Uniqueness of singular solution of semilinear elliptic equation

In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established.     

A stability analysis for a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis.     

一类奇摄动半线性时滞抛物型偏微分方程的渐近解

文中讨论了一类奇摄动时滞抛物型偏微分方程的初边值问题,得到了其形式渐近展开,证明了奇摄动半线性时滞偏微分方程的极大值原理,从而得到了最大值估计及相应的Schuader估计.在此基础上,得到了柱状区域上解的存在唯一性和渐近解的一致有效性.     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

On a semilinear stochastic partial differential equation with double-parameter fractional noises

We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws.The equation is driven by double-parameter fractional noises.In addition,the existence and moment estimate are also obtained for the density of the law of such a solution.     

Existence of positive solution for singular fractional differential equation

In this paper, we establish the existence of a positive solution to a singular boundary value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

Convergence of the method of characteristics in approximate solution of a semilinear partial differential equation of first order

We construct an approximate solution for an initial boundary-value problem of the formu _t (x, t) + a (x, t) u_x (x, t)=b (x, t, u), u (x, 0)=u₀ (x),u (0,t)=u₁ (t) by the method of characteristics. It is proved that the approximate solution converges to the exact one with rate of convergence of second order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1128–1138, August, 1990.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Overdetermined systems of first order partial differential equations with singular solution

We study overdetermined systems of first order partial differential equations with singular solutions. The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.     

Regularities for semilinear stochastic partial differential equations

In this paper, we study the regularities of solutions to semilinear stochastic partial differential equations in general settings, and prove that the solution can be smooth arbitrarily when the data is sufficiently regular. As applications, we also study several classes of semilinear stochastic partial differential equations on abstract Wiener space, complete Riemannian manifold as well as bounded domain in Euclidean space.     

Local solvability of semilinear partial differential equations

Summary It is proved an abstract theorem for local solvability of semilinear equations with complex coefficients. This result is illustrated by nonlinear perturbations of complex principal type and multiple complex characteristics operators.

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Riassunto Si dimostra un teorema astratto per la risolvibilità locale di equazioni semilineari di coefficienti complessi. Il risultato ottenuto viene illustrato mediante perturbationi nonlineari di operatori di tipo principale complesso e caratteristiche compless multiple.

     

On singular partial differential boundary problems

Summary In the first part, the existence of a differential operator is established that is elliptic inside a domain and singular on the boundary. In the second part the invariance of the essential spectrum is established under a large class of perturbations of the original boundary problem. To Giovanni Sansone on his 70^th birthday. This research has partially been supported by a grant of the United States National Science Foundation.     

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

We deal with the numerical approximation of the problem of local stabilization of Burgers equation. We consider the case when only partial boundary measurements are available. An estimator is coupled with a feedback law in order to stabilize the discretized system. Two different feedback laws are compared. Their performance is analyzed in different domains related to idealized cardiovascular geometries, with increasing complexity.     

Positive solutions of a semilinear elliptic equation with singular nonlinearity

This paper is devoted to the study of positive solutions of the semilinear elliptic equation Δu+K(|x|)u−p=0, x∈Rn with n?3 and p>0. Asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has infinitely many positive solutions with fast growth.     

On the solution of a one-dimensional stochastic differential equation with singular drift coefficient

We determine generalized diffusion coefficients and describe the structure of local times for a process defined as a solution of a one-dimensional stochastic differential equation with singular drift coefficient.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 642–655, May, 2004.