one dimensional behavior of singular n dimensional solutions of semilinear heat equations we consider u x t a solution of u t u u p 1 u that blows up at time t where u n 0 t p 1 n 2 p 0 in particular if in addition p 3 then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow up set finally we prove that the mere hypothesis that the blow up set is continuous implies that it is c 1 1 2 for any 0

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

hexagonal circle patterns and integrable systems patterns with constant angles hexagonal circle patterns with constant intersection angles are introduced and studied it is shown that they are described by discrete integrable systems of toda type conformally symmetric patterns are classified circle pattern analogs of holomorphic mappings z sp c and log z are constructed as special isomonodromic solutions circle patterns studied in the paper include schramm s circle patterns with the combinatorics of the square grid as a special case

life span of solutions for a semilinear parabolic problem with small diffusion abstract this paper is concerned with the initial boundary value problem formula where p 1 e 0 is a bounded domain in rn and is a continuous function on it is shown that the blowup time t e of the solution of this problem satisfies t e 1 p 1 1 p as e 0 moreover when the maximum of x is attained at one point we determine the higher order term of t e which reflects the pointedness of the peak of the proof is based on a careful construction of super and subsolutions

a direct one step pressure actualization for incompressible flow with presssure neumann condition we develop a velocity pressure algorithm in primitive variables and finite differences for incompressible viscous flow with a neumann pressure boundary condition the pressure field is initialized by least squares and up dated from the poisson equation in one step without iteration simulations with the square cavity problem are made for several reynolds numbers we obtain the expected displacement of the central vortex and the appearance of secondary and tertiary eddies different geometry ratios and a 3d cavity simulation are also considered

blow up for parabolic and hyperbolic problems with variable exponents abstract in this paper we study the blow up problem for positive solutions of parabolic and hyperbolic problems with reaction terms of local and nonlocal type involving a variable exponent we prove the existence of initial data such that the corresponding solutions blow up at a finite time

global decaying solutions of a focusing energy critical heat equation in r4 abstract we study solutions of the focusing energy critical nonlinear heat equation u t u u 2 u in r 4 we show that solutions emanating from initial data with energy and h 1 norm below those of the stationary solution w are global and decay to zero via the concentration compactness plus rigidity strategy of kenig merle 33 34 first global such solutions are shown to dissipate to zero using a refinement of the small data theory and the l 2 dissipation relation finite time blow up is then ruled out using the backwards uniqueness of escauriaza seregin sverak 17 18 in an argument similar to that of kenig koch 32 for the navier stokes equations

the blow up rate of solutions of semilinear heat equations on caracterise le comportement asymptotique de la solution de l equation de la chaleur semilineaire u t u exp u dans o t u k sur o t u x 0 x k pour xe k 0 dans un voisinage d un point d explosion quand t approche du temps d explosion fini t

numerical analysis of the rescaling method for parabolic problems with blow up in finite time abstract in this work we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u u x t 2 p 1 u x 2 t for that purpose we apply the rescaling method proposed by berger and kohn 1988 to such problems the convergence of the method is proved under some regularity assumption some numerical experiments are given to derive the blow up profile verifying henceforth the theoretical results

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

riccati ermakov systems and explicit solutions for variable coefficient reaction diffusion equations abstract we present several families of nonlinear reaction diffusion equations with variable coefficients including generalizations of fisher kpp and burgers type equations special exact solutions such as traveling wave rational triangular wave and n wave type solutions are shown by means of similarity transformations the variable coefficients are conditioned to satisfy riccati or ermakov systems of equations when the riccati system is used conditions are established so that finite time singularities might occur we explore solution dynamics across multi parameters in the supplementary material we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

the blow up problem for a semilinear parabolic equation with a potential abstract let be a bounded smooth domain in r n we consider the problem u t u v x u p in 0 t with dirichlet boundary conditions u 0 on 0 t and initial datum u x 0 m x where m 0 is positive and compatible with the boundary condition we give estimates for the blow up time of solutions for large values of m as a consequence of these estimates we find that for m large the blow up set concentrates near the points where p 1 v attains its maximum

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

on the behavior of blow up solutions to a parabolic problem with critical exponent abstract the blow up behavior of the solution to a semilinear equation with critical exponent u t u u p 1 u in 0 t u 0 on 0 t u 0 u 0 on is established we obtain that if is star shaped about a and n 3 p n 2 n 2 then lim t t t t u a y t t t exists and equals 0 or in l loc 2 r n where 1 p 1

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

blowup solutions for a reaction diffusion system with exponential nonlinearities abstract we consider the following parabolic system whose nonlinearity has no gradient structure t u u e p v t v v e q u u 0 u 0 v 0 v 0 p q 0 in the whole space r n we show the existence of a stable blowup solution and obtain a complete description of its singularity formation the construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude in particular our analysis uses neither the maximum principle nor the classical methods based on energy type estimates which are not supported in this system the stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem

global existence for a vector valued nonlinear heat equation in this paper we are concerned with the global existence the uniqueness the nonnegativity of classical solutions of a semilinear parabolic coupled system

some blow up problems for a semilinear parabolic equation with a potential the blow up rate estimate for the solution to a semilinear parabolic equation ut u v x u p 1u in 0 t with 0 dirichlet boundary condition is obtained as an application it is shown that the asymptotic behavior of blow up time and blow up set of the problem with nonnegative initial data u x 0 m x as m goes to infinity which have been found in c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006 is improved under some reasonable and weaker conditions compared with c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006

integrable cluster dynamics of directed networks and pentagram maps abstract the pentagram map was introduced by r schwartz more than 20 years ago in 2009 v ovsienko r schwartz and s tabachnikov established liouville complete integrability of this discrete dynamical system in 2011 m glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver using compatibility of poisson and cluster structures and poisson geometry of directed networks on surfaces we generalize glick s construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations the appendix relates the simplest of these discrete maps to the toda lattice and its tri hamiltonian structure

the method of layer potentials for the heat equation in lipschitz cylinders pour l equation de la chaleur dans des cylindres de lipschitz on etudie des problemes aux valeurs limites d initiales avec conditions de neumann et de dirichlet

