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.

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

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 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.

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.

Generalized Solution of A Class of Singularly Perturbed Robin Problems for Nonlinear Reaction Diffusion Equation

A class of singularly perturbed Robin problems for reaction diffusion equation is considered. Under suitable conditions the asymptotic behavior of the generalized solution for the problems are studied.

GENERALIZED SOLUTION OF SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

The singularly perturbed initial boundary value problem for reaction diffusion equation is considered. Under suitable conditions the existence, uniqueness and asymptotic behavior of the generalized solution for the problems are studied.

Generalized Solution of a Kind of Nonparametric Curvature Evolution with Boundary Condition

The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on the reciprocal of the Gauss curvature.

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.

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.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

SOLUTION TO A PARAMETRIC EQUATION WITH GENERALIZED BOUNDARY CONDITION IN TRANSPORT THEORY

This paper deals with the solution of a parametric equation with generalized boundary condiiton in transport theory. It gives the distribution of parameter (so called delta-eigenvalue [1]) with which the equation has non-zero solution. A necessary and sufficient condition for the existence of; he control critical eigenvalue delta0 is established.

Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.

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.

SPIRAL SOLUTION TO THE TWO-DIMENSIONAL TRANSPORT EQUATIONS

The existence of spiral solution for the two-dimensional transport equations is considered in the present paper. Based on the notion of generalized solutions in the sense of Lebesgue-stieltjes integral, the global weak solution of transport equations which includes δ-shocks and vacuum is constructed for some special initial data.

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.