attractors for the nonclassical diffusion equations with fading memory abstract we discuss long time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical the existence and regularity of global attractors in weak topological space and strong topological space are obtained while the forcing term only belongs to h 1 and l 2 respectively the results in this part are new and appear to be optimal corresponding to the forcing term

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities abstract in this paper we discuss continuation properties and asymptotic behavior of regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions a necessary and sufficient condition for global in time existence of regular solutions is given we also formulate sufficient conditions to construct a piecewise regular solutions continuation beyond maximal time of existence for regular solutions applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed in particular global solvability and the existence of a global attractor for u t t d 1 2 u t d u f u in h 0 1 l 2 is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition similar result is obtained for a 2 m th order semilinear parabolic initial boundary value problem in a hilbert space h 2 b j m

mean curvature flow of entire graphs in a half space with a free boundary we study the mean curvature flow of graphs with prescribed contact angle on a fixed smooth hyperplane in euclidean space we obtain long time existence and convergence to a self similar solution of the mean curvature flow orthogonal to the fixed hyperplane

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

exponential attractors for a conserved phase field system with memory we consider a conserved phase field system of caginalp type characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient respectively the model consists of a parabolic integrodifferential equation coupled with a fourth order evolution equation for the phase field this system supplemented with suitable boundary conditions can be interpreted as a dissipative dynamical system in a proper phase space in a previous joint work the last two authors have proved the existence of a global attractor of finite fractal dimension here we show the existence of an exponential attractor by means of the techniques we developed for nonconserved phase field systems with memory

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

asymptotic behavior of flows by powers of the gaussian curvature we consider a 1 parameter family of strictly convex hypersurfaces in mathbb r n 1 moving with speed k alpha where denotes the outward pointing unit normal vector and alpha geqslant 1 n 2 for alpha gt 1 n 2 we show that the flow converges to a round sphere after rescaling in the affine invariant case alpha 1 n 2 our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling

evolution of geometric statistics a model for the description of the geometry in multiphase mechanics is derived and studied equations for the rate of change of mean and gauss curvature are derived these coupled equations show that the evolution of these curvatures depend on the velocity of the interface normal to itself examples are given to show the utility of these equations to apply these concepts to multiphase mechanical situations an ensemble average is applied to the curvature equations equations for the evolution of the volume fraction and the interfacial area density are also derived these equations also depend on the curvatures and the average interfacial velocity examples are given to show that these equations give a description of the geometry of multiphase mechanics that is superior to the present description which describes the force between materials as depending on two geometric parameters the volume fraction and the average radius of the dispersed material these examples also suggest constitutive equations to d

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

dynamics of parabolic problems with memory subcritical and critical nonlinearities in this paper we study the long time behavior of the solutions of non autonomous parabolic equations with memory in cases when the nonlinear term satisfies subcritical and critical growth conditions in order to do this we show that the family of processes associated to original systems with heat source f x t being translation bounded in lloc2 r l2 is dissipative in higher energy space m 0 1 and possesses a compact uniform attractor in m0

isoperimetric inequality for radial probability measures on euclidean spaces abstract we generalize the poincare limit which asserts that the n dimensional gaussian measure is approximated by the projections of the uniform probability measure on the euclidean sphere of appropriate radius to the first n coordinates as the dimension diverges to infinity the generalization is done by replacing the projections with certain maps using this generalization we derive a gaussian isoperimetric inequality for an absolutely continuous probability measure on euclidean spaces with respect to the lebesgue measure whose density is a radial function

attractors for the non autonomous nonclassical diffusion equations with fading memory abstract the long time dynamical behavior of the non autonomous nonclassical diffusion equation with fading memory when nonlinearity is critical is discussed for in the weak topological space h 0 1 l 2 r h 0 1 first the asymptotic regularity of solutions is proven and then the existence of a compact uniform attractor together with its structure and regularity is obtained while the time dependent forcing term is only translation bounded instead of translation compact the result extends and improves some results given in y xiao attractors for a nonclassical diffusion equation acta math appl sin engl ser 18 2002 273 276 c sun m yang dynamics of the nonclassical diffusion equations asympt anal 59 2008 51 81

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

flow by powers of the gauss curvature abstract we prove that convex hypersurfaces in r n 1 contracting under the flow by any power 1 n 2 of the gauss curvature converge after rescaling to fixed volume to a limit which is a smooth uniformly convex self similar contracting solution of the flow under additional central symmetry of the initial body we prove that the limit is the round sphere for 1

convexity and horizontal second fundamental forms for hypersurfaces in carnot groups we use a riemannian approximation scheme to give a characterization for smooth convex functions on a carnot group in the sense of danielli garofalo nhieu or lu manfredi stroffolini in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph