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

the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

exponential attractors for reaction diffusion equations with arbitrary polynomial growth abstract in this paper we study exponential attractors for an equation with arbitrary polynomial growth nonlinearity f and inhomogeneous term g first we prove by the l trajectory method that the exponential attractor in l 2 with g h 1 second by proving the semigroup satisfying discrete squeezing property we obtain the exponential attractor in h 0 1 with g l 2 because the solutions without higher regularity than l 2 p 2 for g belong only to l 2 in the equation the general method by proving the lipschitz continuity between l 2 p 2 and l 2 does not work in our case therefore we give a new method presented in a theorem to obtain an exponential attractor in a stronger topology space i e l 2 p 2 with g g stated in a definition when it is out of reach for the other known techniques

convergence to steady state and attractors for doubly nonlinear equations this paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic type equations for degenerate equations we prove thanks to the lojasiewicz inequality that the solutions to some nonautonomous equations converge to a steady state in the nondegenerate case the existence of exponential attractors is shown by using the l trajectories method

dynamics of second order in time evolution equations with state dependent delay abstract we deal with a class of second order in time nonlinear evolution equations with state dependent delay this class covers several important pde models arising in the theory of nonlinear plates we first prove well posedness in a certain space of functions which are c 1 in time in contrast with the first order models with discrete state dependent delay this result does not require any compatibility conditions the solutions constructed generate a dynamical system in a c 1 type space over delay time interval our main result shows that this dynamical system possesses compact global and exponential attractors of finite fractal dimension to obtain this result we adapt the recently developed method of quasi stability estimates

attractors for nonlinear reaction diffusion systems in unbounded domains via the method of short trajectories article history we consider a nonlinear reaction diffusion equation on the whole space r d we prove the well posedness of the corresponding cauchy problem in a general functional setting namely when the initial datum is uniformly locally bounded in l 2 only then we adapt the short trajectory method to establish the existence of the global attractor and if d 3 we find an upper bound of its kolmogorov s e entropy

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

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

smooth attractors of finite dimension for von karman evolutions with nonlinear frictional damping localized in a boundary layer abstract in this paper dynamic von karman equations with localized interior damping supported in a boundary collar are considered hadamard well posedness for von karman plates with various types of nonlinear damping are well known and the long time behavior of nonlinear plates has been a topic of recent interest since the von karman plate system is of hyperbolic type with critical nonlinearity noncompact with respect to the phase space this latter topic is particularly challenging in the case of geometrically constrained nonlinear damping in this paper we first show the existence of a compact global attractor for finite energy solutions and we then prove that the attractor is both smooth and finite dimensional thus the hyperbolic like flow is stabilized asymptotically to a smooth and finite dimensional set

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

exponential attractors for semigroups in banach spaces abstract let s t t 0 be a semigroup on a banach space x and a be the global attractor for s t t 0 we assume that there exists a t such that s s t is of class c 1 on a bounded absorbing set b 0 a and s b 0 a b 0 a and furthermore the linearized operator l at each point of b 0 a can be decomposed as l k c with k compact and c 1 then we prove the existence of an exponential attractor for the discrete semigroup s n n 1 in the banach space x and then we apply the standard approach of eden et al 1994 9 to obtain the continuous case here b 0 a denotes the 0 neighborhood of a in banach space x and c denotes the norm of the operator c we prove as a simple application the existence of an exponential attractor for some nonlinear reaction diffusion equations with polynomial growth nonlinearity of arbitrary order

large time behavior via the method of l trajectories abstract the method of l trajectories is presented in a general setting as an alternative approach to the study of the large time behavior of nonlinear evolutionary systems it can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics in both cases we prove the existence of a finite dimensional global attractor and the existence of an exponential attractor

necessary and sufficient conditions for the existence of exponential attractors for semigroups and applications abstract first we establish some necessary and sufficient conditions for the existence of exponential attractors by using limit compactness and a measure of non compactness then we provide a new method for proving the existence of exponential attractors we prove the existence of exponential attractors for reaction diffusion equations and 2d navier stokes equations as simple applications

