On approximate solutions to some nonlinear diffusion problems

In the paper an approximate solution to a nonlinear diffusion problem in semiconductor production is considered. Some integral operator methods are used to estimate an error between this approximate solution and the exact one.This paper has been written when the author was a Visiting Professor in the Universidad de Extremadura.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Existence of positive solutions for some problems with nonlinear diffusion

In this paper we study the existence of positive solutions for problems of the type

where is the -Laplace operator and . If , such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases , and , respectively. Also, some systems of equations are considered.

     

NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

一类非线性演化方程初值问题的幂级数解

研究了一类非线性演化方程初值问题.通过不变子空间方法,这类初值问题被约化为常微分方程组的初值问题.这类初值问题是适定的.本文给出了这类初值问题关于时间变量t的幂级数解.     

Hypergeometric series solutions of some anticlastic deformation problems

Zusammenfassung Diese Arbeit befasst sich mit der allgemeinen Behandlung antiklastischer Durchbiegungsprobleme. Von Kármáns Gleichung für grosse Durchbiegungen und antiklastische Deformation wurden mit Hilfe der Frobeniusschen Methode gelöst. Für Platten, deren Dicke proportional einer beliebigen Potenz des Abstandes vom Plattenrand ist, wurden Lösungen in Form allgemeiner hypergeometrischer Funktionen ausgearbeitet. Numerische Resultate für antiklastische Deformation werden für verschiedene Plattendicken präsentiert.

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Nomenclature A _i integration constants - a _ir coefficients of the series - b halfwidth of plate - b _i characteristic numbers, functions ofn, defined by Equation (11) - c _ij =b _j –b _i +1 - p _{F q} generalized hypergeometric function, defined by Equation (18) - H harmonic sum, defined by Equation (23) - n thickness variation parameter - R longitudinal radius of curvature - s _i roots of indicial equation - t thickness of plate - t _o thickness of plate aty=0 - u =Y ₁ ^y - w(y) transverse deflection of a bent strip atx=0 - w _c complementary part of solutionw - w _p particular part of solutionw - w _(oy) initial deviation of middle surface of strip fromx-y plane atx=0 - x, y, z rectangular Cartesian coordinates, defined in Figure 1 - y ₁ =1-y/b - =4-2n - =y ₁ d/dy ₁ - _u =u d/du - - Poisson's ratio     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Power series solutions to basic stationary boundary value problems of elasticity

The four basic stationary boundary value problems of elasticity for the Lamé equation in a bounded domain of ³ are under consideration. Their solutions are represented in the form of a power series with non-positive degrees of the parameter =1/(1–2), depending on the Poisson ratio . The coefficients of the series are solutions of the stationary linearized non-homogeneous Stokes boundary value problems. It is proved that the series converges for any values of outside of the minimal interval with the center at the origin and of radiusr1, which contains all of the Cosserat eigenvalues.Dedicated to Prof. I.Gohberg on the occasion of his 70th birthday     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.     

Existence of solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence of positive, negative and sign-changing solutions for some fourth order semilinear elliptic boundary value problems. We present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct the sign-changing solutions.     

Boundary layer solutions for some nonlinear elastic membrane problems

Summary The method of matched asymptotic expansions is used to describe finite axisymmetric deformations of two thin elastic membrane problems: a circular membrane with a small radial traction at the edge, an annular membrane with a small circular hole at the center. In both problems, the surface load is assumed to be an axial pressure. There is a boundary layer at the outer edge in the first problem, and at the hole in the second problem.

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Zusammenfassung Mit Hilfe der in der singulären Störungsrechnung benutzten Methode einer inneren und äußeren asymptotischen Entwicklung werden endliche, axialsymmetrische Deformationen bei zwei Membranproblemen beschrieben: eine dünne elastische Kreismembran mit kleiner radialer Zugspannung am Rande, und eine Kreisringmembran mit einem kleinen konzentrischen Loch. In beiden Problemen wird die Belastung als ein axial gerichteter Oberflächendruck angenommen. Im ersten Problem zeigt die Lösung am äußeren Rand Grenzschichtverhalten, welches im zweiten Problem am Lochrand vorliegt.

     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.