Constructing Solutions of an Ill-Posed Nonlinear Singularly Perturbed Problem for an Equation of Elliptic Type

Mathematical Notes -     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR NONLINEAR THIRD-ORDER ROBIN BOUNDARY VALUE PROBLEM WITH A TURNING POINT

The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear third- order Robin boundary value problem with a turning point.In order to achieve this aim,existence and uniqueness of the solution for third-order nonlinear Robin boundary value problem is derived first based on the upper and lower solutions method under relatively weaker conditions.In this manner,the goal of this paper is gained by applying the existence and uniqueness results mentioned above.     

Hele-Shaw type problems with dynamical boundary conditions

In this paper, we study the nonlinear evolution equation of Hele-Shaw type with dynamical boundary conditions. That is, the equation ut=Δw+f where u∈H(w) and H is the Heaviside function, with boundary condition μ(x,w)t_∂w+k∇w⋅ν=g, where ν denotes the outward normal vector of the fixed boundary of the domain. We prove existence, uniqueness and some qualitative properties of the solution.     

ON DIRICHLET TYPE SPACES AND α-BLOCH SPACES IN THE UNIT BALL OF Cn

In this paper, the authors study the inclusion relations between Dirichlet type spaces DT and α-Bloch spacesβα by means of higher radial derivative. The strictness and the best possibility of the inclusion relations are shown with constructive methods.Furthermore, they sharpen one of the results when T = n, which proves that a conjecture in [7] is true.     

关于《一类奇异边值问题的正解》的注记

文[4]通过构造反例断言文[1]中定理的必要性证明有误,本文首先指出文[4] 的这个断言不正确,然后对文[4]中定理2.1作了本质性的改进.     

A Nonlinear Drift - Diffusion System with Electric Convection Arising in Electrophoretic and Semiconductor Modeling

A multi-dimensional transient drift-diffusion model for (at most) three charged particles, consisting of the continuity equations for the concentrations of the species and the Poisson equation for the electric potential, is considered. The diffusion terms depend on the concentrations. Such a system arises in electrophoretic modeling of three species (neutrally, positively and negatively charged) and in semiconductor theory for two species (positively charged holes and negatively charged electrons). Diffusion terms of degenerate type are also possible in semiconductor modeling. For the initial boundary value problem with mixed Dirichlet - Neumann boundary conditions and general reaction rates, a global existence result is proved. Uniqueness of solutions follows in the Dirichlet boundary case if the diffusion terms are uniformly parabolic or if the initial and boundary densities are strictly positive. Finally, we prove that solutions exist which are positive uniformly in time and globally bounded if the reaction rates satisfy appropriate growth conditions.     

Intrinsic thermal stability and quenching recovery of thin-film superconductors with thermal boundary resistance

The stability behaviour of a thin-film superconductor under a localized release of thermal disturbance is investigated. Two-dimensional conjugate film/substrate conduction equation with anisotropic thermal conductivity of the film, and Joule heat are employed to investigate effects of substrate and thermal properties on the intrinsic stability and quenching recovery. To consider the thermal boundary resistance between film and substrate, an interfacial-layer model (ILM) with very low diffusivity and an acoustic mismatch model (AMM) are employed. Results show that the thermal boundary resistance influences strongly the intrinsic stability. Thermal boundary resistance increases intrinsic stability if the thermal conductivity of the substrate or the disturbance energy is large. Higher Biot numbers and thermal conductivity ratios of film to substrate in longitudinal direction influence stability favorably. We demonstrate also that operation of a film/substrate system, such as YBCO/MgO, is either intrinsically stable or irrecoverably unstable.The authors wish to express their sincere appreciation to Dr. R. C. Chen for his invaluable advice and suggestions during the course of this paper. This research was supported by the National Science Council of the R. O. C. through grant NSC 83-0401-E-009-006. The computations were performed on the IBM ES/9000 at the National Center For High-Performance Computing.     

Viscosity solution theory of a class of nonlinear degenerate parabolic equations II. Lipschitz continuity of free boundary

In [1] we construct a unique bounded Hölder continuous viscosity solution for the nonlinear PDEs with the evolutionp-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate the Lipschitz continuity of the free boundary of viscosity solution and its asymptotic spherical symmetricity, however, this result does not include the anisotropic case.This research is supported by the National Natural Sciences Foundation of China.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.