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Chunlai Mu,Dengming Liu,Shouming Zhou 대한수학회 2010 대한수학회지 Vol.47 No.6
In this paper, we study the properties of positive solutions for the reaction-diffusion equation [수식] with nonlocal nonlinear boundary condition u (x, t) =∫Ω∫ (x, y) ul (y, t)dy on □Ω×(0, T) and nonnegative initial data u0 (x), where p, q, k, l > 0. Some conditions for the existence and nonexistence of global positive solutions are given.
Mu, Chunlai,Liu, Dengming,Zhou, Shouming Korean Mathematical Society 2010 대한수학회지 Vol.47 No.6
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.
Critical Fujita exponent for a fast diffusive equation with variable coefficients
Zhongping Li,Chunlai Mu,Wanjuan Du 대한수학회 2013 대한수학회보 Vol.50 No.1
Abstract. In this paper, we consider the positive solution to a Cauchy problem in RN of the fast diffusive equation: [수식], with nontrivial, nonnegative initial data. Here [수식}. We prove that [수식] is the critical Fujita exponent. That is, if 1 < q ≤ qc, then every positive solution blows up in finite time, but for q > qc, there exist both global and non-global solutions to the problem.
Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source
Jun Zhou,Chunlai Mu 대한수학회 2009 대한수학회보 Vol.46 No.6
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∈(0,+∞) with q1 < q2. In other words, when q belongs to different intervals (0, q_1), (q_1, q_2), (q2,+∞), 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∈(q_2,+∞), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval (q_1,+∞), while for q ∈ (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 λ will play an important role. In other words, when λ belongs to different interval (0, λ_1) or (λ_1,+∞), where λ_1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties. 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∈(0,+∞) with q1 < q2. In other words, when q belongs to different intervals (0, q_1), (q_1, q_2), (q2,+∞), 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∈(q_2,+∞), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval (q_1,+∞), while for q ∈ (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 λ will play an important role. In other words, when λ belongs to different interval (0, λ_1) or (λ_1,+∞), where λ_1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.
CRITICAL EXPONENTS FOR A DOUBLY DEGENERATE PARABOLIC SYSTEM COUPLED VIA NONLINEAR BOUNDARY FLUX
Mi, Yongsheng,Mu, Chunlai,Chen, Botao Korean Mathematical Society 2011 대한수학회지 Vol.48 No.3
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.
BLOW-UP FOR A NON-NEWTON POLYTROPIC FILTRATION SYSTEM WITH NONLINEAR NONLOCAL SOURCE
Zhou, Jun,Mu, Chunlai Korean Mathematical Society 2008 대한수학회논문집 Vol.23 No.4
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.
CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
Li, Zhongping,Mu, Chunlai,Du, Wanjuan Korean Mathematical Society 2013 대한수학회보 Vol.50 No.1
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.
CRITICAL BLOW-UP AND EXTINCTION EXPONENTS FOR NON-NEWTON POLYTROPIC FILTRATION EQUATION WITH SOURCE
Zhou, Jun,Mu, Chunlai Korean Mathematical Society 2009 대한수학회보 Vol.46 No.6
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.
CRITICAL EXPONENTS FOR A DOUBLY DEGENERATE PARABOLIC SYSTEM COUPLED VIA NONLINEAR BOUNDARY FLUX
Yongsheng Mi,Chunlai Mu,Botao Chen 대한수학회 2011 대한수학회지 Vol.48 No.3
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.
BLOW-UP RATE ESTIMATES FOR A SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH ABSORPTION
Xiang, Zhaoyin,Chen, Qiong,Mu, Chunlai Korean Mathematical Society 2007 대한수학회지 Vol.44 No.4
In this note, we consider a system of two reaction-diffusion equations with absorption, under homogeneous Dirichlet boundary. Using scaling methods, we establish the blow-up rate estimates.