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

stable self similar blowup in energy supercritical yang mills theory we consider the cauchy problem for an energy supercritical nonlinear wave equation that arises in 1 5 dimensional yang mills theory a certain self similar solution w 0 of this model is conjectured to act as an attractor for generic large data evolutions assuming mode stability of w 0 we prove a weak version of this conjecture namely that the self similar solution w 0 is nonlinearly stable phrased differently we prove that mode stability of w 0 implies its nonlinear stability the fact that this statement is not vacuous follows from careful numerical work by bizon and chmaj that verifies the mode stability of w 0 beyond reasonable doubt

profile for the imaginary part of a blowup solution for a complex valued semilinear heat equation abstract in this paper we consider the following complex valued semilinear heat equation t u u u p u c in the whole space r n where p n p 2 we aim at constructing 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 with the following estimates for the final profile 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 as x a note that the imaginary part is non zero and that it blows up also at point a our method 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 up to our knowledge this is the first time where the blowup behavior of the imaginary part is derived in multi dimension

blowup stability at optimal regularity for the critical wave equation abstract we establish strichartz estimates for the radial energy critical wave equation in 5 dimensions in similarity coordinates using these we prove the nonlinear asymptotic stability of the ode blowup in the energy space

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

strichartz estimates in similarity coordinates and stable blowup for the critical wave equation we establish strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time dependent self similar potential as an application we consider the critical wave equation and prove the asymptotic stability of the ode blowup profile in the energy space

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

on a consistency measure for object labeling problems abstract the problem of defining a consistency measure for object labeling problems is addressed the optimal probabilistic measures are derived for both discrete and fuzzy solutions alternative fuzzy consistency measures are suggested based on the error each object has in estimating its label using the labels of its neighbors in general the best estimation is impractical from computational complexity aspects as well as information considerations however if the information available about the problem is probabilities of pairs of events the best estimation is a linear form

blowup behaviour for the nonlinear klein gordon equation we analyze the blowup behaviour of solutions to the focusing nonlinear klein gordon equation in spatial dimensions d ge 2 we obtain upper bounds on the blowup rate both globally in space and in light cones the results are sharp in the conformal and sub conformal cases the argument relies on lyapunov functionals derived from the dilation identity we also prove that the critical sobolev norm diverges near the blowup time

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

finite index theorems for iterated galois groups of cubic polynomials let k be a number field or a function field let f in k x be a rational function of degree d ge 2 and let beta in mathbb p 1 overline k for all n in mathbb n cup infty the galois groups g n beta mathrm gal k f n beta k beta embed into mathrm aut t n the automorphism group of the d ary rooted tree of level n a major problem in arithmetic dynamics is the arboreal finite index problem determining when mathrm aut t infty g infty beta infty when f is a cubic polynomial and k is a function field of transcendence degree 1 over an algebraic extension of mathbb q we resolve this problem by proving a list of necessary and sufficient conditions for finite index this is the first result that gives necessary and sufficient conditions for finite index and can be seen as a dynamical analog of the serre open image theorem when k is a number field our proof is conditional on both the abc conjecture for k and vojta s conjecture for blowups of mathbb p 1 times mathbb p 1 we also use our approach to solve some natural variants of the finite index problem for modified trees

uniqueness of meromorphic functions whose derivatives share four small functions in this paper we will prove some uniqueness theorems of meromorphic functions whose derivatives share four distinct small functions the results in this paper improve those given by r nevanlinna l yang g d qiu and other authors an example is provided to show that the results in this paper are best possible

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

a note on partial sharing of values of meromorphic functions with their shifts abstract we introduce the notion of partial value sharing as follows let e a f be the set of zeros of f z a z where each zero is counted only once and a is a meromorphic function small with respect to f a meromorphic function f is said to share a partially with a meromorphic function g if e a f e a g we show that partial value sharing of f z and f z c involving 3 or 4 values combined with an appropriate deficiency assumption is enough to guarantee that f z f z c provided that f z is a meromorphic function of hyper order strictly less than one and c c

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

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

the blow up boundary for nonlinear wave equations consider the cauchy problem for a nonlinear wave equation du f u in n space dimensions n x with x continuously difierentiable introduction consider the nonlinear wave equation 0 1 uu u t au f u for x 6 r i 0 with the initial data 0 2 u x 0 x ul x 0 g x for x g rn it is well known that if f u is nonnegative and superlinear then in general a solution cannot exist for all times furthermore if t is the supremum of all times s such that a classical solution exists for all 0 t s then sup w jc t oo if t t

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

on the solvability of the cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part we consider one multidimensional version of the cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part depending on the exponent of nonlinearity and spatial dimension of equation we investigate the problem on the nonexistence of global solutions of the cauchy characteristic problem the question on the local solvability of that problem is also considered

uniqueness of meromorphic functions sharing a small function and its applications in this paper we shall study the uniqueness problems of meromorphic functions sharing a small function our results improve or generalize many previous results on value sharing of meromorphic functions

