long time dynamics of kirchhoff wave models with strong nonlinear damping we study well posedness and long time dynamics of a class of quasilinear wave equations with a strong damping we accept the kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the l2 norm of the gradient of the displacement we prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration or even negativity of the stiffness coefficient and the case of a supercritical source term our main results deal with global attractors for strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite dimensional attractor in the non supercritical case this attractor is strong in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals

inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function abstract the purpose of the present paper is to introduce several new classes of meromorphic functions defined by using a meromorphic analogue of the choi saigo srivastava operator for the generalized hypergeometric function and investigate various inclusion properties of these classes some interesting applications involving these and other classes of integral operators are also considered

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

the noor integral and strongly starlike functions abstract let a denote the class of analytic functions f z defined in the unit disc satisfying the condition f 0 f 0 1 0 we introduce some new subclasses of strongly starlike functions defined by the noor integral and study their properties

tempered random attractors for parabolic equations in weighted spaces this paper deals with the asymptotic behavior of solutions of parabolic equations on unbounded domains which contain stochastic terms as well as non autonomous deterministic terms we define a continuous random dynamical system for the equations in weighted sobolev spaces that allow functions to have certain polynomial growth rate at infinity and hence include all bounded solutions we prove pullback asymptotic compactness of solutions as well as the existence and uniqueness of tempered random attractors in the weighted spaces the structures of the tempered attractors are fully characterized by tempered complete solutions in the case where the non autonomous deterministic terms are periodic in time we show that the tempered random attractors are also periodic in time to overcome the difficulty of non compactness of sobolev embeddings on unbounded domains the idea of uniform pathwise estimates on the tails of solutions is employed to show asymptotic compactness

asymptotic dynamics of plate equations with a critical exponent on unbounded domain abstract the long time behavior of plate equations with a critical exponent on the unbounded domain r n is studied we show that there exists a compact global attractor the attractor is characterized as the unstable manifold of the set of stationary points due to the existence of a lyapunov functional

strongly damped wave equation in uniform spaces abstract the cauchy problem for a semilinear strongly damped wave equation is considered in the whole of r n under suitable conditions imposed on nonlinear term which are much like for equations in bounded domains a dissipative semigroup t t of global solutions to this problem is constructed in a reach phase space being the product of the locally uniform spaces existence of an attractor for t t is then established

existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain abstract in this paper we study the long time behavior of solutions for the parabolic equation with non linear laplacian principal part in r n we prove the existence of a global l 2 r n l r n attractor when n p and the existence of a global l 2 r n l n p n p r n attractor when n p

on certain applications of the hadamard product abstract in this paper we give an extension of the ozkan and altintas results o ozkan o altintas applications of differential subordination appl math lett 19 2006 728 734 on the inclusion relationships involving various subclasses of analytic and univalent functions defined in terms of linear operators we show that these classes are closed under convolution with convex functions

asymptotic behavior of solutions for initial boundary value problems for strongly damped nonlinear wave equations abstract we study initial boundary value problems for strongly damped nonlinear wave equations by using improved integral estimates it is proven that the solutions of the problems decay to zero exponentially as time t approaches infinity under a very simple and general assumption regarding the nonlinear term

existence of a global attractor for a p laplacian equation in rn abstract in this paper we prove the existence of the l 2 r n w 1 p r n l q r n global attractor for the p laplacian equation u t div u p 2 u u p 2 u f u g with a more general nonlinear term f f 1 a x f 2 in r n where a l 1 r n l r n and f 1 f 2 satisfy the arbitrary q order polynomial growth condition without any restriction on q p q 2 p 2

some applications of differential subordination and the dziok srivastava convolution operator abstract by making use of the principle of differential subordination and the dziok srivastava convolution operator we introduce and investigate three interesting subclasses of analytic and univalent functions in the open unit disk we obtain their inclusion relationships and also show that these classes are closed under convolution with convex functions the results presented in this paper provide generalizations of several recent works by for example ozkan and altintas 2006 16 trojnar spelina 2008 30 and many other authors

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

some inclusion properties of a certain family of integral operators abstract the authors introduce several new subclasses of analytic functions which are defined by means of a general integral operator i and investigate various inclusion properties of these subclasses many interesting applications involving these and other families of integral operators are also considered

existence of random attractors for weakly dissipative plate equations with memory and additive white noise in this paper we prove the existence of random attractors for the continuous random dynamical systems generated by stochastic weakly dissipative plate equations with linear memory and additive white noise by defining the energy functionals and using the compactness translation theorem

a linear operator and associated families of meromorphically multivalent functions making use of a linear operator which is defined here by means of the hadamard product or convolution the authors introduce and investigate the various properties and characteristics of two novel families of meromorphically multivalent functions they also extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions

finite and infinite dimensional attractors for porous media equations the fractal dimension of the global attractors of porous media equations in bounded domains is studied the conditions which guarantee this attractor to be finite dimensional are found and the examples of infinite dimensional attractors that do not satisfy these conditions are constructed the upper and lower bounds for the kolmogorov e entropy of infinite dimensional attractors are also obtained

global attractors for plate equations with critical exponent in locally uniform spaces abstract a plate equation with critical exponent in locally uniform spaces with a coefficient x belonging to the locally uniform spaces l l u p r n is studied this equation is shown to generate a dissipative semigroup in locally uniform spaces h l u 2 r n l l u 2 r n which possesses global attractors in weighted spaces h 2 r n l 2 r n

global existence of weak solutions for two dimensional semilinear wave equations with strong damping in an exterior domain abstract we consider two dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power type nonlinearity u p if the initial data have a small weighted energy we shall derive a global existence and energy decay results in the case when the power p of the nonlinear term satisfies p 6

on the existence of the compact global attractor for semilinear reaction diffusion systems on rn we show that a class of reaction diffusion systems on rngenerates an asymptotically compact semiflow on the banach space of bounded uniformly continuous functions if such a semiflow is dissipative then a unique non empty compact minimal attractor is known to exist we apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on rnthat contain appropriate weight functions we also state conditions which guarantee that the attractor has finite hausdorff dimension

the immersed interface method for axis symmetric problems and application to the hele shaw flow many physical application problems are axis symmetric using axis symmetric properties many three dimensional problems can be solved efficiently using two dimensional axis symmetric coordinates in this paper the immersed interface method iim in axis symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient solution or flux a staggered grid is used to overcome the pole singularity other challenges include deriving the jump relations in axis symmetric coordinates designing the numerical algorithm when the interface is close to the pole r 0 computing interface quantities such as the normal and tangential directions surface derivatives curvature information etc the numerical algorithm is based on a finite difference discretization and uniform grid in the axis symmetric coordinates the finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface the method is shown to be second order accurate in the infinity norm the developed new iim is applied to the hele shaw flow and compared with the results from the literature

the existence of global attractors for a class of reaction diffusion equations with distribution derivatives terms in rn abstract we consider the existence of global attractors for a class of semilinear reaction diffusion equations u t u u v x g u f u d i h i h in r n r u x 0 u 0 x in r n in the whole space with weighted terms and some distribution derivatives in inhomogeneous terms using the measure of non compactness method we prove the existence of global attractors in proper spaces

global attractors for degenerate parabolic equations on unbounded domains we consider a nonlinear degenerate parabolic equation and show existence of the maximal compact attractor in weighted spacesl1 rn x dx with weight l1 rn going to zero at infinity

global attractors for von karman equations with nonlinear interior dissipation abstract in this paper we study the asymptotic behavior of weak solutions for von karman equations with nonlinear interior dissipation we prove the existence of a global attractor in the space w 2 2 l 2

global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain abstract in this paper we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent we prove the existence regularity and finite dimensionality of a global attractor in w 2 2 r n l 2 r n

on the solvability of the cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part we consider one multidimensional version of the cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part depending on the exponent of nonlinearity and spatial dimension of equation we investigate the problem on the nonexistence of global solutions of the cauchy characteristic problem the question on the local solvability of that problem is also considered

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

certain properties of the dziok srivastava operator the main object of the present paper is to investigate several interesting properties and characteristics of a linear operator h p q s a 1 which was recently introduced and studied in a series of papers by dziok and srivastava appl math comput 103 1999 1 13 adv stud contemp math 5 2002 115 125 integral transform spec funct 14 2003 7 18 the various properties and characteristics of the linear operator h p q s a 1 which are considered in this paper are associated with for example the principle of differential subordination between analytic functions and a certain one parameter family of integral operators

attractors of partial differential evolution equations in an unbounded domain there is a large number of papers in which attractors of parabolic reaction diffusion equations in bounded domains are investigated in this paper these equations are considered in the whole unbounded space and a theory of attractors of such equations is built while investigating these equations specific difficulties arise connected with the noncompactness of operators with the continuity of their spectra etc therefore some new conditions on nonlinear terms arise in this paper weighted spaces are widely applied an important feature of this problem is worth mentioning namely properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ

singularities in hele shaw flows driven by a multipole we study analytically and numerically singularity formation in an interface flow driven by a multipole for a two dimensional hele shaw cell with surface tension our analysis proves that singularity formation is inevitable in the case of a dipole for a multipole of a higher order we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole it is therefore very likely that this solution will develop finite time singularities our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole in addition it is observed that the interface approaches the multipole from directions which can be predicted beforehand we also estimate as a function of time the distance between the finger tip and multipole and the results are in excellent agreement with numerical computations

certain classes of meromorphic multivalent functions in this paper we introduce and investigate two new classes of meromorphic multivalent functions such results as subordination properties coefficient inequalities convolution properties and integral representations are proved several sufficient conditions for meromorphic multivalent starlikeness and convexity are also derived