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

global attractors for small samples and germs of 3d navier stokes equations abstract we prove the existence of a global attractor for the generalized semiflow in the sense of j m ball on the space of small samples of solutions to the 3d incompressible navier stokes equations this way to overcome the possible nonuniqueness of solutions is less radical than that of g sell and does not provide unique solutions on the other hand the existence of the global attractor does not need the unproven hypothesis of continuity of solutions required by ball the extension of this approach to the space of germs of solutions is also discussed

pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions abstract this paper is concerned with the existence of pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions due to the lack of the holder continuity in time of the semigroup and the smoothing property of the difference of two weak solutions we construct a pullback exponential attractor in the original phase space by the l trajectories methods

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

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

long time dynamics for a class of kirchhoff models with memory this paper is concerned with a class of kirchhoff models with memory effects utt 2u div u p 2 u 0 s 2u t s ds ut f u h defined in a bounded domain of rn this non autonomous equation corresponds to a viscoelastic version of kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations under assumptions that the exponent p and the growth of f u are up to the critical range it turns out that the model corresponds to an autonomous dynamical system in a larger phase space by adding a component which describes the relative displacement history then the existence of a global attractor is granted furthermore in the subcritical case this global attractor has finite hausdorff and fractal dimensions

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

attractors for 2d navier stokes models with delays abstract the existence of an attractor for a 2d navier stokes system with delay is proved the theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous however on some occasions the attractors may attract not only in the pullback sense but in the forward one as well also this formulation allows to treat in a unified way terms containing various classes of delay features constant variable distributed delays etc as a consequence some results for the autonomous model are deduced as particular cases of our general formulation

finite dimensional attractors for the kirchhoff models with critical exponents the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models utt u m 1 u ut 2u h ut g u f x it proves that for the subcritical and critical cases 1 m n 2 n 2 with n 6 the above mentioned dynamical system possesses in phase spaces x1 a finite dimensional global attractor and a weak exponential attractor respectively

asymptotic behavior for the quasi geostrophic equations with fractional dissipation in r2 abstract in this paper the surface quasi geostrophic equations qge with fractional dissipation in r 2 are considered our aim is to study the long time behavior of solutions of qge in the subcritical case to this end we investigate the global well posedness and global attractor for qge in h s r 2 via commutator estimates for nonlinear terms a new iterative technique for estimates of higher order derivatives and with the help of a nonlocal damping term besides by using the fractional lieb thirring inequality estimates of the finite hausdorff and fractal dimensions of the global attractor are found

finite dimensional attractors for the kirchhoff models the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models arising in elasto plastic flow utt div u m 1 u ut 2u h ut g u f x by using the method of l trajectories and the operator technique it proves that under subcritical case 1 m n 2 n 2 the above mentioned dynamical system possesses in different phase spaces a finite dimensional weak global attractor and a weak exponential attractor respectively for application the fact shows that for the concerned elasto plastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the fractal dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

a finite dimensional attractor for three dimensional flow of incompressible fluids abstract the asymptotic behavior of the flow for a system of the navier stokes type is investigated in the considered model the viscous part of the stress tensor is generally a nonlinear function of the symmetric part of the velocity gradient provided that the function describing this dependence satisfies the polynomial p 1 growth condition a unique weak solution exists if either p 2 n 2 and u 0 h or p 1 2 n n 2 and u 0 w 1 2 c n h in the first case the existence of a global attractor in h is proved in order to indicate the finite dimensional behavior of the flow at infinity the fractal dimension of the new invariant set composed from all short trajectories with initial value in the attractor is estimated in the l 2 0 h topology having uniqueness only for more regular data in the second case many trajectories can start from the initial value u 0 h this does not allow one to define a semigroup on the space h therefore the set of short trajectories x s closed in l 2 0 h is introduced along with a semigroup working on this set the existence of a global attractor with a finite fractal dimension is then demonstrated