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

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

normal families of meromorphic functions with multiple values let f be a family of meromorphic functions defined in a domain d let 0 be a holomorphic function in d and k be a positive integer suppose that for every function f f f 0 f k 0 and all zeros of f k z have multiplicities at least k 2 k if for k 1 has only zeros with multiplicities at most 2 and for k 2 has only simple zeros then f is normal in d this improves and generalizes the related results of gu fang and chang yang schwick et al

bounded solutions to an energy subcritical non linear wave equation on r3 abstract in this work we consider an energy subcritical semi linear wave equation 3 p 5 t 2 u u x u p 1 u x t r 3 r u t 0 u 0 h s p r 3 t u t 0 u 1 h s p 1 r 3 where s p 3 2 2 p 1 and the function r 3 1 1 is a radial continuous function with a limit at infinity we prove that unless the elliptic equation w x w p 1 w has a nonzero radial solution w c 2 r 3 h s p r 3 any radial solution u with a finite uniform upper bound on the critical sobolev norm u t t u t h s p h s p r 3 for all t in the maximal lifespan must be a global solution in time and scatter

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

the radial wave operator in similarity coordinates we present a rigorous functional analytic setting to study the radial wave equation in similarity coordinates as an application we analyze linear stability of the fundamental self similar solution of the wave equation with a focusing power nonlinearity this is part 1 in a series of papers donninger r asymptotics and analytic modes for the wave equation in similarity coordinates j evol equ 9 511 2009 donninger r nonlinear stability of self similar solutions for semilinear wave equations commun partial differ equ c 2010 american institute of physics doi 10 1063 1 3299302

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

improving consistency and reducing ambiguity in stochastic labeling an optimization approach we approach the problem of labeling a set of objects from a quantitative standpoint we define a world model in terms of transition probabilities and propose a definition of a class of global criteria that combine both ambiguity and consistency a projected gradient algorithm is developed to minimize the criterion we show that the minimization procedure can be implemented in a highly parallel manner results are shown on several examples and comparisons are made with relaxation labeling techniques

classification of singularities for blowing up solutions in higher dimensions consider the cauchy problem p u t u u p when x r n t 0 n 1 u x 0 u 0 x when x r n where p 1 and u 0 x is a continuous nonnegative and bounded function it is known that under fairly general assumptions on u 0 x the unique solution of p u x t blows up in a finite time by which we mean that lim sup sup u x t t t x r n in this paper we shall assume that u x t blows up at x 0 t t and derive the possible asymptotic behaviours of u x t as x t 0 t under general assumptions on the blow up rate

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

the blow up rate for a non scaling invariant semilinear wave equations abstract we consider the semilinear wave equation 1 t 2 u u f u x t r n 0 t with f u u p 1 u log a 2 u 2 where p 1 and a r we show an upper bound for any blow up solution of 1 then in the one space dimensional case using this estimate and the logarithmic property we prove that the exact blow up rate of any singular solution of 1 is given by the ode solution associated with 1 namely u u p 1 u log a 2 u 2 unlike the pure power case g u u p 1 u the difficulties here are due to the fact that equation 1 is not scale invariant

slow blow up solutions for the h 1 mathbb r 3 critical focusing semilinear wave equation given 1 2 and 0 arbitrary we prove the existence of energy solutions of ttu u u5 0 0 1 in r3 1 which blow up exactly at r t 0 as t 0 these solutions are radial and of the form u t 1 2w t r r t inside the cone r t where t t 1 w r 1 r2 3 1 2 is the stationary solution of 0 1 and is a radiation term with r t x t 2 t x t 2 x t 6 dx 0 t 0 outside of the light cone there is the energy bound r t u x t 2 ut x t 2 u x t 6 dx 0 the regularity of u increases with as in our accompanying article on wave maps 10 the argument is based on a renormalization method for the soliton profile w r

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

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

stable blow up dynamics for energy supercritical wave equations we study the semilinear wave equation displaystyle partial t 2 psi delta psi vert psi vert p 1 psi for p 3 with radial data in three spatial dimensions there exists an explicit solution which blows up at t t 0 given by displaystyle psi t t x c p t t frac 2 p 1 where c p is a suitable constant we prove that the blow up described by psi t is stable in the sense that there exists an open set in a topology strictly stronger than the energy of radial initial data that leads to a solution which converges to psi t as t to t in the backward lightcone of the blow up point t r t 0

stable blowup for the cubic wave equation in higher dimensions abstract we consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data we prove that there exists an open set of initial data such that the corresponding solution exists in a backward light cone and approaches the ode blowup profile

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