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

a blowup solution of a complex semi linear heat equation with an irrational power abstract in this paper we consider the following semi linear complex heat equation t u u u p u c in r n with an arbitrary power p p 1 we construct for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a moreover we also describe the asymptotics of the solution by the following final profiles u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a 0 as x a in addition to that since we also have u 1 0 t and u 2 0 t as t t the blowup in the imaginary part shows a new phenomenon unknown for the standard heat equation in the real case a non constant sign near the singularity with the existence of a vanishing surface for the imaginary part shrinking to the origin in our work we have succeeded to extend for any power p where the non linear term u p is not continuous if p is not integer in particular the solution which we have constructed has a positive real part we study our equation as a system of the real part and the imaginary part u 1 and u 2 our work relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

toric networks geometric r matrices and generalized discrete toda lattices we use the combinatorics of toric networks and the double affine geometric r matrix to define a three parameter family of generalizations of the discrete toda lattice we construct the integrals of motion and a spectral map for this system the family of commuting time evolutions arising from the action of the r matrix is explicitly linearized on the jacobian of the spectral curve the solution to the initial value problem is constructed using riemann theta functions

excluding blowup at zero points of the potential by means of liouville type theorems we consider the diffusive hamilton jacobi equation with homogeneous dirichlet conditions and regular initial data it is known from barles dalio 2004 that the problem admits a unique continuous global viscosity solution which extends the classical solution in case gradient blowup occurs we study the question of the possible loss of boundary conditions after gradient blowup which seems to have remained an open problem till now somewhat surprisingly our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena for any smooth bounded domain we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup actually we show that the latter possibility is rather exceptional more generally we show that the set of the points where boundary conditions are lost can be prescribed to be arbitrarily close to any given open subset of the boundary

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

the critical fujita exponent for the fast diffusion equation with potential abstract this paper studies the cauchy problem for positive solutions of the fast diffusion equation with source and quadratically decaying potential u t u m v x u m u p in r n 0 t where 1 2 m n m 1 p 1 n 2 v x x 2 with 0 as x and is the positive root of m m n 2 0 we obtain the critical fujita exponent p c m 2 m n to the problem in the sense that every nontrivial solution blows up in finite time when 1 p p c and there are both global and non global solutions if p p c

blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions abstract this paper concerns with blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space we establish the rate estimates for blow up solutions and prove that the blow up set is r n under proper conditions on initial data furthermore for n 1 more complete conclusions about such two topics are given

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

non commutative integrability of the grassmann pentagram map abstract the pentagram map is a discrete dynamical system first introduced by schwartz in 1992 it was proved to be integrable by schwartz ovsienko and tabachnikov in 2010 gekhtman shapiro and vainshtein studied poisson geometry associated to certain networks embedded in a disc or annulus and its relation to cluster algebras later gekhtman et al and tabachnikov reinterpreted the pentagram map in terms of these networks and used the associated poisson structures to give a new proof of integrability in 2011 mari beffa and felipe introduced a generalization of the pentagram map to certain grassmannians and proved it had a lax representation we reinterpret this grassmann pentagram map in terms of noncommutative algebra in particular the double brackets of van den bergh and generalize the approach of gekhtman et al to establish a noncommutative version of integrability

construction of a stable periodic solution to a semilinear heat equation with a prescribed profile we construct a periodic solution to the semilinear heat equation with power nonlinearity in one space dimension which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude thanks to the geometrical interpretation of the finite dimensional parameters in terms of the blow up time and blow up point we derive the stability of the constructed solution with respect to initial data

liouville arnold integrability of the pentagram map on closed polygons the pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane this map has recently attracted a considerable interest mostly because its connection to a number of different domains such as classical projective geometry algebraic combinatorics moduli spaces cluster algebras and integrable systems integrability of the pentagram map was conjectured by schwartz and proved by the present authors for a larger space of twisted polygons in this article we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons in the case of convex polygons in the real projective plane this result implies the existence of a toric foliation on the moduli space the leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic our proof is based on an invariant poisson structure on the space of twisted polygons we prove that the hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants

integrability of higher pentagram maps we define higher pentagram maps on polygons in mathbb p d for any dimension d which extend r schwartz s definition of the 2d pentagram map we prove their integrability by presenting lax representations with a spectral parameter for scale invariant maps the corresponding continuous limit of the pentagram map in dimension d is shown to be the 2 d 1 equation of the kdv hierarchy generalizing the boussinesq equation in 2d we also study in detail the 3d case where we prove integrability for both closed and twisted polygons and describe the spectral curve first integrals the corresponding tori and the motion along them as well as an invariant symplectic structure

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

the pentagram map and y patterns abstract the pentagram map introduced by r schwartz is defined by the following construction given a polygon as input draw all of its shortest diagonals and output the smaller polygon which they cut out we employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map

integrable deformations of a polygon abstract a new integrable mapping with a simple geometric interpretation is presented this mapping arises from the nonlinear superposition principle for the backlund transformations of some vector evolution equation