emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

twisted groups and locally toroidal regular polytopes in recent years much work has been done on the classification of abstract regular polytopes by their local and global topological type abstract regular polytopes are combinatorial structures which generalize the well known classical geometric regular polytopes and tessellations in this context the classical theory is concerned with those which are of globally or locally spherical type in a sequence of papers the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal here this investigation of locally toroidal regular polytopes is continued with a particular emphasis on polytopes of ranks 5 and 6 for large classes of such polytopes their groups are explicitly identified using twisting operations on quotients of coxeter groups in particular this leads to new classification results which complement those obtained elsewhere the method is also applied to describe certain regular polytopes with small facets and vertex figures

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

solitons and vertex operators in twisted affine toda field theories affine toda field theories in two dimensions constitute families of integrable relativistically invariant field theories in correspondence with the affine kac moody algebras the particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary here these results are extended from the untwisted to the twisted algebras the new soliton solutions and their masses are found by a folding procedure which can be applied to the affine kac moody algebras themselves to provide new insights into their structures the relevant foldings are related to inner automorphisms of the associated finite dimensional lie group which are calculated explicitly and related to what is known as the twisted coxeter element the fact that the twisted affine kac moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions

bubble tower radial solutions in the slightly supercritical brezis nirenberg problem abstract in this paper we consider the brezis nirenberg problem in dimension n 4 in the supercritical case we prove that if the exponent gets close to n 2 n 2 and if simultaneously the bifurcation parameter tends to zero at the appropriate rate then there are radial solutions which behave like a superposition of bubbles namely solutions of the form j 1 k 1 1 m j 4 n 2 y 2 n 2 2 m j 1 o 1 n n 2 n 2 4 where mj and mj o mj 1 for all j these solutions lie close to turning points to the right of the associated bifurcation diagram

hermitian forms and locally toroidal regular polytopes in the classical theory of regular polytopes the structure of a finite or infinite polytope honeycomb is governed by a real quadratic form this form determines a geometry into which the polytope is embedded in such a way that all its symmetries are realized by isometries the symmetry group is the coxeter group associated with the quadratic form cf coxeter 8 these facts have important consequences for other fields in mathematics see for example tits 29 303 the purpose of this paper is to show that much of the correspondence between polytopes and forms remains true for the theory of abstract regular polytopes that is regular incidence polytopes the concept of regular incidence polytopes provides a suitable setting for combinatorial structures resembling the classical regular polytopes cf danzer schulte 131 for related notions see also mcmullen 171 griinbaum 161 dress 14 buekenhout 2 and tits 29 in i163 griinbaum suggested studying abstract regular polytopes whose faces and vertex figures are not necessarily of spherical type he posed the problem of classifying all finite universal or in his notation naturally generated abstract regular polytopes y p see also coxeter shephard ll weiss 31 321 and 24 25 in 20 21 this problem was attacked by twisting operations on coxeter groups and unitary reflexion groups leading to the explicit recognition of the universal i p for many choices of 9 and p in this paper we completely classify all the finite universal polytopes of

singular extremal solutions for supercritical elliptic equations in a ball abstract we study positive singular solutions to the dirichlet problem for the semilinear elliptic equation u f u 0 in the unit ball on r n we assume that f has the form f u u p g u where p n 2 n 2 and g u is a lower order term we first show the uniqueness of the singular solution to the problem and then study the existence of the singular extremal solution in particular we show a necessary and sufficient condition for the existence of the singular extremal solution in terms of the weak eigenvalue of the linearized problem

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

a necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations in their 1968 paper fujita and watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded non negative solutions in particular they showed that if the underlying ode has non unique solutions as characterised via an osgood type condition and the nonlinearity f satisfies a concavity condition then the parabolic pde also inherits the non uniqueness property this concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other similar non uniqueness phenomena in parabolic equations in this paper we provide an elementary proof of non uniqueness for the pde without any such concavity assumption on f an important consequence of our result is that uniqueness of the trivial solution of the pde is equivalent to uniqueness of the trivial solution of the corresponding ode which in turn is known to be equivalent to an osgood type integral condition on f

the cauchy problem for heat equations with exponential nonlinearity abstract the cauchy problem for the semilinear heat equations is studied in the orlicz space exp l 2 r n where any power behavior of interaction works as a subcritical nonlinearity we prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp l 2 r n

higher toroidal regular polytopes abstract quad a regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids the locally toroidal polytopes of rank 4 have been extensively discussed and their classification is now nearly complete in this paper the locally toroidal polytopes of higher rank are investigated again an almost complete classification of these regular polytopes is obtained as well as sporadic examples there are several infinite families

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

singularity formation for the two dimensional harmonic map flow into s 2 we construct finite time blow up solutions to the 2 dimensional harmonic map flow into the sphere s 2 begin aligned u t delta u nabla u 2 u quad text in omega times 0 t u varphi quad text on partial omega times 0 t u cdot 0 u 0 quad text in omega end aligned where omega is a bounded smooth domain in mathbb r 2 u omega times 0 t rightarrow s 2 u 0 bar omega rightarrow s 2 is smooth and varphi u 0 big partial omega given any k points q 1 ldots q k in the domain we find initial and boundary data so that the solution blows up precisely at those points the profile around each point is close to an asymptotically singular scaling of a 1 corotational harmonic map we build a continuation after blow up as a h 1 weak solution with a finite number of discontinuities in space time by reverse bubbling which preserves the homotopy class of the solution after blow up furthermore we prove the codimension one stability of the one point blow up phenomenon

glm equations tau function and scattering data the direct and the inverse scattering problem for affine toda mkdv systems is addressed and is found to develop non standard features within the framework of the inverse scattering method a solution scheme based on the tau function formalism is described the inverse problem is shown to be equivalent to a set of decoupled scalar gelfand levitan marchenko type equations the fredholm grothendieck determinants of the latter are shown to define tau functions in the sense of the kyoto school in particular a simple monodromy formula allows the derivation of trace identities

geometry and integrability of topological antitopological fusion integrability of equations of topological antitopological fusion being proposed by cecotti and vafa describing the ground state metric on a given 2d topological field theory tft model is proved for massive tft models these equations are reduced to a universal form being independent on the given tft model by gauge transformations for massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method also it is shown that the ground state metric together with some part of the underlined tft structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms

locally projective polytopes of type 4 3 3 p this paper attempts to classify the locally projective section regular n polytopes of type 4 3 3 p that is to classify polytopes whose facets are cubes or hemicubes and the vertex figures are spherical or projective polytopes of type 3 3 p with the facets and vertex figures being not both spherical spherical or projective n 1 polytopes of type 3 3 p only exist when p 4 or p 5 and n 1 4 or n 1 2 however some existence and non existence results are obtained for other values of p and n in particular a link is derived between the existence of polytopes of certain types and vertex colourability of certain graphs the main result of the paper is that locally projective section regular n polytopes exist only when p 4 or when p 5 and n 4 or 5

generalized drinfel d sokolov hierarchies ii the hamiltonian structures in this paper we examine the bi hamiltonian structure of the generalized kdv hierarchies we verify that both hamiltonian structures take the form of kirillov brackets on the kac moody algebra and that they define a coordinated system classical extended conformal algebras are obtained from the second poisson bracket in particular we construct the w n l algebras first discussed for the case n 3 and l 2 by polyakov and bershadsky

optimal condition for blow up of the critical lq norm for the semilinear heat equation abstract we shed light on a long standing open question for the semilinear heat equation u t u u p 1 u namely without any restriction on the exponent p 1 nor on the smooth domain we prove that the critical l q norm blows up whenever the solution undergoes type i blow up a similar property is also obtained for the local critical l q norm near any blow up point in view of recent results of existence of type ii blow up solutions with bounded critical l q norm which are counter examples to the open question our result seems to be essentially the best possible result in general setting this close connection between type i blow up and critical l q norm blow up appears to be a completely new observation our proof is rather involved and requires the combination of various ingredients it is based on analysis in similarity variables and suitable rescaling arguments combined with backward uniqueness and unique continuation properties for parabolic equations as a by product we obtain the nonexistence of self similar profiles in the critical l q space such properties were up to now only known for p p s and in radially symmetric case for p p s where p s is the sobolev exponent

