• Bulletin of the Korean Mathematical Society
• /
• v.46 no.6
• /
• pp.1159-1173
• /
• 2009

This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents $q_1,\;q_2\;{\in}\;(0,+{\infty})$) with $q_1\;{<}\;q_2$. In other words, when q belongs to different intervals (0, $q_1),\;(q_1,\;q_2),\;(q_2,+{\infty}$), the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval (0, $q_2$]. However, when q ${\in}\;(q_2,+{\infty}$), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval ($q_1,+{\infty}$), while for q ${\in}\;(0,\;q_1$), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = $q_1$ is concerned, the other parameter ${\lambda}$ will play an important role. In other words, when $\lambda$ belongs to different interval (0, ${\lambda}_1$) or (${\lambda}_1$,+${\infty}$), where ${\lambda}_1$ is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

• Journal of applied mathematics & informatics
• /
• v.28 no.5_6
• /
• pp.1171-1184
• /
• 2010

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for nonlinear m-point boundary value problem for the following second-order dynamic equations on time scales $({\phi}(u^{\Delta}))^{\nabla}+a(t)f(t,\;u(t))=0$, t $\in$ (0, T), $u(0)=\sum\limits^{m-2}_{i=1}a_iu(\xi_i)$, $\phi(u^{\Delta}(T))=\sum\limits^{m-2}_{i=1}b_i{\phi}(u^{\Delta}(\xi_i))$, where $\phi$ : R $\rightarrow$ R is an increasing homeomorphism and positive homomorphism and ${\phi}(0)=0$. In [27], we obtained the existence results of the above problem for an increasing homeomorphism and positive homomorphism with sign changing nonlinearity. The purpose of this paper is to supplement with a result in the case when the nonlinear term f is nonnegative, and the most point we must point out for readers is that there is only the p-Laplacian case for increasing homeomorphism and positive homomorphism due to the sign restriction. As an application, one example to demonstrate our results are given.