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Uniqueness of singular solution of semilinear elliptic equation  [PDF]
Lai Baishun,Luo Qing
Mathematics , 2010,
Abstract: In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established
Asymptotic stability of singular solution to nonlinear heat equation  [PDF]
Dominika Pilarczyk
Mathematics , 2009,
Abstract: In this paper, we discuss the asymptotic stability of singular steady states of the nonlinear heat equation in the weighted Lebesgue norms.
Existence and Uniqueness of Very Singular Solution of a Degenerate Parabolic Equation with Nonlinear Convection  [cached]
Fang ZhongBo,Piao Daxiong,Wang Jian
Boundary Value Problems , 2009,
Abstract: We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: satisfying a specific decay rate: with and . Here and . Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term defined on the half line .
Singular Behavior of the Solution to the Stochastic Heat Equation on a Polygonal Domain  [PDF]
Felix Lindner
Mathematics , 2013,
Abstract: We study the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a bounded polygonal domain O in R^2. It is shown that the solution u can be decomposed into a regular part u_R and a singular part u_S which incorporates the corner singularity functions for the Poisson problem. Due to the temporal irregularity of the noise, both u_R and u_S have negative L_2-Sobolev regularity of order s<-1/2 in time. The regular part u_R admits spatial Sobolev regularity of order r=2, while the spatial Sobolev regularity of u_S is restricted by r<1+\pi/\gamma, where \gamma is the largest interior angle at the boundary of O. We obtain estimates for the Sobolev norm of u_R and the Sobolev norms of the coefficients of the singularity functions. The proof is based on a Laplace transform argument w.r.t. the time variable. The result is of interest in the context of numerical methods for stochastic PDEs.
Existence and Uniqueness of Very Singular Solution of a Degenerate Parabolic Equation with Nonlinear Convection  [cached]
Zhong Bo Fang,Daxiong Piao,Jian Wang
Boundary Value Problems , 2009, DOI: 10.1155/2009/415709
Abstract: We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: (|f′|p 2f′)′+βrf′+αf+(fq)′=0 satisfying a specific decay rate: lim r→∞rα/βf(r)=0 with α:=(p 1)/(pq 2p+2) and β:=(q p+1)/(pq 2p+2). Here p>2 and q>p 1. Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term ut=(|ux|p 2ux)x+(uq)x defined on the half line [0,+∞).
New results on pathwise uniqueness for the heat equation with colored noise  [PDF]
Thomas Rippl,Anja Sturm
Mathematics , 2012,
Abstract: We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in q dimensional space ($q \geq 1$) with a singular correlation kernel. The noise coefficient is H\"older continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the H\"older coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper.
Well posedness and unconditional non uniqueness for a 2D semilinear heat equation  [PDF]
Slim Ibrahim,Rym Jrad,Mohamed Majdoub,Tarek Saanouni
Mathematics , 2010,
Abstract: We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space $H^1(\R^2)$. The uniqueness part is non trivial although it follows Brezis-Cazenave's proof \cite{Br} in the case of monomial nonlinearity in dimension $d\geq3$. Next, %Following Caffarelli-Vasseur \cite{cv}, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative energy blows up in finite time. Lastly, we show that the unconditional result is lost once we slightly enlarge the Sobolev space $H^1(\R^2)$. The proof consists in constructing a singular stationnary solution that will gain some regularity when it serves as initial data in the heat equation. The Orlicz space appears to be appropriate for this result since, in this case, the potential term is only an integrable function.
The Tychonoff uniqueness theorem for the G-heat equation  [PDF]
Qian Lin
Mathematics , 2010, DOI: 10.1007/s11425-011-4184-y
Abstract: In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat equation.
Fractional Integro-Dierential Equations of Mixed Type with Solution Operator and Optimal Controls  [cached]
Wichai Wittayakiattilerd,A. Chonwerayuth
Journal of Mathematics Research , 2011, DOI: 10.5539/jmr.v3n3p140
Abstract: Local and global existence and uniqueness of mild solution for the fractional integro-differential equations of mixed type with delay are proved by using a family of solution operators and the contraction mapping principle on Banach space. The Bolza optimal control problem of a corresponding controlled system is solved. The Gronwall lemma with singular and time lag is derived to be tool for obtaining a priori estimate. In addition, the application to the fractional nonlinear heat equation is shown.
EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR SECOND ORDER SINGULAR DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
奇异二阶微分方程积分边值问题正解的存在唯一性

ZHANG Xingqiu,
张兴秋

系统科学与数学 , 2010,
Abstract: By applying upper and lower solutions method together with maximal principle, the existence and uniqueness of positive solutions for a class of second order singular differential equations with integral boundary value problems are investigated. Sufficient conditions for the existences and uniqueness of $C0,1]$ positive solution as well as $C^10,1]$ are given. The nonlinearity $f(t,x)$ may be singular at $t=0,1$ and $x=0$.
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