global structure of self similar solutions in a semilinear parabolic equation abstract the aim of this paper is to investigate the structure of radial solutions for a semilinear elliptic equation u y u 2 u p 1 u p 1u 0 y rn which is related to forward self similar solutions of a semilinear parabolic equation vt v v p 1v t x 0 rn we study the existence uniqueness and parameter dependency of rapidly decaying solutions with a prescribed number of zeros

structure of positive radial solutions to scalar field equations with harmonic potential abstract we study a nonlinear elliptic equation with a harmonic potential u x 2 u u p 1 u 0 x r n e which is related to standing waves for a nonlinear schrodinger equation the existence of positive solution for e in some energy space can be shown by the standard variational method under n 1 n and 1 p n 2 n 2 however the uniqueness is a delicate problem in this paper noting that all positive solutions of e in the energy space are radially symmetric about the origin we study the structure of positive radial solutions to e and show the uniqueness of positive solution with finite energy when n 3

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

the asymptotic behaviour of the heat equation in a twisted dirichlet neumann waveguide abstract we consider the heat equation in a straight strip subject to a combination of dirichlet and neumann boundary conditions we show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of t 1 2 the proof employs similarity variables that lead to a non autonomous parabolic equation in a thin strip contracting to the real line that can be analysed on weighted sobolev spaces in which the operators under consideration have discrete spectra a careful analysis of its asymptotic behaviour shows that an added dirichlet boundary condition emerges asymptotically at the switching point breaking the real line in two half lines which leads asymptotically to the 1 2 gain on the spectral lower bound and the t 1 2 gain on the decay rate in the original physical variables this result is an adaptation to the case of strips with twisted boundary conditions of previous results by the authors on geometrically twisted dirichlet tubes

a note on nonexistence of global solutions to ut δu 12x u λu h x t u p abstract we consider the inequality ut u 1 2 x u u h x t u p for p 1 r posed in r n r n 1 we show that in certain growth conditions there is an absence of global weak solutions

an lp theory of invariant manifolds for parabolic partial differential equations on rd abstract we study the problem about the existence of finite dimensional invariant manifolds for nonlinear heat equations of the form u u f u u on r d 1 we show that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero there exist finite dimensional invariant manifolds which control the long time asymptotics of solutions we consider the problem for these equations in the framework of weighted sobolev spaces of l p type the l p theory of this problem gives the l estimate of the long time asymptotics of solutions under natural assumptions on the nonlinear term f and their initial data

determinant bundles and virasoro algebras we consider the interplay of infinite dimensional lie algebras of virasoro type and moduli spaces of curves suggested by string theory we will see that the infinitesimal geometry of determinant bundles is governed by virasoro symmetries the mumford forms are just invariants of these symmetries the representations of virasoro algebra define twisted d modules on moduli spaces thesed modules are equations on correlators in conformal field theory

hitchin s connection and differential operators with values in the determinant bundle for a family of smooth curves we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves there exists a so called theta line bundle on the family of moduli spaces when the kodaira spencer map of the family of curves is an isomorphism we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves as an application the so called hitchin connection on the direct image of powers of the theta line bundle is derived naturally from the identification theorem a logarithmic extension to certain singular stable curves is also presented in this paper

weighted sobolev spaces with applications to singular nonlinear boundary value problems abstract variational methods are used in a weighted sobolev space to prove the existence of solutions for a certain class of singular nonlinear ordinary differential equations

decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity abstract we show that the solutions of a semilinear system of elastic waves in an exterior domain with a localized damping near infinity decay in an algebraic rate to zero we impose an additional condition on the lame coefficients it seems that this restriction cannot be overcome by using the two finite speed propagation of the elastic model since we do not assume compact support on the initial data and because the dissipation does not have compact support the decay rates obtained for the total energy of the linear problem and the l 2 norm of the solution improve previous results for the semilinear problem the decay rates in this paper seem to be the first contribution mainly in the context of initial data without compact support and localized dissipation

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

movement of hot spots on the exterior domain of a ball under the neumann boundary condition abstract we consider the cauchy neumann problem of the heat equation in the exterior domain of a ball in r n and study the movement of hot spots of the solution as t

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

radial symmetry of self similar solutions for semilinear heat equations abstract the symmetry properties of positive solutions of the equation u 1 2 x u 1 p 1 u u p 0 in r n where n 2 p n 2 n was studied it was proved that u must be radially symmetric about the origin provided u x o x 2 p 1 as x and that there exist non radial solutions u satisfying lim sup x x 2 p 1 u x 0

some pseudoprimes and related numbers having special forms we give an example of a pseudoprime which is itself of the form 2n 2 answering a question posed by a rotkiewicz show that lehmer s example of an even pseudoprime having three prime factors is not unique and answer a question of benkoski concerning the solutions of 2n 2 1 mod n

self similar solutions to a parabolic system modeling chemotaxis we study the forward self similar solutions to a parabolic system modeling chemotaxisut u u v vt v u in the whole space r2 where is a positive constant using the liouville type result and the method of moving planes it is proved that self similar solutions u v must be radially symmetric about the origin then the structure of the set of self similar solutions is investigated as a consequence it is shown that there exists a threshold in r2u for the existence of self similar solutions in particular for 0 1 2 there exists a self similar solution u v if and only if r2u 8

on a problem of a rotkiewicz pour des entiers positifs a k 2 fixes arbitraires il existe une infinite d entiers composes n tels que a n k 1 mod n

