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Abstracts and Talk Materials
Dynamical Systems in Studies of Partial Differential Equations
September 24 - 28, 2012

Vortex crystals are configurations of vortices that maintain their basic shapes for long times, while perhaps rotating or translating rigidly in space. It is common to observe these patterns in nature and experiments, where the underlying dynamics are governed, for example, by the two-dimensional Euler or Navier-Stokes equations. One can often take advantage of the localized structure of vortices to derive equations of motion that are much simpler than the original PDE. In this work, we use a point vortex model and a generalized'' point vortex model (the latter allowing for positive core size) to study the existence and stability of a number of inviscid and weakly viscous vortex crystals, respectively. We complement this analysis with comparisons of our results to crystals observed in an electron column experiment as well as in hurricane eyes and related numerical simulations.

Keywords of the presentation: Navier-Stokes, metastabilty, hypocoercitiy

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time scale, and then dominate the dynamics for very long time intervals. We propose a dynamical systems explanation of the metastability of an explicit family of physically relevant quasi-stationary solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation with small viscosity on the torus. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. This is joint work with C. Eugene Wayne.

Slowly Non-Dissipative PDEs and their Non-Compact Global Attractors
December 31, 1969

This poster will present recent results in the area of global attractor theory for two types of slowly non-dissipative PDEs. We use a variety of PDE and dynamical systems techniques to obtain explicit decompositions of the non-compact global attractor structures for classes of asymptotically symmetric and asymptotically asymmetric grow-up equations. We present a number of these techniques, and address how inertial manifold theory, the Fučik spectrum, Poincaré compactification and the Conley index at infinity allow us to analyze behavior approaching and within the realm at infinity.

KAM theory for conformally symplectic systems
December 31, 1969

We present a KAM theory for systems that transport a symplectic form into a multiple of itself. The KAM theorem is written in an a-posteriori format and has many practical consequences. The theorem allows to produce numerical schemes that are remarkably efficient and work close to the breakdown of analyticity.

Vortex filament interactions and Hamiltonian PDEs
September 24, 2012

Keywords of the presentation: vortex dynamics, KAM theory

Abstract: Over the past period of a decade and more, mathematicians in the PDE community have developed techniques for the phase space analysis of the dynamics of many model nonlinear Hamiltonian PDEs. These results include constructions of KAM invariant tori, Birkhoff normal forms and Nekhoroshev stability theorems, and constructions of cascade orbits. In this talk I will describe some applications and extensions of these ideas to a problem in fluid dynamics concerning the interaction of two near-parallel vortex filaments in three dimensions. In addition, as well as generalizations of this problem, I will speculate about further applications of the techniques of Hamiltonian PDEs to other nonlinear systems of fluid dynamics, in the form of a class of conservative nonlinear evolution problems of physical significance.

Keywords of the presentation: transversality, hyperbolicity, Kupka-Smale, singular nodal set,Sard-Smale, unique continuation

The system arises in the theory of condensates.(We also discuss a related population system). We obtain limiting equations (a scalar equation if we originally had a system of two equations),discuss bounds for solutions which are independent of the large parameter and discuss when we can use properties of solutions of the limit system to obtain properties of solutions of the original equations.

Quasi-periodic solutions in some ill-posed Hamiltonian PDE's
September 25, 2012

Keywords of the presentation: quasi-periodic solutions, KAM theory, ill-posed equations.

We present an a-posteriori KAM theorem for whiskered tori in Hamiltonian PDE's. We do not need to assume that the equations define an evolution (just that the forward (resp. backwards) evolution is defined in the stable (resp. unstable) space and that these spaces are complementary with the tangent and the symplectic conjugate. This is joint work with Y. Sire and closely related to work with E. Fontich and Y. Sire.

Keywords of the presentation: fast diffusion, extinction in finite time, asymptotic behavior

We consider positive solutions of the Cauchy problem for the fast diffusion equation. Sufficient conditions for extinction of solutions in finite time are well known. We shall discuss results on the asymptotic behavior of solutions near the extinction time obtained in collaboration with John R. King, Juan Luis Vazquez, Michael Winkler and Eiji Yanagida.