geometry of phase space and solutions of semilinear elliptic equations in a ball we consider the problem u u p u in in b u 0 in b u 0 on b where b denotes the unit ball in r n n 3 a 0 and p 1 merle and peletier showed that for p n 2 n 2 there is a unique value a 0 such that a radial singular solution exists this value is the only one at which an unbounded sequence of classical solutions of 1 may accumulate here we prove that if additionally p 0 0 in b u 0 on b where f s s p s q 1 q p and p satisfies the same condition as above

null vectors in integrable field theory the form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine gordon model this space has to be isomorphic to that of the corresponding minimal model of conformal field theory we describe the subspaces which correspond to the verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion neveu schwarz or ramond depending on the particular primary field the description of null vectors relies on the relation between form factors and deformed hyper elliptic integrals the null vectors correspond to the deformed exact forms and to the deformed riemann bilinear identity in the operator language the null vectors are created by the action of two operators linear in the fermion and quadratic in the fermion we show that by factorizing out the null vectors one gets the space of operators with the correct character in the classical limit using the operators and we obtain a new very compact description of the kdv hierarchy we also discuss a beautiful relation with the method of whitham

a limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth abstract we study radial solutions of the semilinear elliptic equation u f u 0 under rather general growth conditions on f we construct a radial singular solution and study the intersection number between the singular solution and a regular solution an application to bifurcation problems of elliptic dirichlet problems is given to this end we derive a certain limit equation from the original equation at infinity using a generalized similarity transformation through a generalized cole hopf transformation all the limit equations can be reduced into two typical cases i e u u p 0 and u e u 0

nonuniqueness for a critical nonlinear heat equation with any initial data abstract we establish for any initial data u 0 l n n 2 r n with n 3 the existence of an infinite number of solutions of the cauchy problem for the nonlinear heat equation t u u u 2 n 2 u in 0 t r n u 0 u 0 in r n in the class c 0 t l n n 2 r n

semilinear elliptic equations and supercritical growth where p p the aim of the present paper is to understand for the case p pc the behaviour of solutions to 1 i with large supremum norm to this end 0 is restricted to be the unit ball f3c r3 as in this case the behaviour of solutions to 1 1 can be described in some detail according to the results proved in gidas ni and nirenberg s all positive solutions of 1 1 must be radial when q b with the scaling iid ilip in the case n 3 problem 1 1 reduces to the following ordinary differential equation problem

vertex operator representation of the soliton tau functions in the an 1 toda models by dressing transformations we study the relation between the group algebraic approach and the dressing symmetry one to the soliton solutions of the an 1 toda field theory in 1 1 dimensions originally solitons in the affine toda models were found by olive turok and underwood single solitons are created by exponentials of elements which ad diagonalize the principal heisenberg subalgebra alternatively babelon and bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine gordon model in this paper we show the equivalence between these two methods to construct solitons in the an 1 toda models

existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case abstract in this paper we consider a class of semilinear elliptic dirichlet problems in a bounded regular domain with cylindrical symmetry involving concave convex nonlinearities with supercritical growth using a new sobolev embedding theorem and variational method we show the existence of two positive solutions of the problem additionally we study the quasilinear elliptic equation and obtain a similar result

non uniqueness for an energy critical heat equation on mathbb r 2 r2 we construct a singular solution of a stationary nonlinear schrodinger equation on mathbb r 2 with square exponential nonlinearity having linear behavior around zero in view of trudinger moser inequality this type of nonlinearity has an energy critical growth we use this singular solution to prove non uniqueness of mild solutions for the cauchy problem of the corresponding semilinear heat equation the proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space

an 1 toda solitons and the dressing symmetry we present an elementary derivation of the solitonlike solutions in the an 1 toda models which is an alternative to the previously used hirota method the solutions of the underlying linear problem corresponding to the n solitons are calculated this enables us to obtain explicit expression for the element which by dressing group action produces a generic soliton solution in the particular example of monosolitons we suggest a relation to the vertex operator formalism previously used by olive turok and underwood our results can also be considered as generalization of the approach to the sine gordon solitons proposed by babelon and bernard

the heat equation with singular nonlinearity and singular initial data abstract we study the existence uniqueness and regularity of positive solutions of the parabolic equation u t u a x u q b x u p in a bounded domain and with dirichlet s condition on the boundary we consider here a l b l and 0 q 1 p the initial data u 0 u 0 is considered in the space l r r 1 in the main result 0 q 1 we assume a b 0 a e in and we assume that u 0 d for some 0 we find a unique solution in the space c 0 t l r l loc 0 t l

phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball abstract we consider the following problem 0 1 p u u f u r 0 u 0 in b and u 0 on b where b is the unitary ball in r n merle and peletier considered the classical laplace case p 2 and proved the existence of a unique value 0 for which a radial singular positive solution exists assuming f u r u q 1 and q 2 2 n n 2 then dolbeault and flores proved that if q 2 but q is smaller than the joseph lundgren exponent then there is an unbounded sequence of radial positive classical solutions for 0 1 which accumulate at 0 again for p 2 we extend both merle peletier and dolbeault flores results to the p laplace setting with the technical restriction 1 p 2 and to more general nonlinearities f which may have more complicated dependence on u and may be spatially non homogeneous then we reproduce the results also for similar bifurcation problems where the linear term u is replaced by a superlinear and subcritical term of the form r u u q 2 our analysis relies on a generalized fowler transformation and profits of invariant manifold theory and it allows to discuss radial nodal solutions too

the emden fowler equation on a spherical cap of sn abstract let s n r n 1 n 3 be the unit sphere and let s s n be a geodesic ball with geodesic radius 0 we study the bifurcation diagram u r 2 of the radial solutions of the emden fowler equation on s s n u u p 0 in s u 0 on s u 0 in s where p 1 among other things we prove the following for each p p s n 2 n 2 there exists 0 such that the problem has a radial solution for and has no radial solution for 0 moreover this solution is unique in the space of radial functions if is close to if p s p p jl then there exists such that the problem has infinitely many radial solutions for where p jl 1 4 n 4 2 n 1 if n 11 if 2 n 10 asymptotic behaviors of the bifurcation diagram as p and p 1 are also studied

a bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions we consider the following semilinear elliptic equation where b 1 is the unit ball in d d 3 0 and p 0 firstly following merle and peletier we show that there exists an eigenvalue p such that has a solution p w p satisfying lim x 0 w p x secondly we study a bifurcation diagram of regular solutions to it follows from the result of dancer that has an unbounded bifurcation branch of regular solutions that emanates from u 0 0 here using the singular solution we show that the bifurcation branch has infinitely many turning points around p when 3 d 9 we also investigate the morse index of the singular solution in the d 11 case

tau functions and dressing transformations for zero curvature affine integrable equations the solutions of a large class of hierarchies of zero curvature equations that includes toda and kdv type hierarchies are investigated all these hierarchies are constructed from affine twisted or untwisted kac moody algebras g their common feature is that they have some special vacuum solutions corresponding to lax operators lying in some abelian up to the central term subalgebra of g in some interesting cases such subalgebras are of the heisenberg type using the dressing transformation method the solutions in the orbit of those vacuum solutions are constructed in a uniform way then the generalized tau functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest weight representations of g such definition of tau functions applies for any level of the representation and it is independent of its realization vertex operator or not the particular important cases of generalized mkdv and kdv hierarc

regular polytopes from twisted coxeter groups and unitary reflexion groups abstract in this paper we discuss the construction of regular incidence polytopes p by twisting operations on coxeter groups and quotients of coxeter groups particular attention is paid to the polytopes obtained from the unitary complex groups generated by reflexions of period 2 in particular this leads to the explicit recognition of the universal regular incidence polytopes p 1 p 2 in a number of interesting cases of regular incidence polytopes p 1 and p 2

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure