Space-time Estimates for Parabolic Type Operator and Application to Nonlinear Parabolic Equations

In this present paper we establish space-time estimates of solutions for linear parabolic type equations based on classical multipliers theory or operator semigroup theory. According to space-time estimates we first construct suitable work space L^q(0, T; L^P), moreover we study the Cauchy problem and initial boundary value problem for semilinear parabolic equation in L^q(0, T; L^P) type space.     

Threshold Results for Semilinear Parabolic Equations with Nonlinear Boundary Conditions

This paper deals with the following semilinear parabolic equations with nonlinear boundary conditions u_t - Δu = f(u) - λu,x ∈ Ω, t > 0 \frac{∂u}{∂n} = g(u), \qquad x ∈ ∂Ω, t > 0 It is proved that every positive equilibrium solution is a threshold.     

带双参数的半线性椭圆方程正解存在与不存在性以及一个预值结论

研究带有双参数的半线性椭圆方程在Robin边界下正解的存在与不存在性.证明了对任意的边界参数c(0c∞),存在λ*=λ*(c)∞,当0λ≤λ*,方程存在一个最小解u_λ,而任意其它的解是对应抛物方程整体解存在与不存在的一个预值.     

抛物型方程初边值问题解的最大模估计

通过构造合适函数变换的方法,建立n维抛物型方程第二初边值问题古典解的最大模估计,并由此得到该定解问题解的唯一性和稳定性.     

一类强耦合抛物型方程组初边值问题解的整体存在性

In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbedding theorem, we obtain the global existence of solutions.     

The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0     

Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

带非局部源的退化奇异半线性抛物方程的爆破

本文研究带齐次Dirichlet边界条件的非局部退化奇异半线性抛物方程ut-(xαux)x=∫0af(u)dx在(0,a)×(0,T)内正解的爆破性质,建立了古典解的局部存在性与唯一性.在适当的假设条件下,得到了正解的整体存在性与有限时刻爆破的结论.本文还证明了爆破点集是整个区域,这与局部源情形不同.进而,对于特殊情形:f(u)=up,p>1及,f(u)=eu,精确地确定了爆破的速率.     

具非线性扩散项的脉冲抛物方程的强迫振动性

研究一类具非线性扩散项的脉冲抛物型方程解的强迫振动性问题,借助平均技巧和脉冲微分不等式,建立了该类方程在Dirichlet边值条件下所有解强迫振动的若干新的充分判据.结论充分表明振动是由脉冲效应引起的.     

Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations

In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equations. Accepted 17 March 1997     

Blow Up for Solutions of Nonlinear Doubly Degenerate Parabolic Equation

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

一类非线性中立型抛物微分方程边值问题解的振动性

讨论了一类非线性中立型抛物微分方程,得到了该类方程的两类边值问题解振动的充分条件.     

半线性抛物方程初边值问题解的Blow up

本文考虑了一类半线性抛物型方程的第三类非线性初边值问题,在某些假设条件下,证明了其解在有限时间内Blow up.     

The First Boundary Value Problem for General Parabolic Monge-Ampere Equation

In this note we consider the first boundary value problem for a general parabolic Monge-Ampere equation u_t - log det(D_{ij}u) = f(x, t, u,D_2u) in Q, \quad u = φ(x, t) on ∂, Q It is proved that there exists a unique convex in x solution to the problem from C^{1+β,2+β/2}(\overline{Q}) under certain structure aod smoothness conditions (H3) - (H7).     

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

0IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblemforthefailliliarequationwherefiisaboundeddomaininR"withsmoothboulldaryoff,p22isacollstantalldlp(z,u)l5of'(tl" 'forsomea20andc>0.FOrp(x,ti)=Itll"'u,theauthorsofpaper[4,7llolwereillterestedillllollllegativesolutionandhadobtainedfollowillgresults(alsosee[12]).1)If25a 2相似文献   

含奇异项的半线性抛物方程组的Cauchy问题

本文考察含奇异项的半线性抛物方程组的Ｃａｕｃｈｙ问题,算出了该问题的猝灭临界指标和猝灭临界维数．     

带奇性的非线性抛物方程边值问题

本文讨论非线性奇异抛物方程第一边值问题解的存在性、唯一性、稳定性以及当t充分大时解的渐近性态．利用先验估计的方法得到：存在唯一的光滑正解，解在L^1范数意义下连续依赖于初值．t充分大时，||u-u↑-||L^2收敛于一个常数．     

Threshold Behavior for Second Initial Boundary Value Problem of F-N Equations

Under biologically reasonable assumptions the threshold behavior for second initial boundary value problem of F-N equation is proved.     

Uniqueness of Solutions for Semiconductor Equations with Avalanche Term

In this paper, we consider the initial and mixed boundary value problems for the semiconductor equations with avalanche term, the uniqueness of the weak solution for the semiconductor equation has been proved.