infinitely many solutions for a hemivariational inequality involving the p x laplacian abstract this paper deals with a hemivariational inequality involving the p x laplacian by using nonsmooth variational methods and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct non negative solutions of the problem in both cases one at the origin and the other at infinity under suitable hypotheses

a poisson equation formulation for pressure calculations in penalty finite element models for viscous incompressible flows the calculation of pressures when the penalty function approximation is used in finite element solutions of laminar incompressible flows is addressed a poisson equation for the pressure is formulated that involves third derivatives of the velocity field the second derivatives appearing in the weak formulation of the poisson equation are calculated from the c0 velocity approximation using a least squares method the present scheme is shown to be efficient free of spurious oscillations and accurate examples of applications are given and compared with results obtained using mixed formulations

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

infinitely many solutions for a p x laplacian equation in rn abstract this paper deals with a p x laplacian equation in r n by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose w 1 p x r n norms tend to zero to infinity respectively under weaker hypotheses about nonlinearity at zero at infinity respectively

local boundedness of general minimizers with nonstandard growth abstract we prove the local boundedness of local minimizers of f x u u d x where f is subject to the general structural conditions z p x b x y r x g x f x y z z p x b x y r x g x

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

global existence and asymptotic behavior of solutions to a nonlocal fisher kpp type problem abstract in this work we consider a nonlocal fisher kpp reaction diffusion problem with neumann boundary condition and nonnegative initial data in a bounded domain in r n n 1 with reaction term u 1 m t where m t is the total mass at time t with the help of poho aev s identity the non existence of nontrivial stationary solutions with dirichlet boundary conditions is being shown when 1 and the initial mass is greater than or equal to one the problem has nonnegative classical solutions while if the initial mass is less than one then the problem admits global solutions for n 1 2 with any 1 2 or n 3 with any 1 1 2 n moreover the asymptotic convergence to the solution of the heat equation is proved

uniform blow up rate for diffusion equations with localized nonlinear source abstract in this short paper we investigate blow up rate of solutions of reaction diffusion equations with localized reactions we prove that the solutions have a global blow up and the rate of blow up is uniform in all compact subsets of the domain

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

existence of three solutions for p x laplacian equations abstract in this paper using b ricceri s three critical points theorem and variational methods we study the solutions of the p x laplacian equations with neumann or dirichlet boundary condition on a bounded domain and obtain three solutions under appropriate hypotheses

a segregated finite element solution for non isothermal flow abstract a segregated finite element algorithm for the solution of the supg formulation of the incompressible steady state navier stokes equations for non isothermal flow is presented in this paper the method features equal order interpolation for all the flow variables the simpler and simplest algorithms are employed and the sets of non symmetric linear equations are solved by means of the preconditioned conjugate gradient squared solver whilst the preconditioned conjugate gradient solver is used to solve the sets of symmetric linear equations three cases are used as a basis for the study i e the flow in a square cavity the flow between parallel plates and natural convection in a square cavity the effect that the choice of the values of the relaxation parameters has on the number of iterations which is needed to obtain a converged solution is investigated in each case there is an optimum combination which is amongst others a function of the grid and the flow parameters a non optimum choice may lead either to an unnecessary number of iterations or the solution becoming unstable from the results it is concluded that the method performs well when the optimum combination of the values for the relaxation parameters is used however much can still be done to improve the robustness of the method

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

holder continuity for nonlinear elliptic problem in musielak orlicz sobolev space abstract under appropriate assumptions on the n function the de giorgi process is presented by the tools recently developed in musielak orlicz sobolev space to prove the holder continuity of fully nonlinear elliptic problems as the applications the holder continuity of the minimizers for a class of the energy functionals in musielak orlicz sobolev spaces is proved and furthermore the local holder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided

uniform blow up profiles for nonlinear and nonlocal reaction diffusion equations abstract in this paper we investigate the blow up properties of positive solutions to the following integro parabolic equations u t u e u 1 e u d x where 0 and 1 and x r n x r for the radially symmetric and non increasing initial data we give a complete classification in terms of global and single point blow up according to the parameters and moreover the blow up rates are also determined in each case particularly for the special case and n 2 it will be proved that the blow up rate at x 0 is faster than that at the other point x 0 this seems to be a new phenomenon for this kind of problem

total versus single point blow up of solutions of a semilinear parabolic equation with localized reaction abstract in this paper we study the initial boundary value problem of a semilinear parabolic equation with localized reaction p u t u u p u q x t x t b 0 t u x t 0 x t b 0 t u x 0 u 0 x x b where b is a unit ball in r n x b and p q 0 for the case x 0 we completely classify blow up solutions of p into total blow up cases and single point blow up cases according to the values p and q moreover we give the blow up rates of solutions near the blow up time for the other case x 0 we show total blow up and single point blow up of solutions of p in some cases depending on p and q

a nonlocal reaction diffusion equation and its relation with fujita exponent abstract this paper is concerned with a type of nonlinear reaction diffusion equation which arises from the population dynamics the equation includes a certain type reaction term u 1 r n u d x of dimension n 1 and 0 an energy methods based proof on the existence of global solutions is presented and the qualitative behavior of solution which is decided by the choice of is exhibited more precisely for 1 1 1 2 p where p is the exponent appears in sobolev s embedding theorem defined in 3 the equation admits global solution for any nonnegative initial data especially in the case of n 2 and 1 the exponent 1 2 n is exactly the well known fujita exponent the global existence result obtained in this paper shows that by switching on the nonlocal effect i e from 0 to 0 the solution s behavior differs distinctly that s from finite time blow up to global existence

