Heat transfer in composite materials with Stefan–Boltzmann interface conditions

In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan–Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one‐dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Complete solution of the electrostatic equilibrium problem for classical weights

In this paper we complete the solution of the electrostatic equilibrium problem for all classical weight functions. In particular we solve this problem for classical generalized Bessel, Jacobi on (0, +∞) and pseudo-Jacobi weights. In addition we present an elementary unified way of dealing with the electrostatic equilibrium problem, which depends on the weights satisfying the Pearson differential equation.     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

The Solvability for a Class of Nonlinear Integrodifferential Equations of Parabolic Type

In this paper we consider the well-posedness for a class of nonlinear integrodifferential equations of parabolic type. We use integral estimates to deduce an a priori estimate in the classical space C^{2+α,1+\frac{α}{2}}. The existence of the solution is established by means of the continuity method which is similar to a parabolic initial and boundary value problem. Moreover, the continuous dependence upon the data and the uniqueness of the solution are obtained. Finally, the results are generalized into a class of nonlinear integrodifferential systems.     

The Neumann Problem for Parabolic Hessian Quotient Equations

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations.We show that the k-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also solutions of the classical Neumann problem converge to a translating solution.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

Optimal consumption of the finite time horizon Ramsey problem

In this paper, we study the stochastic Ramsey problem related to an economic growth model with the CES production function in a finite time horizon. By changing variables, the Hamilton-Jacobi-Bellman equation associated with this optimization problem is transformed. By the viscosity solution technique, we show the existence of a classical solution of the transformed Hamilton-Jacobi-Bellman equation, and then give an optimal consumption policy of the original problem.     

Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors

In this paper we study a free boundary problem modelling the growth of nonnecrotic tumors. The main trait of this free boundary problem is that it is essentially multidimensional, so that its well-posedness is hard to establish by using the usual methods in the classical theory of free boundary problems. In this paper we use the functional analysis method based on the theory of analytic semigroups to prove that this problem has a unique local solution in suitable function spaces. Continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper.     

带有动力学条件的高维一相Stefan问题

本文我们证明了带有动力学条件的高维一相Ｓｔｅｆａｎ总题局部古典解的存在唯一性。     

The Diffraction Problem and Verigin Problem of Quasilinear Parabolic Equation in Divergence Form for the One-dimensional Case

In this paper, we consider the flow of two immiscible fluids in a onedimensional porous medium (the Verigin problem) and obtain a quasilinear parabolic equation in divergence form with the discontinuous coefficients. We prove first the existence and uniqueness of locally classical solution of the diffraction problem and then prove the existence of local solution of the Verigin problem.     

Evolution of hypersurfaces in $ {\Bbb R}^N$ by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

Evolution of hypersurfaces in \Bbb RN {\Bbb R}^N by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

Muskat Problem with Surface Tension

In this paper, we consider Muskat problem with surface tension at the free boundary. Under the natural conditions, we prove tbe existence of classical solution locally in time by Schauder fixed point theorem.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

On the existence of a global classical solution of the boundary value problem for □u−μu+aum=0 in the interior domain

In this paper we consider the boundary value problem for a semilinear equation □u(t, x)−μu(t, x)+au^m(t, x)=0, μ>0, a∈ℜ in the interior domain. We find a time global classical solution with exponential decay property by using singular hyperbolic equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Mathematical analysis of a continuous version of statistical model for criminal behavior

This paper is concerned with a reaction-advection-diffusion system, which is a continuous version of statistical model for criminal activity proposed by Short et al (Math. Models Methods Appl. Sci., 8 (2008), 1249-1267). From very recently results, it is known that the classical solution of the corresponding initial-boundary value problem exists globally. On the other hand, Rodriguez (Phys. D, 260 (2013), 191-200) asserted the uniform-in-time boundedness of classical solution; however, there is a gap in the proof. In this work, we first solve this open problem by establishing some crucial a priori estimates. Then, invoking the uniform boundedness, we present that the classical solution converges exponentially to the steady state. Our results are consistent with the experiment observation and numerical simulation in the existing literatures.     

Boundary integral method for Stokes flow past a porous body

In this paper we obtain an indirect boundary integral method in order to prove existence and uniqueness of the classical solution to a boundary value problem for the Stokes–Brinkman-coupled system, which describes an unbounded Stokes flow past a porous body in terms of Brinkman's model. Therefore, one assumes that the flow inside the body is governed by the continuity and Brinkman equations. Some asymptotic results in both cases of large and, respectively, of low permeability are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal combinational quota‐share and excess‐of‐loss reinsurance policies in a dynamic setting

In this paper, we describe a large insurance company's surplus by a Brownian motion with positive drift, which is the approximation of a classical risk process. The problem of minimizing the probability of ruin by controlling the combinational quota‐share and excess‐of‐loss reinsurance strategy is considered. We show that the optimal combinational reinsurance strategy must be the pure excess‐of‐loss reinsurance strategy. Moreover, we give an explicit solution for the optimal reinsurance strategy. Copyright © 2006 John Wiley & Sons, Ltd.