the phase plane picture for a class of fourth order conservative differential equations we study the bounded solutions of a class of fourth order equations u u f u 0 0 we show that when is not too large then the paths in the u u plane of two bounded solutions do not cross moreover the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase plane and a continuation theorem shows that they fill up part of the phase plane we apply these results to the extended fisher kolmogorov efk equation a fourth order model equation for bi stable systems the uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary efk equation correspond exactly to those of the classical second order fisher kolmogorov equation besides we establish the asymptotic stability of the heteroclinic solution of the efk equation

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

double integral characterizations of harmonic bergman spaces abstract recently li et al have characterized except for a critical case the weighted bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions in this paper we extend those characterizations to the case of weighted harmonic bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case we also investigate the possibility of extensions to the half space setting our observations reveal an interesting half space phenomenon caused by the unboundedness of the half space

derivatives of harmonic bergman and bloch functions on the ball on the setting of the unit ball of euclidean n space we investigate properties of derivatives of functions in the harmonic bergman space and the harmonic bloch space our results are 1 size estimates of derivatives of the harmonic bergman kernel 2 gleason s problem and 3 characterizations in terms of radial tangential and ordinary derivative norms in the course of proofs some reproducing formulas are found and estimated

harmonic bergman functions on half spaces we study harmonic bergman functions on the upper half space of rn among our main results are the bergman projection is bounded for the range 1 p oo certain nonorthogonal projections are bounded for the range 1 p oo the dual space of the bergman ll space is the harmonic bloch space modulo constants harmonic conjugation is bounded on the bergman spaces for the range 1 p oo the bergman norm is equivalent to a normal derivative norm as well as to a tangential derivative norm

a nonlinear fourth order diffusion problem convergence to the steady state and non negativity of solutions the paper deals with nonlinear diffusion both time dependent and time independent the spatial terms in the partial differential equation p d e contain a second order nonlinear part where the non negative diffusivity depends on the dependent variable and a fourth order linear part in the context of non null time independent boundary conditions convergence of the unsteady to the steady state is established the second part of the paper discusses criteria on data ensuring non negativity of the solutions this is done for the steady state irrespective of the spatial dimension and it is done for the unsteady state for the one dimensional rectilinear case only using a result from the first part of the paper

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

a note on the fractal dimension of attractors of dissipative dynamical systems inotherwords thehausdorandfractaldimensionshavethesameupperbound turningtotheapplicationswerstnotethatinalltheexamplesofpartialdierentialequationsconsideredin 8 thefunctionf d isconcave secondly inthetwoparticularexamples areaction diusionsystemandtwo dimensionalnavier stokessystem weobtainf d explicitlyand consequently weobtainexplicitupperboundsofthefractaldimensionoftheattractorintermsofthephysicalparameters finally we observe that theorem 2 1 can be obtained by a careful analysis ofthecorrespondingresultsin 8 however weprovetheorem2 1generalizingtotheinnite dimensionalcasethemethodof 2 wherethekaplan yorkeestimateforthefractaldimensionofthelorenzattractorhasbeenobtained 2 main estimatesupposethatasemigroups

projections for harmonic bergman spaces and applications on the setting of general bounded smooth domains in rn we construct l1 bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic bergman spaces in addition we show that those projections satisfy sobolev lp estimates of any order even for p 1 among applications are gleason s problems for the harmonic bergman sobolev and little bloch functions on star shaped domains with strong reference points

berezin transform and toeplitz operators on harmonic bergman spaces abstract for an operator which is a finite sum of products of finitely many toeplitz operators on the harmonic bergman space over the half space we study the problem does the boundary vanishing property of the berezin transform imply compactness this is motivated by the axler zheng theorem for analytic bergman spaces but the answer would not be yes in general because the berezin transform annihilates the commutator of any pair of toeplitz operators nevertheless we show that the answer is yes for certain subclasses of operators in order to do so we first find a sufficient condition on general operators and use it to reduce the problem to whether the berezin transform is one to one on related subclasses

