599-1619
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• 2018

In this paper we study the initial value problem of the non-relativistic Vlasov-Darwin system with generalized variables (VDG). We first prove local existence and uniqueness of a nonnegative classical solution to VDG in three space variables, and establish the blow-up criterion. Then we show that it converges to the well-known Vlasov-Poisson system when the light velocity c tends to infinity in a pointwise sense.

• Journal of the Korean Mathematical Society
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• v.47 no.6
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• pp.1317-1328
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• 2010

In this paper, we study the properties of positive solutions for the reaction-diffusion equation $u_t$ = $\Delta_u+{\int}_\Omega u^pdx-ku^q$ in $\Omega\times(0,T)$ with nonlocal nonlinear boundary condition u (x, t) = ${\int}_{\Omega}f(x,y)u^l(y,t)dy$ $\partial\Omega\times(0,T)$ and nonnegative initial data $u_0$ (x), where p, q, k, l > 0. Some conditions for the existence and nonexistence of global positive solutions are given.