holder continuity of quasiminimizers with nonstandard growth abstract we show the holder continuity of quasiminimizers of the energy functionals f x u u d x with nonstandard growth under the general structure conditions z p x b x y r x g x f x y z z p x b x y r x g x the result is illustrated by showing that weak solutions to a class of a b harmonic equations div a x u u b x u u are quasiminimizers of the variational integral of the above type and thus are holder continuous our results extend work by chiado piat and coscia 1997 fan and zhao 2000 and giaquinta and giusti 1984

limits as p x of p x harmonic functions abstract in this note we study the limit as p x of solutions to p x u 0 in a domain with dirichlet boundary conditions our approach consists in considering sequences of variable exponents converging uniformly to and analyzing how the corresponding solutions of the problem converge and which equation is satisfied by the limit

solution of nearly incompressible field problems using a generalized finite element approach abstract it is well known that for problems approaching incompressible limits standard finite element approaches suffer from suboptimal convergence due to a phenomenon known as locking we analyze this phenomenon and present a robust generalized finite element formulation that enables optimal solution convergence this approach is explored for the two dimensional incompressible elasticity equations with a variable poisson s ratio as well as the two dimensional lid driven cavity stokes flow problem using a penalty pressure formulation

blow up behavior for a nonlinear heat equation with a localized source in a ball in this paper we consider the initial boundary value problem of a semilinear parabolic equation with local and non local localized reactions in a ball ut u up uq x t in b r where p q 0 b r x rn x 1 there exist blow up solutions of this problem for large initial data we treat the radially symmetric and one peak non negative solution u x t u r t r x of this problem we give the complete classification of total blow up phenomena and single point blow up phenomena according to p and q i if q 1 or p q 2 then single point blow up occurs whenever solutions blow up ii if 1 p q both phenomena total blow up and single point blow up occur depending on the initial data iii if p 1 q total blow up occurs whenever solutions blow up iv if max p q 1 every solution exists globally in time

a multiplicity result for p x laplacian problem in rn abstract in this paper we discuss the solutions of the following p x laplacian problem div u p x 2 u u p x 2 u f x u x r n where 1 p p x p n based on the theory of the variable exponent sobolev spaces w 1 p x r n we get that there exist at least two non trivial weak solutions

a monotone streamline upwind finite element method for convection dominated flows a new streamline upwind approximation for treating advection dominatcd transport is presented the proposed method is quite simple to implement with existing finite element methods for this work the method is used in a two dimensional finite element computer program the new method is applied to several test cases that have been presented in previous articles to evaluate the stability and accuracy of proposed methods for treating advection dominatcd flows the results demonstrate that the new method is able to predict the test cases with a significant increase in accuracy compared to previous techniques in addition these results demonstrate conclusively that the method does not exhibit the non physical spatial oscillations that have plagued previous methods

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems

finite element solution of the navier stokes equations for incompressible flow using a segregated algorithm abstract a segregated finite element algorithm for the solution of the supg formulation of the incompressible steady state navier stokes equations is investigated in this paper the method features equal order interpolation for all the flow variables the simplest algorithm is employed which results in symmetric coefficient matrices for the momentum equations the same iterative linear equation solver can therefore be employed for the solution of the momentum and pressure equations the effect of applying the supg weighting functions only to the convective terms is compared with the effect of applying the functions to all the terms in the momentum equations the effect of using an iterative linear equation solver as opposed to a direct linear equation solver is also discussed

a free boundary problem for the p x laplacian abstract we consider the optimization problem of minimizing 1 p x u p x x u 0 d x in the class of functions w 1 p with u 0 w 0 1 p for a given 0 0 and bounded w 1 p is the class of weakly differentiable functions with u p x d x we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

existence and multiplicity results for some eigenvalue problems involving variable exponents abstract variational techniques are used to get some existence and multiplicity results for some eigenvalue problems involving variable exponents

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

nodal solutions of p x laplacian equations abstract the paper is concerned with the existence of nodal radial solutions for the p x laplacian equation div u p x 2 u a x u p x 2 u u q x 2 u in u w 0 1 p x it is proved that for any given nonnegative integer k the problem has a pair of solutions which has exactly k nodes when a p and q satisfy suitable conditions

eigenvalues of the p x laplacian neumann problems abstract we study the eigenvalues of the p x laplacian operator with zero neumann boundary condition on a bounded domain where p x is a continuous function defined on the domain with p x 1 we show that similarly to the p laplacian case the smallest eigenvalue of the problem is 0 and it is simple and the supremum of all the eigenvalues is infinity however unlike the p laplacian case for very general variable exponent p x the first eigenvalue is not isolated that is the infimum of all positive eigenvalues of the problem is 0 we also study some properties of the set of functions having p x average value zero

supg stabilized finite element resolution of the navier stokes equations applications to water treatment engineering in this paper an analysis of the viscous incompressible flow has been carried out from the very definition of the governing equations up to the resolution of some practical problems passing through the comprehensive study of the stabilized finite element techniques used in their resolution as a consequence of this analysis a code based upon a realistic interpretation of the forces has been written which allows for the modelling of the open channel flow with optimum results in the resolution of some benchmark and real flow problems related with the wastewater industry

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100