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

existence of ground states with exponential decay for semi linear elliptic equations in ofrn abstract in this paper we prove the existence of a positive solution tending exponentially to zero at infinity for a class of semi linear elliptic equations with nonconstant coefficients in ofr n

on the congruence 2n k 1 mod n it is shown that there are infinitely many positive integers k such that the congruence 2n k 1 mod n has infinitely many solutions n

asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance abstract there are wide classes of nonlinear evolution equations which possess invariant properties with respect to a scaling and translations if a solution is invariant under the scaling then it is called a self similar solution which is a candidate for the asymptotic profile of general solutions at large time in this paper we establish an abstract framework to find more precise asymptotic profiles by shifting self similar solutions suitably

the evolution to localized and front solutions in a non lipschitz reaction diffusion cauchy problem with trivial initial data abstract in this paper we establish the existence of spatially inhomogeneous classical self similar solutions to a non lipschitz semi linear parabolic cauchy problem with trivial initial data specifically we consider bounded solutions to an associated two dimensional non lipschitz non autonomous dynamical system for which we establish the existence of a two parameter family of homoclinic connections on the origin and a heteroclinic connection between two equilibrium points additionally we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

on a class of nonlinear elliptic equations with fast increasing weight and critical growth abstract we are concerned with the existence of rapidly decaying solutions for the equation div k x u k x x u q 2 u k x u 2 2 u x r n where n 3 2 q 2 2 n n 2 0 is a parameter k x exp x 4 2 and the number is given by 2 2 q 2 2 we obtain a positive solution if 2 q 2 and a sign changing solution if q 2 the existence results depend on the values of the parameter in the proofs we apply variational methods

existence of minimizers for eigenvalues of the dirichlet laplacian with a drift abstract this paper deals with the eigenvalue problem for the operator l x with dirichlet boundary conditions we are interested in proving the existence of a set minimizing any eigenvalue k of l under a suitable measure constraint suggested by the structure of the operator more precisely we prove that for any c 0 and k n the following minimization problem min k quasi open set e x 2 2 d x c has a solution

on two signed solutions to a second order semi linear parabolic partial differential equation with non lipschitz nonlinearity abstract in this paper we establish the existence of a 1 parameter family of spatially inhomogeneous radially symmetric classical self similar solutions to a cauchy problem for a semi linear parabolic pde with non lipschitz nonlinearity and trivial initial data specifically we establish well posedness for an associated initial value problem for a singular two dimensional non autonomous dynamical system with non lipschitz nonlinearity additionally we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as

on a decay property of solutions to the haraux weissler equation abstract we give a sufficient condition that non radial h 1 solutions to the haraux weissler equation should belong to the weighted sobolev space h 1 r n where is the weight function exp x 2 4 our result provides in some sense a connection between the solutions obtained by ode method and those by variational approach in the space h 1 r n

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

global existence and blowup for sign changing solutions of the nonlinear heat equation abstract in this paper given 0 2 n we prove the existence of a function with the following properties the solution of the equation u t u u u on r n with the initial condition u 0 is global on the other hand the solution with the initial condition u 0 blows up in finite time if 0 is either sufficiently small or sufficiently large

on the congruence 2 n 2 1 mod n there exist infinitely many positive integers n such that 2 2 1 mod n in the monograph 5 i proposed the following problem problem 18 p 138 let a k 1 be fixed positive integers do there exist infinitely many composite n such that a 1 put a 2 k 2 in the above problem since by fermat s theorem 2p l 1 mod p for odd primes p if 2 2 1 mod n and n 2 n must be composite r matuszewski and p rudnicki with the aid of the computer k 202 in warsaw checked that below 4208 such integers do not exist the following theorem holds congruence 2m 1 mod2m 1 it follows that 2m v v m 1 mod2m 1 22 1 3 s 1 mod2m 1 and 2 2 1 modn for n 2m 1 thus 2 2 1 mod n for n 2 1 where 0 4700063497 suppose now that 2 2 1 mod and n 8 let be a primitive factor of the number 2 2 1 a prime factor of 2 1 is said to be primitive if it does not divide any of the numbers 2m 1 for m 1 2 n 1 by a theorem of k zsigmondy 7 such a prime factor exists for any n 6 and is of the form nt 1 now we shall show that nx np is also a solution of the congruence 2 1 2 1 mod we have p 2 n 2 k 1 where k is a positive integer and p 2 3 n and 1 thus

a semilinear heat equation with singular initial data we first prove existence and uniqueness of non negative solutions of the equation in in the range 1 p n when initial data u x 0 a x 2 p 1 x 0 for a 0 it is proved that the maximal and minimal solutions are self similar with the form where g g a satisfies after uniqueness is proved the asymptotic behaviour of solutions of is studied in particular we show that the case for a 0 is also considered and a sharp decay rate of the above equation is derived in the final we reveal existence of solutions of the first and third equations above which change sign