Keywords of the presentation: Nonlinear partial differential equations, gradient systems, energy dissipation

We study the local energy dissipation in gradient-like nonlinear partial differential equations on unbounded domains. Our basic assumption, which happens to be satisfied in many classical examples, is a pointwise upper bound on the energy flux in terms of the energy dissipation rate. Under this hypothesis, we derive a simple and general bound on the integrated energy flux which implies that, in low space dimensions, our "extended dissipative system" has a gradient-like dynamics in a suitable averaged sense. In particular, we can estimate the time spent by any trajectory outside a neighborhood of the set of equilibria. As an application, we study the long-time behavior of solutions to the two-dimensional vorticity equation in an infinite cylinder. This talk is based on a joint work with S. Slijepcevic (Zagreb)

Traveling waves in high Lewis number combustion model
December 31, 1969

High Lewis number combustion model is a model that is used to describe propagation of fronts in the process of burning of high density liquid fuels. Both existence and stability of such fronts are important issues in applications and, at the same time, are challenging mathematical problems. I will present new results that prove the existence of fronts and show that, depending on the value of the exothermicity parameter, these fronts can be either convectively or absolutely unstable.

We establish the existence of certain invariant probability measures that can be associated with the time averaged observation of a broad and generic class of dissipative dynamical systems via the notion of a generalized Banach limit. This work extends a framework originally developed for the study of statistically stationary solutions of the Navier-Stokes equations.

We apply these abstract results by considering some concrete dynamical systems where the semigroup is known to be non-compact thus addressing examples not covered under previous results in this direction.

Qualitative properties of some reaction-diffusion waves
September 25, 2012

Keywords of the presentation: traveling waves, reaction-diffusion equations, qualitative properties

The usual notions of reaction-diffusion waves or fronts can be viewed as examples of generalized transition waves. These new notions involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. The existence of transition waves has been proved in various contexts where the standard notions of waves make no sense anymore. In this talk, I will focus on some uniqueness and further qualitative properties of the transition waves, including the existence and uniqueness of the mean speed for somes classes of equations. The talk is also based on some joint works with H. Berestycki and H. Matano.

Keywords of the presentation: water-wave problem, periodic waves, transverse instability, dimension-breaking bifurcation

The gravity-capillary water-wave problem concerns the irrotational flow of a perfect fluid in a domain bounded below by a rigid bottom and above by a free surface under the influence of gravity and surface tension. In the case of large surface tension the system has a family of traveling two-dimensional periodic waves for which the free surface has a periodic profile in the direction of propagation and is homogeneous in the transverse direction. We show that these periodic waves are linearly unstable under spatially inhomogeneous perturbations which are periodic in the direction transverse to propagation. As a consequence, the periodic waves undergo a dimension-breaking bifurcation generating a family of spatially three-dimensional solutions which are periodic in both the direction of propagation and the transverse direction.

Accelerated Fronts in a two stage invasion problem
December 31, 1969

We study wavespeed selection in a staged invasion process. That is, we study a model in which an unstable homogeneous state is replaced via an invading front with a secondary state. This secondary state is also unstable and, in turn, replaced by a stable homogeneous state via a secondary invasion front. In particular, we are interested in the selected wavespeed of the secondary front and under what conditions does the in uence of the primary front increase this speed. We nd three regimes: a locked regime where both fronts travel at the same speed, a pulled regime where the secondary front travels at the linear spreading speed associated to the intermediate state, and an accelerated regime where the selected speed is between these two speeds. We show that the transition to locked fronts can be described by the crossing of a resonance pole in the linearization about the primary front. In addition, we use properties of this resonance pole to derive an estimate for the selected wavespeed in the accelerated case and to determine the transition between accelerated and pulled fronts

Foliations provide a nonlinear coordinate system that decouples two variables in an ODE or PDE. Through each (base) point in phase space there is a pair of leaves: one each in the (un)stable foliation. If the base point is a steady state, the leaves are the invariant stable and unstable manifolds. Under a spectral gap condition the unstable manifold is an inertial manifold. The intersection of a leaf in the stable foliation and the inertial manifold is the unique initial condition through which the trajectory attracts at an exponential rate the trajectory through the base point of the leaf. We present a unified approach to compute all these objects.