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

dynamics of strongly damped wave equations in locally uniform spaces attractors and asymptotic regularity this paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces after proving the global well posedness we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded in h 2 lu r n x h 1 lu r n subset which attracts exponentially every initial h 1 lu r n x l 2 lu r n bounded set with respect to the h 1 lu r n x l 2 lu r n norm then we show there is a h 1 lu r n x l 2 lu r n h 1 r n x h 1 r n global attractor which reflects the strongly damped property of u t to some extent

extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations abstract we consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors we then show some applications of the abstract scheme to various evolutionary problems from odes and retarded functional differential equations to parabolic and hyperbolic pdes in particular we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in r n with nonlinearities depending on the gradient of the solution we consider as well systems of reaction diffusion equations in r n and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in r n we further discuss some nonlocal and quasilinear problems as well as the fourth order cahn hilliard equation

on integral operators abstract let fn z z 1 z n 1 n no and f 1 n be defined such that f n f 1 n z 1 z where denotes convolution hadamard product let f be analytic in the unit disc e we define a new operator inf f 1 n f which is analogous to one defined by ruscheweyh using this operator the classes m n are defined a function f analytic in e is in m n if and only if inf is close to convex the properties of f m n are discussed in some detail it is shown that m n m n 1 for n no and for n 0 1 m n consists entirely of univalent functions closure properties of some integral operators defined on m n are also given

singularities in hele shaw flows we study analytically and numerically singularities in an interface flow driven by surface tension and a sink in a two dimensional hele shaw cell our analysis proves that singularity formation is inevitable in generic situations our numerical simulations suggest that the hele shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out a corner is formed right at the tip of the interface that touches the sink

attractors for partly dissipative reaction diffusion systems in rn in this paper we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2 n 2 n l l 2000 academic press

classes of analytic functions associated with the generalized hypergeometric function using the generalized hypergeometric function we introduce and study a class of analytic functions with negative coefficients coefficients estimates distortion theorems extreme points and the radii of convexity and starlikeness for this class are given relevant connections of these results with those in several earlier investigations are indicated

on certain subclass of meromorphic close to convex functions in the present paper we introduce and investigate a certain new subclass mk s l a b of meromorphic close to convex functions such results as inclusion relationships coefficient inequalities and convolution property are derived relevant connections of the results presented here with those obtained in earlier works are also pointed out

regularity of random attractors for a degenerate parabolic equations driven by additive noises abstract we study the regularity of random attractors for a class of stochastic degenerate parabolic equations with the leading term involving a diffusion variable which many be non smooth or unbounded without any restrictions on the upper growth order p of the nonlinearity except that p 2 we show that the associated random dynamical system admits a unique compact random attractor in the space d 0 1 2 d n l d n for any 2 2 p 2 where d n is an arbitrary bounded or unbounded domain in r n n 2

limiting behavior for strongly damped nonlinear wave equations in this paper we investigate both the existence and the limiting behavior for the equation utt aut au t u ut where a is a sectorial operator is periodic in t and satisfies certain regularity and growth assumptions in most results on limiting behavior we will assume a has compact resolvent we consider the equation as an abstract ode defined on a paired space x x 0 1 with regard to the limiting behavior one of our principal results will be to show that if there is a bounded set in one of the spaces considered for which all points or trajectories enter into and remain then there is a set j consisting of very smooth functions defined on all of the spaces considered which is the maximum compact invariant set uniformly asymptotically stable connected and having very strong attractivity properties in all these spaces we will often show it attracts all points in a bounded set uniformly we will give a few sharper results for the case where a the work is motivated by recent papers of webb and fitzgibbon and applies techniques found in recent papers by the author

inclusion properties of certain classes of meromorphic functions associated with the wright generalized hypergeometric function the purpose of the present paper is to introduce several new classes of meromorphically univalent functions which are defined here by means of a linear operator also we study various inclusion properties of these classes

a cauchy integral approach to hele shaw problems with a free boundary the case of zero surface tension in this paper we study a nonlinear and nonlocal free boundary dynamics the hele shaw problem without surface tension when the fluid domain is either bounded or unbounded the key idea is to use a global quantity the cauchy integral of the free boundary to capture the motion of the boundary this cauchy integral is shown to be linear in time the free boundary at a fixed time is then recovered from its cauchy integral at that time the main tool in our analysis ischerednichenko s theorem concerning the inverse properties of the cauchy integrals as products of our approach we establish the short time existence and uniqueness of classical solutions for analytic initial boundaries we also show the non existence of classical solutions for all smooth but non analytic initial boundaries when there is a sink at either a finite point or at infinity when the fluid domain is bounded all solutions except the circular one break down before all the fluid is sucked out from the sink regularity results are also obtained when there is a source at a finite point or at infinity

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s