parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

rate distortion function for a heegard berger problem with two sources and degraded reconstruction sets in this paper we investigate an instance of the heegard berger problem with two sources and arbitrarily correlated side information sequences at two decoders in which the reconstruction sets at the decoders are degraded specifically two sources are to be encoded in a manner that one of the two is reproduced losslessly by both the decoders and the other is reproduced within some prescribed distortion level at one of the two decoders we establish a single letter characterization of the rate distortion function for this model in particular we show that the optimal coding scheme for this setting is one in which the common description to be recovered by both the decoders should allow to involve all or part of the source that is to be reproduced at only one decoder furthermore we also generalize our result to the setting in which the source component that is to be recovered by both users is reconstructed in a lossy fashion under the requirement that all terminals i e the encoder and both the decoders can share an exact copy of the compressed version of this source component i e a common encoder decoder reconstruction constraint for this model as well we establish a single letter characterization of the associated rate distortion function

extreme amenability of abelian l0 groups we show that for any abelian topological group g and arbitrary diffused submeasure every continuous action of l0 g on a compact space has a fixed point this generalizes earlier results of herer and christensen glasner furstenberg and weiss and farah and solecki this also answers a question posed by farah and solecki in particular it implies that if h is of the form l0 r then h is extremely amenable if and only if h has no nontrivial characters which gives an evidence for an affirmative answer to a question of pestov the proof is based on estimates of chromatic numbers of certain graphs on z n it uses tools from algebraic topology and builds on the work of farah and solecki

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

global bifurcation of periodic solutions to ordinary differential equations abstract general homotopy continuation and bifurcation results are discussed and proved for a class of semiflows these results and further conley homotopy index methods are applied to obtain a type of global bifurcation for periodic solutions of second order ordinary differential equations

lp bounds for rate distortion with variable side information we consider a rate distortion problem with side information at multiple decoders several upper and lower bounds have been proposed for this general problem or special cases of it we provide an upper bound for general instances of this problem which takes the form of a linear program by utilizing random binning and simultaneous decoding techniques 1 and compare it with the existing bounds we also provide a lower bound for the general problem which was inspired by a linear programming lower bound for index coding and show that it subsumes most of the lower bounds in literature using these upper and lower bounds we explicitly characterize the rate distortion function of a problem that can be seen as a gaussian analogue of the odd cycle index coding problem

on a free boundary problem for the curvature flow with driving force we study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles our first main result concerns the classification of solutions every solution falls into one of the three categories namely area expanding area bounded and area shrinking types we then study in detail the asymptotic behavior of solutions in each category among other things we show that solutions are asymptotically self similar both in the area expanding and the area shrinking cases while solutions converge to either a stationary solution or a traveling wave in the area bounded case we also prove results on the concavity properties of solutions one of the main tools of this paper is the intersection number principle however in order to deal with solutions with free boundaries we introduce what we call the extended intersection number principle which turns out to be exceedingly useful in handling curves with moving endpoints

an entire solution of a bistable parabolic equation on r with two colliding pulses we consider semilinear parabolic equations of the form equation a ut uxx f u x r t i where i 0 i 0 or i i solutions defined for all x t r2 x t r2 are referred to as entire solutions assuming that f c1 r f c1 r is of a bistable type with stable constant steady states 0 and 0 0 we show the existence of an entire solution u x t u x t of the following form for t t u t u t has two humps or pulses one near the other near as t increases the humps move toward the origin x 0x 0 eventually colliding and forming a one hump final shape of the solution with respect to the locally uniform convergence the solution u t u t is a heteroclinic orbit connecting the stable steady state 0 to the unstable ground state of the equation uxx f u 0uxx f u 0 we find the solution u as the limit of a sequence of threshold solutions of the cauchy problem for equation a

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

extreme amenability of l0 a ramsey theorem and levy groups we show that l0 h is extremely amenable for any diffused submeasure and any solvable compact group h this extends results of herer christensen and of glasner and furstenberg weiss proofs of these earlier results used spectral theory or concentration of measure our argument is based on a new ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods using this work we give an example of a group which is extremely amenable and contains an increasing sequence of compact subgroups with dense union but which does not contain a levy sequence of compact subgroups with dense union this answers a question of pestov we also show that many levy groups have non levy sequences answering another question of pestov

global stability of traveling waves for an area preserving curvature flow with contact angle condition abstract we consider an evolving plane curve with two endpoints that can move freely on the x axis with generating constant contact angles we discuss the asymptotic behavior of global in time solutions when the evolution of this plane curve is governed by the area preserving curvature flow equation the main result shows that any moving curve converges to a traveling wave if the moving curve starts from an embedded convex curve and remains bounded in global time

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

on free actions minimal flows and a problem by ellis we exhibit natural classes of polish topological groups g such that every continuous action of g on a compact space has a fixed point and observe that every group with this property provides a solution in the negative to a 1969 problem by robert ellis as the ellis semigroup e u of the universal minimal g flow u being trivial is not isomorphic with the greatest g ambit further refining ouir construction we obtain a polish topological group g acting freely on the universal minimal flow u yet such that s g and e u are not isomorphic we also display polish topological groups acting effectively but not freely orn their universal minimal flows in fact we can produce examples of groups of all three types having any prescribed infinite weight our examples lead to dynamical conclusions for some groups of importance in analysis for instance both the full group of permutations s x of an infinite set equipped with the pointwise topology and the unitary group u x of an infinite dimensional hilbert space with the strong operator topology admit no free action on a compact space and the circle 1 forms the universal minimal flow for the topological group homeo 1 of orientation preservinig haomeomorphisms it also follows that a closed subgroup of an amenable topological group need not be amenable

broadcasting with side information bounding and approximating the broadcast rate index coding has received considerable attention recently motivated in part by applications such as fast video on demand and efficient communication in wireless networks and in part by its connection to network coding optimal encoding schemes and efficient heuristics were studied in various settings while also leading to new results for network coding such as improved gaps between linear and non linear capacity as well as hardness of approximation the problem of broadcasting with side information a generalization of the index coding problem begins with a sender and sets of users and messages each user possesses a subset of the messages and desires an additional message from the set the sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message the fundamental parameter of interest is the broadcast rate the average communication cost for sufficiently long broadcasts though there have been many new nontrivial bounds on by bar yossef 2006 lubetzky and stav 2007 alon 2008 and blasiak 2011 there was no known polynomial time algorithm for approximating within a nontrivial factor and the exact value of remained unknown for all nontrivial instances using the information theoretic linear program introduced in blasiak 2011 we give a polynomial time algorithm for recognizing instances with 2 and pinpoint precisely for various classes of graphs e g various cayley graphs of cyclic groups further extending ideas from ramsey theory we give a polynomial time algorithm with a nontrivial approximation ratio for computing finally we provide insight into the quality of previous bounds by giving constructions showing separations between and the respective bounds in particular we construct graphs where is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse

source coding problems with conditionally less noisy side information a computable expression for heegard and berger s rate distortion function has eluded information theory for nearly three decades heegard and berger s single letter achievability bound is well known to be optimal for physically degraded side information however it is not known whether the bound is optimal for arbitrarily correlated side information general discrete memoryless sources in this paper we consider a new setup where the side information at one receiver is conditionally less noisy than that at the other the new setup includes degraded side information as a special case and it is motivated by the literature on degraded and less noisy broadcast channels our key contribution is a converse proving the optimality of heegard and berger s achievability bound in a new setting where the side information is conditionally less noisy and one distortion function is deterministic the less noisy setup is also generalized to two different successive refinement problems

on mean curvature flow with driving force for symmetric motion with singular initial hypersurface abstract we consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface we study this problem by level set method we give some criteria to judge whether the interface evolution is fattening or non fattening

maharam s problem we construct an exhaustive submeasure that is not equivalent to a measure this solves problems of j von neumann 1937 and d maharam 1947

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

mean curvature flow of graphs in warped products let m be a complete riemannian manifold which either is compact or has a pole and let be a positive smooth function on m in the warped product m r we study the flow by the mean curvature of a locally lipschitz continuous graph on m and prove that the flow exists for all time and that the evolving hypersurface is c for t 0 and is a graph for all t moreover under certain conditions the flow has a well defined limit

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

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

blow up rates of solutions of initial boundary value problems for a quasi linear parabolic equation abstract we consider initial boundary value problems for a quasi linear parabolic equation k t k 2 k k with zero dirichlet boundary conditions and positive initial data it has known that each of solutions blows up at a finite time with the rate faster than t t 1 in this paper it is proved that sup k t t t 1 log log t t 1 as t t under some assumptions our strategy is based on analysis for curve shortening flows that with self crossing brought by s b angenent and j j l velazquez in addition we prove some of numerical conjectures by watterson which are keys to provide the blow up rate

on amenability and groups of measurable maps abstract we show that if g is an amenable topological group then the topological group l 0 g of strongly measurable maps from 0 1 into g endowed with the topology of convergence in measure is whirly amenable hence extremely amenable conversely we prove that a topological group g is amenable if l 0 g is

approaching an extinction point in one dimensional semilinear heat equations with strong absorption abstract this paper deals with the cauchy problem ut uxx up 0 0 u x 0 u0 x 0 and u x t 0 appear when t is close to t we describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached we also give conditions under which some of these behaviours actually occur

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

transition operators of diffusions reduce zero crossing if u t x is a solution of a one dimensional parabolic secondorder linear partial differential equation pde then it is known that under suitable conditions the number of zero crossings of the function u t decreases that is does not increase as time t increases such theorems have applications to the study of blow up of solutions of semilinear pde time dependent sturm liouville theory curve shrinking problems and control theory we generalise the pde results by showing that the transition operator of a possibly time inhomogenous one dimensional diffusion reduces the number of zero crossings of a function or even suitably interpreted a signed measure our proof is completely probabilistic and depends in a transparent manner on little more than the sample path continuity of diffusion processes

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

rate distortion when side information may be absent the problem is considered of encoding a discrete memoryless source when correlated side information may or may not be available to the decoder it is assumed that the side information is not available to the encoder the rate distortion function r d l d 2 is determined where d 1 is the distortion achieved with side information and d 2 is the distortion achieved without it a generalization is made to the case of m decoders each of which is privy to its own side information an appropriately defined d admissible rate for this general case is shown to equal r d when the side information sources satisfy a specified degradedness condition explicit results are obtained in the quadratic gaussian case and in the binary hamming case

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3