A joint work with Yu-Min Chung.

This poster introduces a striking correspondence between the dynamics of the scalar parabolic equations and the ones of ODE's. It presents a list of dynamical properties of the PDE, depending of the domain and symmetries of the non-linearity, and draws a parallel with the dynamical properties of some classes of ODE's. This correspondence is helpful for predicting which dynamical behaviours can be expected in some classes of scalar parabolic equations. Several recent results of this poster have been obtained in joint works with G. Raugel and P. Brunovsky.

Long-time behavior of quasi-static droplet evolution
September 27, 2012

Keywords of the presentation: line dynamics, asymptotic behavior

We investigate the properties of a model describing the motion of liquid drops sitting on a flat surface. Here we consider the so-called quasi-static approximation model, where the speed of the contact line between the fluid and the surface is much slower than the capillary relaxation time.

We show that, under a geometric constraint on the initial configuration, the solution is globally well-posed, and the support of the solution uniformly converges to a ball. We will also discuss the asymptotic behavior of the droplet evolution on tilted surface. This is joint work with William Feldman and Antoine Mellet.

An open problem in nonlinear dispersive equations is the asymptotic completness conjecture which states that any initial data eventually converges to a superposition of coherent states. A big obstacle in solving the conjecture is the fact that the set of coherent states is not completely known. I will present recent results which not only find all coherent (bound) states in one dimensional nonlinear Schroedinger equation, and analyse their stability but also makes significant progress in higher dimensions. This is joint work with Vivek Natarajan (U. of Illinois).

Viscous Profiles for Shock Waves in Isentropic Magnetohydrodynamics
December 31, 1969

Standing planar waves in isentropic magnetohydrodynamics are governed by a five-dimensional gradient-like system, which depends parametrically on two constants of integration and three viscosity coefficients. Depending on the constants of integration, there exist up to four rest points. A heteroclinic orbit connecting two such rest points of this system corresponds to a viscous profile for the standing shock wave with these rest points as end states. In the typical case that there are four rest points we show how the respective ratios of the viscosity coefficients determine which of the rest points are connected by heteroclinic orbits. We have used the Conley index and geometric singular perturbation theory to prove the results.

Nonlinear Eigenvalues, Keldysh's Theorem, and the Evans function
December 31, 1969

We show how to use an abstract Keldysh's Theorem (that gives a formula for the inverse of a holomorphic operator valued function) in order to look for the number and location of unstable eigenvalues of differential operators that appear when one linearizes partial differential equations about such special solutions as traveling waves. Relations with the Evans function are also described.

Regulatory network process cellular signals in time and space enabling cell self-organization. Kinase cascades can emerge from receptors and transmit signals from the cell membrane to the nucleus. Usually, for the simplicity, the biology analyst assumes that the kinase activity happens on the cell with shape of the ball. But as biology shows, in reality, there are many cells with the shape of finite cylinder. In this work, we just propose the model of kinase activity on the cell of finite cylinder. ONE QUENSTION is that how to describe the kinase activity on the cell surface i.e mathematically how to impose the boundary conditions. Inspired by the case of neuron and its long axon, we only give the nonzero flux of the kinase on one of circular faces of the cylinder. Based on the model, we provide the steady state solution, i.e the concentration of the kinase and the analysis on how the kinase concentration varies according to the change of the parameters (diffusion coefficient, reaction rate etc.). We demonstrate the critical value (bifurcation value) of some parameter and the bifurcation from the steady state zero solution. We also give the axially symmetric solution (concentration) of the steady state equation in series form. Our partial parameter analysis shows that the concentration of kinase on the finite cylinder is probably not the monotone function of the parameter (dephosphorylation rate) as that the case with the cell of the ball shows.

