fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

on the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical sobolev exponent abstract this paper is concerned with a semilinear parabolic equation involving critical sobolev exponent in a ball or in r n the asymptotic behavior of unbounded radially symmetric nonnegative global solutions which do not decay to zero is given the structure of the space of initial data is also discussed

fast inversion of triangular toeplitz matrices an algorithm requiring 0 n log n operations is proposed for the inversion of triangular toeplitz matrices this algorithm is faster than existing techniques even for relatively small values of n and can easily be written in fortran

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

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

fast and numerically stable algorithms for discrete cosine transforms abstract in this paper we derive fast and numerically stable algorithms for discrete cosine transforms dct of radix 2 length which are based on real factorizations of the corresponding cosine matrices into products of sparse almost orthogonal matrices of simple structure these algorithms are completely recursive are simple to implement and use only permutations scaling with 2 butterfly operations and plane rotations rotation reflections our algorithms have low arithmetic costs which compare with known fast dct algorithms further a detailed analysis of the roundoff errors for the presented dct algorithms shows their excellent numerical stability which outperforms a real fast dct algorithm based on polynomial arithmetic

the existence of bounded solutions of a semilinear elliptic equation on etudie l existence de la solution non constante bornee de l equation differentielle w y w 2 w p 1 w w p 1 0 dans r n n2 avec pp c ou p c n 2 n 2 est l exposant de l espace de sobolev critique pour r n

on the uniqueness of l1 continuation after blowup abstract this paper is concerned with the uniqueness of l 1 continuation beyond blowup for a cauchy problem of a semilinear heat equation p u t u u p in r n 0 t u x 0 u 0 x 0 in r n with p 1 0 t and u 0 l r n here we say that u is an l 1 solution of p if u c 0 t l loc 1 r n with u l loc p r n 0 t satisfies p in the distributional sense in the case of p s p p jl a counter example for the uniqueness of radial l 1 solution of p after blowup was given in m fila n mizoguchi multiple continuation beyond blow up differential integral equations 20 2007 671 680 where p s and p jl are the exponent of sobolev and of joseph and lundgren respectively we give a sufficient condition for the uniqueness of l 1 continuation beyond blowup for p p jl in the radial case if for an l 1 solution u of p there exists a sequence u n of classical solutions of p such that u 0 n u 0 in l r n as n for the sequence u 0 n of initial data and that u n t u t in l loc p r n as n for t 0 t then u is called a limit l 1 solution based on the sufficient condition we prove the uniqueness of limit l 1 solution with radial symmetry after blowup for p p jl

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

representations of toeplitz plus hankel matrices using trigonometric transformations with application to fast matrix vector multiplication abstract representations of real toeplitz and toeplitz plus hankel matrices are presented that involve real trigonometric transformations dct dst dht and diagonal matrices these representations can be used for fast matrix vector multiplication in particular it is shown that the multiplication of an n n toeplitz plus hankel matrix by a vector requires only 4 transformations of length n plus o n operations

global unbounded solutions of the fujita equation in the intermediate range we consider the fujita equation u t vardelta u u p on mathbb r n with n ge 3 we prove the existence of global unbounded solutions for p s p p jl where p s n 2 n 2 and begin aligned p jl left begin array l quad l frac n 2 2 4n 8 sqrt n 1 n 2 n 10 text if n 10 infty text if n le 10 end array right end aligned previously it was known that global unbounded solutions exist for p ge p jl whereas for p p s there are no such solutions

are the coefficients of a polynomial well conditioned functions of its roots we show that the coefficients of a polynomial can be rather illconditioned functions of its given roots in certain cases however the coefficients are well conditioned e g if the roots lie symmetrically in a closed angular sector for an angle less than centered at the origin of the complex plane we introduce and study condition numbersb l k for the sensitivity of the polynomial coefficienta k to changes in the rootx l and develop an algorithm to compute the condition numbers and to decide whether a given set of roots will generate the polynomial coefficients in an ill way this algorithm is an essential preliminary for certain numerical pole placement algorithms in linear control theory

factorization of analytic functions by means of koenig s theorem and toeplitz computations by providing a matrix version of koenig s theorem we reduce the problem of evaluating the coefficients of a monic factor r z of degree h of a power series f z to that of approximating the first h entries in the first column of the inverse of an n times n toeplitz matrix in block hessenberg form for sufficiently large values of n this matrix is reduced to a band matrix if f z is a polynomial we prove that the factorization problem can be also reduced to solving a matrix equation phi x 0 for an h times h matrix x where phi is a matrix power series whose coefficients are toeplitz matrices the function phi is reduced to a matrix polynomial of degree 2 if f z is a polynomial of degreen and h geq n 2 these reductions allow us to devise a suitable algorithm based on cyclic reduction and on the concept of displacement rank for generating a sequence of vectors vec v 2 j that quadratically converges to the vector vec v having as components the coefficients of the factor r z in the case of a polynomial f z of degree n the cost of computing the entries of vec v 2 j given vec v 2 j 1 is o n log n theta n arithmetic operations where theta n o n log 2 n is the cost of solving an n times n toeplitz like system in the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation from the numerical experiments performed with several test polynomials and power series the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root finders based on recursive splitting strategies applications to solving spectral factorization problems and markov chains are also shown

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

blow up of solutions of nonlinear parabolic inequalities we study nonnegative solutions u x t of the nonlinear parabolic inequalities au u t u u in various subsets of r n x r where a n 2 n and a 0 1 are constants we show that changing the value of a in the open interval 0 1 can dramatically affect the blow up of these solutions

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

a superfast solver for sylvester s resultant linear systems generated by a stable and an anti stable polynomial abstract we develop a superfast method for the solution of n m n m sylvester s resultant linear systems associated with two real polynomials a z and c z of degree n and m respectively where a z is a stable polynomial i e all its roots lie inside the unit circle whereas c z is an anti stable polynomial i e z m c z 1 is stable the proposed scheme proceeds by iteratively constructing a sequence of increasing approximations of the solution it is based on a blend of ideas from structured numerical linear algebra computational complex analysis and linear operator theory each iterative step can be performed in o n m log n m arithmetic operations by combining fast polynomial arithmetic based on fft with displacement rank theory for structured matrices in addition the resulting process is shown to be quadratically convergent right from the start since the approximation error at the j th iteration is o r 2 j where r max a i 0 i min c i 0 i denotes the separation ratio between the spectrum of a z and c z finally we report and discuss the results of many numerical experiments which confirm the effectiveness and the robustness of the proposed algorithm

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

a fast and stable algorithm for splitting polynomials abstract this paper concerns the fast numerical factorization of degree a b polynomials in a neighborhood of the polynomial xa we want to obtain the so called splitting of one such polynomial i e a degree a factor with roots close to zero and a degree b factor with roots close to infinity an important application of splitting is complete polynomial factorization or root finding a new algorithm for splitting polynomials is presented this algorithm requires o dlog 1 1 floating point operations with o log 1 1 bits of precision as far as complexity is concerned this is the fastest algorithm known by the authors for that problem

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

an l blow up estimate for a nonlinear heat equation on decrit une borne superieure au taux d explosion des solutions de l equation de la chaleur non lineaire u t u u p ou u 0 p 1

threshold and generic type i behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume that n 3 and p ps n 2n 2 our first goal is to analyze a threshold behavior for solutions with initial data u0 v where v c h1 and v 0 v 0 it is known that there exists 0 such that the solution converges to 0 as t if 0 with j j 1 2 k the blow up is complete and of type i with a flat local profile our method is based on a combination of the zero number principle and energy estimates in the second part of the paper we employ the very same idea to show that the constant solution attains the smallest rescaled energy among all non zero stationary solutions of the rescaled equation using this result we derive a sharp criterion for no blow up

existence of type ii blowup solutions for a semilinear heat equation with critical nonlinearity abstract we consider the blowup rate of solutions for a semilinear heat equation u t u u p 1 u x r n t 0 with critical power nonlinearity p n 2 n 2 and n 3 first we investigate the profiles of backward self similar solutions by making use of the variational method and then by employing the intersection comparison argument with a particular self similar solution we derive the criteria of the blowup rate of solutions assuming the positivity of solutions in backward space time parabola in particular we show the existence of the so called type ii blowup solutions for the cauchy dirichlet problems on suitable shrinking domains in the case n 3

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

initial blow up of solutions of semilinear parabolic inequalities abstract we study classical nonnegative solutions u x t of the semilinear parabolic inequalities 0 u t u u p in 0 1 where p is a positive constant and is a bounded domain in r n n 1 we show that a necessary and sufficient condition on p for such solutions u to satisfy a pointwise a priori bound on compact subsets k of as t 0 is p 1 2 n and in this case the bound on u is max x k u x t o t n 2 as t 0 if in addition is smooth u satisfies the boundary condition u 0 on 0 1 and p 1 2 n then we obtain a bound for u on the entire set as t 0

asymptotic behavior at infinity of the stationary solution to a semilinear heat equation in this paper we investigate the asymptotic behavior at infinity of the backward self similar solution of the differential equation u t u e u x t 0 where is a ball with the dirichlet boundary or r n 3 n we prove that under some reasonable condition at infinity every radial symmetric nontrivial bounded above solution of the equation y y n 1 y y 2 y e 1 0 tends to minus infinity as y this equation comes from the scaled ignition model furthermore log y 2 converges to a constant for sufficiently large y this result extends the similar one in lacey 1993 for an arbitrary solution which is bounded above and for dimension 3 n in space

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

boundedness of global solutions of a supercritical parabolic equation we study radially symmetric classical solutions of the dirichlet problem for a heat equation with a supercritical nonlinear source we give a sufficient condition under which blow up in infinite time cannot occur this condition involves only the growth rate of the source term at infinity we do not need the homogeneity property which played a key role in previous proofs of similar results we also establish the blow up rate for a class of solutions which blow up in finite time

nonexistence results for a semilinear heat equation with bounded potentials abstract we study the liouville type theorem for positive solutions to the semilinear heat equation u t u v u p in r n r we show that the nonexistence of positive solutions holds for any p 1 and for any v bounded from below by a positive constant in the superlinear case p 1 we give some sufficient conditions on the potential and the exponent which guarantee the nonexistence of positive solutions in the whole space

jacobson s conjecture morita duality and related questions in j jacobson asked a question that gave rise to the so called jacobson s conjecture if r is a left noetherian ring with jacobson radical j then nn e mr j 0 herstein h and jategaonkar jl constructed counterexamples to show that this conjecture is not true in j3 jategaonkar asked if jacobson s conjecture holds for left noetherian rings with a left morita duality the main purpose of this paper is to give a negative answer to this question in section 1 we give the basic tools for our investigations these are elementary facts concerning trivial extensions and generalized triangular matrix rings many of them are already known in section 2 we give counterexamples to the above mentioned question using some commutative algebra and the results in section 1 we prove proposition 2 5 that a large class of generalized trangular matrix rings provides counterexamples to this question the ring r 8 where v is a maximal discrete valuation domain of rank 1 and q is its field of quotients is a typical ring of such a class of course these rings are not local but have at least two distinct left maximal ideals next proposition 2 11 we provide a local counterexample this is a suitable trivial extension ring in section 3 we get some more results about jacobson s conjecture for left noetherian rings we show that in some particular cases it holds for example if r is a left principal ideal domain with a left morita duality or if r is noetherian on both sides and rr is linearly compact in the discrete topology then jacobson s conjecture holds the last one of these facts when r has also a left morita duality has already been proved in another context and using a more elaborate approach in j4

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

finite time blow up for the fractional critical heat equation in rn abstract we consider the fractional critical heat equation u t s u u n 2 s n 2 s in r n 0 t u x 0 u 0 x in r n where 4 s n 6 s 0 s 1 for sufficiently small t 0 and given k distinct points q 1 q k we show that there exists an initial condition u 0 such that the solution u x t blows up at these k distinct points as t t more precisely the blow up profile around each concentration point takes the form of sharply scaled bubble and u t l r n t t n 2 s 6 s n as t t

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

on blow up space jets for the navier stokes equations in r3 with convergence to euler equations formation of blow up singularities for the navier stokes equations ut u u p u and div u 0 in r3 0 t is studied in cylindrical polar coordinates r z in r3 their restriction to the linear subspace w2 span 1 z is shown to be consistent using links with blow up theory for nonlinear reaction diffusion partial differential equations the following questions are under scrutiny ii introducing a self similar blow up swirl mechanism with the angular swirl divergences t ln t t and t t t as t t where r is a parameter ii existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional global blow up of the z component of the velocity iii as an intrinsic part of the construction convergence of the above rescaled patterns as t t to smooth blow up self similar solutions of the corresponding three dimensional euler equations ut u u p and div u 0 in r3 0 t which iv are also shown to admit single point blow up in

single point gradient blow up on the boundary for diffusive hamilton jacobi equations in planar domains consider the diffusive hamilton jacobi equation u t u u p p 2 on a bounded domain with zero dirichlet boundary conditions which arises in the kpz model of growing interfaces it is known that u remains bounded and that u may blow up only on the boundary in this paper under suitable assumptions on omega subset mathbb r 2 and on the initial data we show that the gradient blow up singularity occurs only at a single point x 0 in partial omega this is the first result of this kind in the study of problems involving gradient blow up phenomena in general domains of mathbb r n we also obtain results on nondegeneracy and localization of blow up points

wiener hopf and spectral factorization of real polynomials by newton s method abstract we present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle respectively the approach is based on solving a nonlinear system by newton s method the jacobi matrix is a polynomial of a companion matrix and thus a bezoutian times the inverse of a triangular toeplitz matrix after getting deeper understanding of the displacement structure of the bezoutian each newton step can be reduced to a few applications of discrete fourier or cosine transforms and the lu decomposition of a conceived linear system with a cauchy like matrix these structural features are employed to design a fast algorithm which shows excellent numerical behavior for example in the case of the extremely ill conditioned test polynomial 2 z n k 0 n 1 z k 2 k 1 n z k of the degree 2 n 20000 we obtain the factorization after 22 newton steps with an error of 3 4 10 8 in the euclidean norm the execution time on a laptop being a couple of minutes the local quadratic convergence of our newton method is proved and explicit convergence balls are given on the basis of estimates for the lipschitz constant which occurs in kantorovich s theorem we also design and test two superfast versions of our method which for polynomials of degree 20000 typically need about half a minute to produce an initial coefficient vector and then perform the newton iteration within 1 s

blow up rate of type ii and the braid group theory a solution u of a cauchy problem or a cauchy dirichlet problem for a semilinear heat equation u t u u p with nonnegative initial data u 0 is said to undergo type ii blow up at t t if lim sup t t 1 p 1 u t let be the radially symmetric singular steady state of the cauchy problem suppose that u 0 l is a radially symmetric function such that u 0 and uo t change sign at most finitely many times by application of the braid group theory we determine the exact blow up rate of solution with initial data u 0 which undergoes type ii blow up in the case of p p jl where p jl is the exponent of joseph and lundgren

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

initial blow up rates and universal bounds for nonlinear heat equations we establish a universal upper bound on the initial blow up rate for all positive classical solutions of the dirichlet problem for the nonlinear heat equation ut u upon 0 t where p 1 and is a smoothly bounded domain in rn namely we show that u t c p t t on 0 t 2 for some n p 0 this is proved for all subcritical p i e p n 2 n 2 if n 4 and under a stronger assumption on p if n 5 as a consequence we improve the known results on universal bounds for global solutions furthermore if p n 3 n 1 then we may take n 1 2 and we show that this value of is optimal interestingly the rate can be faster than the previously known maximal initial blow up rate of the cauchy problem applications to universal blow up estimates at t t are given the neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical p up to dimension n 6

polynomial factorization through toeplitz matrix computations abstract the problem of polynomial factorization is translated into the problem of constructing a wiener hopf factorization and three algorithms are designed for the solution of the latter problem these algorithms are based on solving linear systems with large but finite circulant and toeplitz matrices the algorithms are of low complexity and perhaps most importantly they are extremely lucid an upper bound for the condition number of the problem of polynomial factorization is given in terms of the condition number of a certain toeplitz matrix

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

resultant matrices and inversion of bezoutians abstract the goal of this paper is to construct inverses of toeplitz and hankel bezoutians we obtain quite explicit formulas which are also specified to cases where the bezoutians possess additional symmetries the construction relies on a result about the nullspace of generalized resultant matrices which can be described by means of the solutions of corresponding bezout equations

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

single point blow up for parabolic systems with exponential nonlinearities and unequal diffusivities abstract we study positive blowing up solutions of systems of the form u t 1 u e p v v t 2 v e q u with 1 2 0 and p q 0 we prove single point blow up for large classes of radially decreasing solutions this answers a question left open in a paper of friedman and giga 1987 where the result was obtained only for the equidiffusive case 1 2 and the proof depended crucially on this assumption

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

blow up rate for a porous medium equation with power nonlinearity abstract the present paper is concerned with the blow up rate of a solution to the following cauchy problem u t u m u p x t r n 0 t with the bounded initial function u x 0 u 0 x 0 where t is the blow up time when u 0 x has a small oscillation around a radial function we establish the growth rate estimate of the form u t l r n c t t 1 p 1 t 0 t where the constant c is independent of t variable and the parameters m and p are expected to be 1 m p m n 2 n 2 although this result is clear it seems that no rigorous proof can be found up until now

type ii blow up for the four dimensional energy critical semi linear heat equation abstract we consider the energy critical four dimensional semi linear heat equation t u u u 3 0 we show the existence of type ii finite time blow up solutions and give a sharp description of the corresponding singularity formation these solutions concentrate a universal bubble of energy in the critical topology u t r 1 q r t u in h 1 where the blow up profile is given by the talenti aubin soliton q r 1 1 r 2 8 and with speed t t t log t t 2 as t t our approach uses a robust energy method approach developed for the study of geometrical dispersive problems raphael and rodnianski 2012 18 merle et al 2011 15 and lies in the continuation of the study of the energy critical harmonic heat flow raphael and schweyer 2011 19 and the energy critical four dimensional wave equation hillaret and raphael 2010 5

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

a sufficient condition for type i blowup in a parabolic elliptic system abstract this paper is concerned with blowup of positive solutions to a cauchy problem for a parabolic elliptic system u t u u v in r n 0 t 0 v u in r n 0 t we say that a solution u v blows up at t t if lim sup t t u t with l norm in r n when a solution u v blows up at t t the blowup is called of type i if lim sup t t t t u t and of type ii otherwise it was shown in herrero and velazquez 1996 12 and mizoguchi and senba 2007 20 that there exist radial type ii blowup solutions for n 2 or n 11 when 3 n 9 type ii blowup solutions were given by formal analysis in brenner et al 1999 2 and herrero et al 1997 10 we give a sufficient condition for a solution to exhibit type i blowup in the case of 3 n 9

isolated singularities of nonlinear parabolic inequalities we study c 2 1 nonnegative solutions u x t of the nonlinear parabolic inequalities 0 le u t delta u le u lambda in a punctured neighborhood of the origin in bf r n times 0 infty when n ge 1 and lambda 0 we show that a necessary and sufficient condition on for such solutions u to satisfy an a priori bound near the origin is lambda le frac n 2 n and in this case the a priori bound on u is u x t o t n 2 quad text as x t to 0 0 t 0 this a priori bound for u can be improved by imposing an upper bound on the initial condition of u