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

persistence of travelling wave solutions of a fourth order diffusion system in this paper the extended burgers huxley equation with the fourth order derivative is considered first the convergence to the uniform steady state is proved which means the solution of the equation with positive initial data will remain positive for time t sufficiently large then the persistence of the travelling wave solution for the extended equation on the unbounded domain is investigated we have proved that this solution will persist under small perturbation of the equation

explicit estimates on the torus for the sup norm and the dissipative length scale of solutions of the swift hohenberg equation in one and two space dimensions abstract in this work we have obtained explicit and accurate estimates of the sup norm for solutions of the swift hohenberg equation she in one and two space dimensions by using the best so far available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations we have evaluated in an explicit manner the values of the sup norm of the solutions of the she in addition we have calculated the so called time averaged dissipative length scale associated to the above solutions

tomography of convex and star bodies new definitions of a star body and its radial function are given these are used in studying sections of convex and star bodies in conjunction with the dual mixed volumes of lutwak the spherical radon transform and the i chord functions when the body contains the origin in its interior and i 0 the latter equals the sum of the values of the ith power of the radial function at antipodal points in the unit sphere special cases of results obtained include the following a simple construction is given of non spherical convex bodies of constant i section i e such that each i dimensional section through an interior point has the same i dimensional volume it is shown that two star bodies containing the origin in their interiors and with analytic boundaries must be equal up to reflection in the origin if the volumes of their sections by i dimensional subspaces for two different values of i are equal examples are given to show that the last statement is no longer necessarily true if the boundaries are only infinitely differentiable or if the bodies are convex polytopes it is also proved that a star body which has constant i section for two different values of i with respect to the same interior point must be a ball centered at that point this answers the dual of the old question as to whether there exist non spherical convex bodies of constant width and constant brightness new results are also obtained concerning the determination of convex polytopes by the volumes of their sections through one or more points

polyharmonic bergman spaces and bargmann type transforms abstract in the main result of the paper we prove the decomposition of polyharmonic bergman spaces over the upper half plane into spaces of polyanalytic functions then we introduce the decomposition of polyharmonic bergman spaces into the orthogonal sum of its true polyharmonic bergman subspaces and we state isometric isomorphisms between the different true polyharmonic bergman spaces this allows us to define the k th harmonic hilbert component of a polyharmonic bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic bergman spaces the harmonic complex fourier transform is introduced in order to give an explicit description of the cartesian and the laguerre harmonic components of the images of a bargmann type transform for the true polyharmonic bergman spaces finally it is proved that the polyharmonic bergman space of order j is isometric isomorphic to 2j copies of the corresponding hardy space

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

positivity and the attractor dimension in a fourth order reaction diffusion equation in this paper we investigate the semilinear partial differential equation ut fuxxxx uxx u 1 u2 with a view particularly to obtaining some insight into how one might establish positivity preservation results for equations containing fourth order spatial derivatives the maximum principle cannot be applied to such equations however progress can be made by employing some very recent best possible interpolation inequalities due to the third named author in which the interpolation constants are both explicitly known and sharp these are used to estimate the lx distance between u and 1 during the evolution a positivity preservation result can be obtained under certain restrictions on the initial datum we also establish an explicit two sided estimate for the fractal dimension of the attractor which is sharp in terms of the physical parameters

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

homotopy classes for stable periodic and chaotic patterns in fourth order hamiltonian systems we investigate periodic and chaotic solutions of hamiltonian systems in 4 which arise in the study of stationary solutions of a class of bistable evolution equations under very mild hypotheses variational techniques are used to show that in the presence of two saddle focus equilibria minimizing solutions respect the topology of the configuration plane punctured at these points by considering curves in appropriate covering spaces of this doubly punctured plane we prove that minimizers of every homotopy type exist and characterize their topological properties

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

estimates of the harmonic bergman kernel on smooth domains abstract we obtain optimal size estimates of the harmonic bergman kernel and its derivatives on smooth domains based on these estimates we derive mapping properties of the harmonic bergman projection on lebesgue spaces and lipschitz spaces