We study a three-dimensional model of cellular electrical activity, which is written as a pseudodi erential equation on a closed surface Gamma in R^3 coupled with a system of ordinary di erential equations on Gamma. We prove global existence/uniqueness for this system for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin- Huxley kinetics. One of the main diculties is the absence of a max- imum principle, which we overcome in part by introducing the notion quasipositivity. We also discuss asymptotic smoothing and the depen- dence of the solution on the size of the domain.

Keywords of the presentation: traveling wave, diffusion equation, intersection-number, omega-limit set

In this talk, I will discuss the front propagation for solutions of one-dimensional diffusion equations on the whole line:

u_t = u_{xx} + f(x,u).

Here f(x,u) is a smooth function that is periodic in x and satisfies f(x,0) = 0. We consider a large class of nonlinearities f including multistable ones. Our analysis reveals some new dynamics where the asymptotic profile of the solution is not characterized by a single front, but by a layer of several fronts traveling at different speeds, which we call a propagating terrace".

This is joint work with Thomas Giletti and Arnaud Ducrot.

Oscillons near Hopf bifurcations
December 31, 1969

Oscillons are planar, spatially localized, temporally oscillating, radially symmetric structures. They have been observed in various experimental contexts, including fluid systems, granular systems, and chemical systems. Oscillons often arise near forced Hopf bifurcations, which are modeled mathematically with the forced complex Ginzburg-Landau (FCGL) equation. We present a proof of the existence of oscillons in the forced planar complex Ginzburg-Landau equation through a geometric blow-up analysis. Our analysis is complemented by a numerical continuation study of oscillons in the forced Ginzburg-Landau equation using Matlab and AUTO.

A topological perturbation theorem for infinite dimensional systems
September 28, 2012

Keywords of the presentation: Nonlinear Dynamics, Conley Theory

Conley theory has proven to be a useful tool for the analysis of global dynamics. Its power arises from the fact that the index provides information about the existence and structure of dynamics, but remains constant as long as isolation is preserved. In particular, it allows one to compute or model the dynamics using one - preferably simple to understand - system and draw conclusions about the dynamics of other systems.

The motivation for the work discussed is that there are a variety of problems which arise in study of the dynamics of PDEs for which the natural phase space of the known system is different from the natural phase space of the system of interest. Attempting to apply Conley theory in this setting immediately leads to two problems: (1) one needs to be able to deduce isolation in one phase space from different phase space, and (2) one needs to be able to argue efficiently that the Conley index remains unchanged. I will discuss an approach to dealing with these two problems.

This is joint work with G. Raugel.

For example, we observe the spiral waves and target patterns in Belousov-Zhabotinskii reaction. In photosensitive BZ reaction with feedback, the locallized patterns are also observed. Many researchers have regarded these patterns by the curves without thickness and have treated such patterns mathematically. In this presentation, using the wave front interaction model proposed by Zykov and Showalter, we show the existence of traveling spots in the plane and the rotating waves in the disk as two dimensional patterns. I will also explain the existence of the rotating spirals.

This talk is based on several join works with Professors J.-S. Guo, J.-C. Tsai, C.-C. Wu and my Ph.D student Y.Y. Chen.

Micropolar fluid equations, which were introduced by Eringen in 1966, are a significant step toward generalization of the standard Navier-Stokes model. By their very nature, they take into account the structure of the media they describe, which plays crucial role in modeling for some well-known fluids, e.g. animal blood or liquid crystals. In particular, the deviancy from the Navier-Stokes model becomes highly apparent in micro-scale when at least one dimension of the domain is only a few times larger than the size of the molecules (e.g. blood vessels, lubricants). Although not all the aspects of physical experiments have been fully explained, it is justified to assume that the surface stresses and the internal degrees of freedom of particles are the deciding factors for properties of fluid motion.

In this work we show the existence of strong solutions to micropolar fluid equations in cylindrical domains. We do not assume any smallness on the initial or the external data but only for their derivative alongside the axis of the cylinder.

