A Nonlinear Degenerate Parabolic Equation

ＡＮｏｎｌｉｎｅａｒＤｅｇｅｎｅｒａｔｅＰａｒａｂｏｌｉｃＥｑｕａｔｉｏｎ￥ＪｉａｎｇＹｉｎｇｊｉｅ（ＪｉａｏｚｕｏＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｗｅｓｔｕｄｙａｎｏｎｌｉｎｅｒｄｅｇｅｎｅｒａｔｅｐａｒａｂｏｌｉｃｅ...     

一类带非线性边界条件的非线性抛物方程的解的整体存在性

该文考虑带非线性边界条件的非线性抛物方程的正整体解的存在性与非存在性。通过使用上下解技巧，得到了所有正解整体存在的充分必要条件。作者所构造的上下解具有相同的形式且计算简便。     

一类具有非线性边界条件的拟线性抛物方程解的全局存在性和爆破

研究了一类具有非线性边界条件的拟线性方程组解的整体存在性和爆破.通过构造不同类型的上、下解并利用M-矩阵的基本性质,给出了非负解整体存在性的充要条件.借助这些新结果,给出了Fuiita型临界曲线,把最近的结果推广到了更一般的方程.     

一类具有非线性内部源和边界流的双重退化抛物型方程

This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form (un)t = (|ux|m-1ux)x up in (0, 1) × (0, T) subject to nonlinear boundary source (|ux|m-1ux)(1,t) = uq(1,t), (|ux|m-1ux)(0,t) = 0, and positive initial data u(x,0) = uo(x), where the parameters m, n, p, q ＞ 0.It is proved that the problem possesses global solutions if and only if p ≤ n and q≤min{n, m(n 1)/ m 1}.     

Doubly Degenerate Parabolic Equation with Nonlinear Inner and Boundary Sources

This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form (un)t = (|ux|m-1ux)x up in (0, 1) × (0, T) subject to nonlinear boundary source (|ux|m-1ux)(1,t) = uq(1,t), (|ux|m-1ux)(0,t) = 0, and positive initial data u(x,0) = uo(x), where the parameters m, n, p, q ＞ 0.It is proved that the problem possesses global solutions if and only if p ≤ n and q≤min{n, m(n 1)/ m 1}.     

具有正边界值的一类退化抛物方程组解的爆破与全局有界

该文研究具有正边界值条件的一类非局部退化抛物型方程组.借助于上下解方法和分段函数,获得了方程组解的全局有界与爆破准则.结果表明,正的边界值ε_0在确定方程组解的爆破中起着关键的作用.     

一类退化抛物方程的边值问题

In this paper, we study the boundary value problem (BVP) for a degenerate parabolic equation. By introducing a proper notion of weak solutions, we prove the uniqueness and existence of weak solutions of the problem. The localization of weak solutions is also discussed, which plays a key role in the proof of the uniqueness.     

Regularity Results for a Strongly Degenerate Parabolic Equation

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.     

Source Type Solutions of a Fourth Order Degenerate Parabolic Equation

In this paper, we study a generalized thin film equation which is relevant to capillary driven flows of thin films of power-law fluids. We prove that the generalized thin film equation in dimension d ≥ 2 has a unique C^1 source type radial self-similar nonnegative solution if 0 〈 n 〈 2p - 1 and has no solution of this type if n ≥ 2p - 1.     

Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.     

半无界空间上退缩抛物型方程解的存在性与爆破

讨论半无界空间上退缩抛物型方程解的存在性与爆破性质.证明了在小初值时解是全局存在的,在大初值时局部解在有限时刻发生爆破.     

Traveling Wave Fronts of a Degenerate Parabolic Equation with Non-divergence Form

We study the traveling wave solutions of a nonlinear degenerate parabolic equation with non-divergence form. Under some conditions on the source, we establish the existence, and then discuss the regularity of such solutions.     

Blow Up for Solutions of Nonlinear Doubly Degenerate Parabolic Equation

Ｂｌｏｗ　Ｕｐ　ｆｏｒ　Ｓｏｌｕｔｉｏｎｓ　ｏｆ　Ｎｏｎｌｉｎｅａｒ　Ｄｏｕｂｌｙ　Ｄｅｇｅｎｅｒａｔｅ　Ｐａｒａｂｏｌｉｃ　ＥｑｕａｔｉｏｎＢｌｏｗＵｐｆｏｒＳｏｌｕｔｉｏｎｓｏｆＮｏｎｌｉｎｅａｒＤｏｕｂｌｙＤｅｇｅｎｅｒａｔｅＰａｒａｂｏｌｉｃ...     

一个带非线性项的退化四阶抛物方程弱解的存在性

讨论一个带非线性项的退化四阶抛物方程的初边值问题,在一些初值的假定下,证明了弱解的存在性.     

非局部退化拟线性抛物型方程组解的爆破和整体存在性

该文采用弱上下解方法和正则化技巧,研究了一类非局部退化抛物型方程组解的爆破和整体存在性,给出了爆破指标,并对非退化情形m=n=1,p_1=q_1=0,p_2q_21给出了一致爆破速率.     

一类拟线性退化抛物方程的Cauchy问题

对方程α（xx）u＋uαyu-αtu=f（·,u）,（x,y,t）∈R^2×（0,T）的Cauchy问题,给出了一个整体BV-弱解的定义,证明了该弱解的连续性.     

带有梯度项的退化抛物方程的爆破

考虑了带有梯度项和变指标项的非线性退化抛物方程u_t=△u~m+μ|▽u|~(p(x))(μ0)非负解的爆破性质.使用特征函数方法和不等式技巧,得到了其齐次Dirichlet问题非负解在有限时刻爆破的充分条件.     

双退缩抛物型方程解的一个性质

在Ｑ＝Ｇ×（０，Ｔ）考虑一类双退缩抛物型方程，在满足较一般的结构条件下，证明了如果它的解在抛物边界等于零，那么必是平凡解．     

带非线性边界条件的非线性抛物型方程组

本文讨论带非线性边界条件的抛物型方程组ｕｔ＝Δｕｍ，ｖｔ＝Δｖｍ，ｘ∈Ω，ｔ＞０，ｕｎ＝ｖｐ，ｖｎ＝ｕｑ，ｘ∈Ω，ｔ＞０，ｕ（ｘ，０）＝ｕ０（ｘ）δ＞０，ｖ（ｘ，０）＝ｖ０（ｘ）δ＞０，ｘ∈Ω（Ｉ）解的整体存在性和在有限时刻爆破问题．其中ｍ，ｐ，ｑ＞０，ΩＩＲＮ是有界光滑区域，δ＞０可以充分小．     

一类退化非线性抛物型方程整体解的衰减估计

J.L.L ions用紧致性方法证明了一类退化非线性抛物型方程初边值问题整体解的存在唯一性,但解的衰减性很少有人考虑.应用M.N akao建立的差分不等式研究了整体解的衰减估计.