holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

gradient estimates for degenerate diffusion equations ii we establish upper and lower bounds for various norms of solutions and their gradients for the equation u t div u m 1 u in n in terms of the norms of the initial data based on the l estimate of u we conclude that u x t is lipschitz continuous in space time for all t 0 whenever u x 0 is in l 1 n

regularization by kinetic undercooling of blow up in the ill posed stefan problem it is well known that the one dimensional supercooled stefan problem possesses solutions that blow up in finite time the asymptotics of such solutions have been analyzed by herrero and velazquez european j appl math 7 1996 pp 119 150 here we consider the effect of kinetic undercooling as a regularizing mechanism to prevent the formation of such singularities and study the continuation of the solution through the near blow up regime the asymptotics of solutions and interfaces are described for small values of the kinetic undercooling parameter it is shown that in this limit the interface jumps over an interval determined by the latent heat and by the initial data specifically in dimensionless variables if the temperature profile at blow up is denoted by u x t c where t c is the finite blow up time then the interface jumps over the interval in which u x t c lambda where lambda is the latent heat

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

boundary estimates for solutions of nonlinear degenerate parabolic systems on etablit la regularite partout jusqu a la frontiere pour des solutions faibles de systemes paraboliques du type u l t div du p 2 du l f l x t u du dans t l 1 2 m u t r m m n p max 1 2n n 2 est un ouvert borne de r n n 2 0 t t 0 t

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

the stefan problem with small surface tension the stefan problem with small surface tension e is considered assuming that the classical stefan problem with e 0 has a smooth free boundary r we denote the temperature of the solution by 00 and consider an approximate solution 00 su for the case where e 7 0 e small we first establish the existence and uniqueness of u and then investigate the effiect of u on the free boundary r it is shown that small surface tension affects the free boundary r radically differently in the two phase problem than in the one phase problem 0 introduction in the classical formulation of the two phase stefan problem the temperatures ow and si of water and ice satisfy the following conditions on the interface tm xn t 0 0 1 vx 0w vx vx ivxo ot 0 2 ow oi the first condition is the cbnservation of energy the functions sw oi further satisfy the heat equation in the water and ice sets respectively as well as initial and boundary conditions on the fixed portions of the boundary for definiteness we shall take in this paper the initial geometry to be as in figure 1 namely the fixed boundary 0d is surrounded by water region g go d and the water is surrounded by ice the fixed boundary does not change in time but the free boundary the water ice interface will of course change with time the one phase stefan problem arises when oi 0 in the ice region or ow 0 in the water region taking for definiteness the case oi 0 the interface conditions are 0 3 vx ow vx4 st received by the editors october 20 1989 1980 mathematics subject classification 1985 revision primary 35r35 secondary 35k20 35k85 this work is partially supported by national science foundation grant dms 86 12880 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

boundary and entropy of space homogeneous markov chains we study the poisson boundary equiv representation of bounded harmonic functions of markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations this automorphism group is locally compact but not necessarily discrete or unimodular the main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups while however the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other various new examples are given as applications including a description of the poisson boundary for random walks on vertex transitive graphs with infinitely many ends and on the diestel leader graphs

on the two phase stefan problem with interfacial energy and entropy abstract the classical stefan theory for the melting of a solid or the freezing of a liquid is too simplistic to describe phenomena such as supercooling in which a liquid supports temperatures below its freezing point or superheating the analog for solids or dendritic growth in which simple shapes evolve to complicated tree like structures this paper develops a general theory for two phase phenomena of this type it develops partial differential equations satisfied in the phase regions and free boundary conditions satisfied on the interface between phases and gives arguments which indicate that the resulting boundary value problems predict the formation of dendrites keywords equilibrium liapunoo functions mathematical physics

singular parabolic equations with measures as initial data abstract in this paper we study the cauchy problem for the singular evolution p laplacian equations with gradient term and source on the assumption of measures as initial conditions for the supercritical case q p 1 p n we obtain that for every nonnegative solution there exists a nonnegative radon measure as initial trace and has some local regularity

well posedness for the classical stefan problem and the zero surface tension limit we develop a framework for a unified treatment of well posedness for the stefan problem with or without surface tension in the absence of surface tension we establish well posedness in sobolev spaces for the classical stefan problem we introduce a new velocity variable which extends the velocity of the moving free boundary into the interior domain the equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature solutions to the classical stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation in which the moving free boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem with our methodology we simultaneously find the required stability condition for well posedness and obtain new estimates for the regularity of the moving free boundary finally we prove that solutions of the stefan problem with positive surface tension sigma converge to solutions of the classical stefan problem as sigma to 0

power concavity on nonlinear parabolic flows our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p laplace equation called power concavity and that the power is determined by the homogeneity of the parabolic operator in the parabolic p laplace equation for the density u the concavity of u p 2 p is considered which indicates why the log concavity has been considered in heat flow p 2 in addition the long time existence of the classical solution of the parabolic p laplacian equation can be obtained if the initial smooth data has concavity and a nondegenerate gradient along the initial boundary 2004 wiley periodicals inc

l estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms abstract we study the nonnegative solutions of the viscous hamilton jacobi problem u t u u q 0 u 0 u 0 in q t 0 t where q 1 0 t 0 and r n or is a smooth bounded domain and u 0 l r r 1 or u 0 m b we show l decay estimates valid for any weak solution without any conditions as x and without uniqueness assumptions as a consequence we obtain new uniqueness results when u 0 m b and q n 2 n 1 or u 0 l r and q n 2 r n r we also extend some decay properties to quasilinear equations of the model type u t p u u 1 u u q 0 where p 1 0 and u is a signed solution

propagation profile of support for evolution p laplacian with convection in half space abstract consider the cauchy dirichlet problem in half space for a one dimensional evolution p laplacian with convection for p 2 and pay attention to the interface t sup x u x t 0 it is well known that lim t t in the absence of the convection while the inclusion of the first order term may change the property of finite or infinite speed of propagation in this paper it will be shown that the nonlinear convection plays a very important role to the evolution of t for the convection with promoting diffusion the fast propagation phenomenon occurs i e u x t 0 whenever t 0 if the convection is strong enough otherwise t remains finite and non localized while under the convection with counteracting diffusion if the convection is strong enough localization even shrinking and extinction appears otherwise t keeps non localized in addition it is found that the time related boundary data are significant also to the behavior of solutions the decay or incremental rates of the boundary data affect not only the contraction or expansion of the supports but also the propagation speed of the interface