Keywords of the presentation: nonlinear elliptic system, symmetry, nonexistence, Liouville-type theorem

We give sufficient conditions ensuring that any positive classical solution (u,v) of an elliptic system in the whole n-dimensional space has the symmetry property u=v. As an application, we improve some known results on Sobolev-critical elliptic systems of Schrodinger type. Our techniques apply to some supercritical problems as well. We also obtain new Liouville-type theorems for non-cooperative systems. Moreover, we provide several counterexamples which indicate that our assumptions are in a sense necessary.

This is a joint work with Philippe Souplet.

Invariance of the Gibbs measure for the periodic quartic gKdV.
December 31, 1969

We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on H^s for s less than 1/2, and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, Inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below H^{1/2}.

Pattern selection in the wake of fronts
September 25, 2012

Keywords of the presentation: Turing bifurcation; Hopf bifurcation; pattern formation; localized structures

Motivated by precipitation patterns such as Liesegang rings, we discuss patterns formed in the wake of moving fronts. It turns out that strikingly simple kinetic mechanisms can lead to a plethora of patterns, from simple bands and stripes to spirals and helices. We'll discuss a number of mathematical problems that arise when one tries to predict which wavenumbers and what type of pattern would arise in the wake of fronts. Applications include elementary chemistry, patterns in bacteria colonies, and self-assembly of nano-scale textures.

Keywords of the presentation: Nonlinear dispersive equations, Scattering, Concentration-Compactness, Invariant Manifolds, Hyperbolic Dynamics

This talk will survey some recent developments in the theory of nonlinear dispersive evolution equations, with emphasis on a qualitative description of the global-in-time dynamics of solutions. We will present the method of concentration compactness which has lead to important advances during the past six years. These results cannot be obtained by perturbative techniques. In the elliptic setting, concentration compactness was introduced into the calculus of variations by P. L. Lions in the 1980s. The main idea is to exhibit the action of non-compact symmetry groups as the only possible obstruction to compactness. For evolution equations, such ideas were developed by Hajer Bahouri and Patrick Gerard in the late 1990s, with independent work by Frank Merle and Luis Vega at about the same time. In 2006 Carlos Kenig and Frank Merle introduced a method into nonlinear dispersive equations which allows one to obtain global existence and scattering results for nonlinear evolution equations by means of a contradiction argument based on induction on energy. Roughly speaking, the idea is to show that if the desired result fails, then it does so at a minimal energy. Using concentration compactness, more precisely, a Bahouri-Gerard decomposition, one then constructs a solution at that minimal energy with pre-compact trajectory in the energy space. The fi nal part of the argument, which is typically based on virial-type identities, excludes the existence of such a rigid object. This method has been applied to different classes of equations. In particular, it was used in joint work with Joachim Krieger to establish global existence and scattering for large data wave maps from 2+1 dimensions into hyperbolic space. We will also present a qualitative description of the flow of focusing nonlinear wave-type equations at energies near the ground state energy. This latter work, joint with Kenji Nakanishi, combines the theory of invariant manifolds in this case generated by a ground state soliton with a one-dimensional exponential instability in the linearized flow { with some of these concentration-compactness and virial ideas, to exhibit a center-stable manifold as a surface separating an open region of finite-time blowup from another one leading to global existence and scattering to a free wave. A crucial dynamical ingredient in this work is the exclusion of almost homoclinic orbits associated with the ground state.

Space time resonances and scattering of the Zakharov system.
September 27, 2012

Keywords of the presentation: Resonances

We will present a new extension of the space time resonance method to show global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in $3$ space dimensions. The wave component is shown to decay pointwise at the optimal rate of $t^{-1}$, whereas the S component decays almost at a rate of $t^{-7/6}$. This is a joint work with Z. Hani and F. Pusateri.

On models of short pulse type in continuous media
December 31, 1969

We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Assuming general Kramers-Kronig or Sellmeier formulas for the permittivity and permeability, we derive two short-pulse equations (SPEs) for the high- and low-frequency band gap respectively in one space dimension. Then we generalize this model into two dimensional case, and give two 2D SPEs, we will discuss robustness of line breather solutions, and evolution of localized 2D initial data.

Nondegeneracy of blow-up points for the parabolic Keller-Segel System
September 26, 2012

Keywords of the presentation: randomization, Gibbs measure, well-posedness

The center theme of this talk is the effect that randomization on the initial data set has on questions of global well-posedness for a variety of evolution equations.

I will start by recalling the notion of Gibbs measure for certain periodic dispersive equations in Hamiltonian form, a work that goes back to Lebowitz-Rose-Speer. I will continue with a short summary of the work of Bourgain, who proved invariance of the Gibbs measure for certain NLS equation and an almost sure global well-posedness as a consequence. I will then continue by illustrating how randomization can be effectively used even when an Hamiltonian structure is not present and as a consequence a Gibbs measure cannot be defined. I will illustrate in this context results proved for example for the Navier-Stokes and wave equations in the supercritical regime.

The linear stability for the subsonic solitary waves of the one-dimensional Benney-Luke equation is investigated rigorously and the critical wave speeds are computed explicitly. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.

Keywords of the presentation: determinig modes, determining nodes, feedback control, Navier-Stokes equations, data assimilations, dissipative eviolution equations, determinig forms

In this talk we will implement the notion of finite number of determining parameters for the long-time dynamics of the Navier-Stokes equations (NSE), such as determining modes, nodes, volume elements, and other determining interpolants, to design finite-dimensional feedback control for stabilizing their solutions. The same approach is found to be applicable for data assimilations. In addition, we will show that the long-time dynamics of the NSE can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named "determining form", which is governed by a globally Lipschitz vector field. The NSE are used as an illustrative example, and all the above mentioned results hold also to other dissipative evolution PDEs.

Blow-up for the quantum Boltzmann equation
September 27, 2012

The Boltzmann-Nordheim equation describes the evolution of the distribution of energies of a quantum weakly interacting many particle system. It has been conjectured in the physical literature that the solutions of the Boltzmann-Nordheim equation might blow-up in a finite time. Such blow-up is expected to be related to the dynamical formation of Bose-Einstein condensates. In this talk it will describe a large class of initial data for which it is possible to prove blow-up in finite time for the corresponding solutions of the Boltzmann-Nordheim equation. (Joint work with M. Escobedo).

Grain Boundaries in Swift-Hohenberg equations
December 31, 1969

We study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of di erent orientation. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angle between the roll solutions. The main difficulty stems from possible interactions of the primary modes with other resonant modes. We develop a normal form analysis and a singular perturbation approach to treat resonances

Keywords of the presentation: Asymptotic Behavior of Singular Solutions

We consider singular solutions of a parabolic partial differential equation with power nonlinearity. It is known that in some range of parameters, this equation has a family of singular steady states with ordered structure. Our concern is the existence of time-dependent singular solutions and their asymptotic behavior, the convergence of solutions to singular steady states. Our method is based on the analysis of a related linear equation with a singular coefficient and the comparison principle. This is a joint work with Masaki Hoshino and Shota Sato.

Stationary measures of Fokker-Planck equations
September 25, 2012

Keywords of the presentation: Stationary measures, Fokker-Planck equations

We consider white noise perturbations of a system of ordinary differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence results of stationary measures in a general domain including the entire space will be presented for both non-degenerate and degenerate noises. Limiting behaviors of stationary measures will be discussed along with applications to problems of stochastic stability and bifurcations.

Our paper is concerned about maximum principle and radial symmetry of viscosity solution for fully nonlinear equation. We obtain the radial symmetry for non-proper fully nonlinear elliptic equation in Euclidean space under certain asymptotic rate at infinity. New maximum principles for fully nonlinear equation are established. Our results apply to Pucci's extremal operators, Bellman or Isaccs equations. Radial symmetry and the corresponding maximum principle in a punctured ball are shown. We also study the symmetry of viscosity solution for fully parabolic equations.

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