Global Existence for a Parabolic-hyperbolic Free Boundary Problem Modelling Tumor Growth

In this paper we study a free boundary problem modelling tumor growth, proposed by A. Friedman in 2004. This free boundary problem involves a nonlinear second-order parabolic equation describing the diffusion of nutrient in the tumor, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells, respectively. By applying Lp theory of parabolic equations, the characteristic theory of hyperbolic equations, and the Banach fixed point theorem, we prove that this problem has a unique global classical solution.     

On the approximation to solutions of operator equations by the least squares method

We consider the equation Au = f, where A is a linear operator with compact inverse A ^–1 in a separable Hilbert space . For the approximate solution u _n of this equation by the least squares method in a coordinate system {e _k } _k that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A ( (B) = (A)), we give a priori estimates for the asymptotic behavior of the expressions r _n = u _n – u and R _n = Au _n – f as n . A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.__________Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 85–90, 2005Original Russian Text Copyright © by M. L. GorbachukSupported by CRDF and Ukrainian Government Joint Grant UM1-2567-OD03.Translated by V. M. Volosov     

Decay and scattering of small solutions for Rosenau equations

We study the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. For a class of nonlinearity of the perturbation, the global small solution was obtained, and the decay and scattering for small amplitude solution are established.     

A three-dimensional steady-state tumor system

The growth of tumors can be modeled as a free boundary problem involving partial differential equations. We consider one such model and compute steady-state solutions for this model. These solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions. Linear and nonlinear stability for these solutions are determined numerically.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

磁层-电离层耦合过程中等离子体粒子运动的周期轨

运用重合度理论探讨了一类非线性问题的周期解,然后将其应用于磁层-电离层耦合过程中等离子体粒子运动模型的周期解问题的研究,得到了一定条件下该模型存在周期轨的结果. 关键词： 非线性 等离子体 周期解     

服务中断的M/M/1重试排队模型的稳态解

研究服务中断的M/M/1重试排队模型的稳态解,证明当α+μ>λ时0不足该模型主算子的特征值.由此推出该模型不存在稳态解.     

Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints

This paper studies the solution stability of a parametric optimal control problem governed by single linear elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a parametric programming problem and a parametric variational inequality, we obtain sufficient conditions under which the solution map to an elliptic optimal control problem is lower semicontinuous.