A PRIORI ESTIMATE FOR MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions

EXISTENCE OF GENERALIZED SOLUTIONS TO A HYPERBOLIC MODEL OF COMBUSTION

In this paper we obtain the existence of the generalized solutions to the Cauchy problem for a model of combustion provided that the function f is of nonconvexity and initial values lie in the bounded, measurable class.

MAXIMUM PRINCIPLES FOR GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Under the assumption that the growth order of the free term to satisfy the natural growth condition with respect to gradient of the generalized solutions, the maximum principle is proved for the bounded generalized solution of quasi_linear elliptic equations.

A PRIORI ESTIMATES TO THE MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH ANISOTROPIC GROWTH CONDITIONS

In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.

Existence, Uniqueness and Blow-up of Generalized Solutions to General Nonlinear Filtration Equations

In this paper, we consider nonnegative solutions to Cauchy problem for the general nonlinear filtration equations ut -Dj （α^ij （x, t, u）Diψ（u）） ＋b^i （t, u）Diu＋C（x, t, u） = 0, and obtain the existence, uniqueness and blow-up in finite time of these solutions under some structure conditions.

Generalized solutions to the Benjamin-Ono equations in sense of Colombeau

This paper discusses the existence and uniqueness of the generalized solution in the sense of Colombeau to the Benjamin-Ono (B-O) equation and the relationship between the new generalized solution and the classical solution.

Generalized Dirac Equation with Vector Structural Coefficient and Its Generalized Solutions in Biquaternions Algebra

On the base of differential biquatemions algebra and theories of generalized functions the biquaternionic wave equation of general type is considered under vector representation of its structural coefficient. Its generalized decisions in the space of tempered generalized functions are constructed. The elementary twistors and twistor fields are built and their properties are investigated. Introduction. The proposed by V.P. Hamilton quatemions algebra [1] and its complex extension - biquaternions algebra are very convenient mathematical tool for the description of many physical processes. At presence these algebras have been actively used in in the work of various authors to solve a number of problems in electrodynamics, quantum mechanics, solid mechanics and field theory. The properties of these algebras are actively studied in the framework of the theory of Clifford algebras. In the papers [2, 3] the differential algebra of biquatemions has been elaborated for construction of generalized solutions of the biquaternionic wave （biwave） equations. The particular types of biwave equations were considered, which are equivalent to the systems of Maxwell and Dirac equations and their generalizations, their biquaternionic decisions also were constructed. Here the biwave equation is considered with vector structural coefficient which is biquaternionic generalization of Dirac equations. Their generalized solutions in the space of tempered distributions are defined and their properties are researched.

THE HOLDER CONTINUITY OF GENERALIZED SOLUTIONS OF A CLASS QUASILINEAR PARABOLIC EQUATIONS

Let A and B satisfy the structural conditions (2), the local Holder continuity interior to Q = G X (0, T) is proved for the generalized solutions of quasilinear parabolic equations as follows: u(t) - divA(x, t, u, del u) + B(x, t, U, del u) = 0.

The Generalized Solutions of One Order Periodic Boundary Value Problems in Banach Space

In this paper, the authors study the periodic boundary value problems of a class of nonlinear integro-differential equations of mixed type in Banach space with Caratheodory＇s conditions. We arrive at the conclusion of the existence of generalized solutions between general- ized upper and lower solutions, and develop the monotone iterative technique to find generalized extremal solutions as limits of monotone solution sequences in Banach space.

SOME PROPERTIES OF GENERALIZED SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS

In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth order of B(x, u, ζ) with respect to ζ is supercritical. Many aspects can be solved satisfactorily.

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie--Biicklund symmetries, are characterized. To construct functionally gener- alized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Bcklund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.

Optimal Conditions on Generalized Solutions of Coupled Nonlinear Diffusion Systems

This paper studies coupled nonlinear diffusion equations with more general nonlinearities, subject to homogeneous Neumann boundary conditions. The necessary and sufficient conditions are obtained for the existence of generalized solutions of the system, which extend the known results for nonlinear diffusion systems with more special nonlinearities.

A CLASS OF SINGULARLY PERTURBED GENERALIZED BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR ELLIPTIC EQUATION OF HIGHER ORDER

The singularly perturbed generalized boundary value problems far the quasi- linear elliptic equation of higher order are considered. Under suitable conditions, the existence, uniqueness and asymptotic behavior of the generalized solution for the Dirichlet problems are studied.

UNIQUENESS OF GENERALIZED SOLUTION FOR THE CAUCHY PROBLEM OF TRANSPORTATTION EQUATIONS

In this paper, we prove the uniqueness of generalized solution defined by Lebesgue-Stieltjes integral for the Cauchy problem of transportation equations. Our results are based on the discussions for linear system with discontinuous coefficient.

WEAK SOLUTIONS OF MONGE-AMPRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

This paper concerns the weak solutions of some Monge-Amp^re type equa- tions in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPèRE EQUATION

In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.