• 대한수학회지
• /
• 제37권5호
• /
• pp.793-803
• /
• 2000

An impulsive semilinear parabolic equation subject to Robin boundary condition is considered. We prove that for certain classes of impulsive sources and continuous initial data, the solutions of the problem under consideration blow up in the desired time interval.

• Journal of applied mathematics & informatics
• /
• 제32권3_4호
• /
• pp.547-554
• /
• 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.

• 대한수학회보
• /
• 제55권4호
• /
• pp.1161-1178
• /
• 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.

• 충청수학회지
• /
• 제23권4호
• /
• pp.773-784
• /
• 2010

The existence of solutions in Besov spaces for nonlinear heat equations having transcendental nonlinearity: $$\frac{\partial}{{\partial}t}u-{\Delta}u=F(u)$$ is investigated. In particular, it is proved the local existence and blow-up phenomena of the solutions in Besov spaces for nonlinear heat equations corresponding to two transcendental nonlinear functions $F(u){\equiv}{\mid}u{\mid}e^{u^2}$ and $F(u){\equiv}e^u$ of rapid growth.

• 대한수학회논문집
• /
• 제31권3호
• /
• pp.567-573
• /
• 2016

We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove finite time blow-up of the solution.

• 충청수학회지
• /
• 제31권3호
• /
• pp.287-308
• /
• 2018

This paper deals with blow-up phenomena for an initial boundary value problem of a quasilinear parabolic equation with time-dependent coefficient in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions for which the solution u(x, t) exists globally or blows up at some finite time $t^*$. Moreover, some upper and lower bounds for $t^*$ are derived in higher dimensional spaces. Some examples are presented to illustrate applications of our results.

• 대한수학회논문집
• /
• 제23권4호
• /
• pp.529-540
• /
• 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
• /
• 제27권5_6호
• /
• pp.1119-1132
• /
• 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.

• 대한수학회보
• /
• 제55권1호
• /
• pp.267-296
• /
• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• 호남수학학술지
• /
• 제19권1호
• /
• pp.61-69
• /
• 1997