spectral collocation solutions to multiparameter mathieu s system our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of mathieu s system as a multiparameter eigenvalue problem mep the reduced wave equation for small deflections is solved directly without approximations introduced by the classical mathieu functions we show how for moderate values of the cut off collocation parameter the qr algorithm and the arnoldi method may be applied successfully while for larger values the jacobi davidson method is the method of choice with respect to convergence accuracy and memory usage

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

pseudodifferential operators on manifolds with foliated boundaries abstract let x be a smooth compact manifold with boundary for smooth foliations on the boundary of x admitting a resolution in terms of a fibration we construct a pseudodifferential calculus generalizing the fibred cusp calculus of mazzeo and melrose in particular we introduce certain symbols leading to a simple description of the fredholm operators inside the calculus when the leaves of the fibration resolving the foliation are compact we also obtain an index formula for fredholm perturbations of dirac type operators along the way we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of dirac type operators a result of independent interest generalizing the well known formula of bismut and cheeger

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

spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems in numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables in several orthogonal coordinate systems separation of variables applied to the helmholtz laplace or schrodinger equation leads to a multiparameter eigenvalue problem mep important cases include mathieu s system lame s system and a system of spheroidal wave functions although multiparameter approaches are exploited occasionally to solve such equations numerically meps remain less well known and the variety of available numerical methods is not wide the classical approach of discretizing the equations using standard finite differences leads to algebraic meps with large matrices which are difficult to solve efficiently the aim of this paper is to change this perspective we show that by combining spectral collocation methods and new efficient numerical methods for algebraic meps it is possible to solve such problems both very efficiently and accurately we improve on several previous results available in the literature and also present a matlab toolbox for solving a wide range of problems

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

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

second root vectors for multiparameter eigenvalue problems of fredholm type a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied a basis for the second root subspace at eigenvalues of fredholm type is computed in terms of the underlying multiparameter system a self adjoint version of this result is given under a weak definiteness condition and sturm liouville and finitedimensional examples are considered

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 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

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

nonlinear multiparameter eigenvalue problems linearised and nonlinearised solutions abstract the paper focuses on a nonlinear multiparameter eigenvalue problem called p this problem involves n spectral parameters and also depends on n 2 numerical parameters where n 2 is an integer if these numerical parameters vanish p degenerates into n linear one parameter eigenvalue problems called p i 0 i 1 n in connection with p one can consider another n nonlinear one parameter eigenvalue problems called p i the problems p i have eigenvalues with as well as without linear counterparts the paper suggests to consider p i as nonunperturbed instead of p i 0 using properties of eigenvalues of p i one manages to prove existence of eigentuples of p among the eigentuples found in this way there are eigetuples with as well as without linear counterparts results of the paper are found with a nonclassical approach applications of the found results to nonlinear optics are shown

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

the index of the spinc dirac operator on the weighted projective space and the reciprocity law of the fourier dedekind sum abstract the main purpose of this paper is to give a geometric interpretation of the reciprocity law of the fourier dedekind sum given by m beck and s robins in fact the v index of the spin c dirac operator on the weighted projective space is equal to the dimension of the space of all weighted homogeneous polynomials of given degree and this equality gives precisely the beck robins reciprocity law in this equality the fourier dedekind sums appear as the localization terms of the v index of the spin c dirac operators and have a relationship to the eta invariants of lens spaces

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

multiparameter spectral theory and separation of variables multiparameter spectral theory generalizes the classical spectral theory of linear operators to n linear operators linked by n spectral parameters the spectral parameters arise as separation constants when for example separation of variables techniques are used to solve boundary value problems for partial differential equations in this paper we discuss some of the fundamental aspects of the theory such as solvability commutativity and abstract representation theory many of these concepts are of importance to modern developments in separation of variables techniques the ideas are illustrated in application to differential equations and special functions of mathematical physics

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

the eta function of the localised dirac operator for the universal cover of sl2 r abstract recently brodzki niblo plymen and wright determined a closed explicit description of the spectrum of the dirac operator d for the universal cover of sl 2 r localised at a representations in the principal series and in the discrete series 4 using this description we introduce an eta function loc s d for the localised dirac operator and we study its analytic properties we show that the localised eta function has an analytic extension to the complex plane with simple poles at negative integers and at s 1 and it is regular at s 0 and we give a formula for loc 0 d we discuss a possible eta invariant for the dirac operator

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

the residue of the global η function at the origin where i dvol x l denotes the riemannian measure on m using the calculus of pseudo differential operators depending upon a complex parameter which was developed by seeley 9 one can show that both q s x p and r s p can be meromorphically extended to the complex plane with simple poles at s m n u for n 0 the residues of l s x p at the poles for n 0 m are given by local formulas in the jets of the total symbol of p we define

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

order structures and the heat equation abstract in this paper we shall introduce the notion of order structure and use it to study properties of a solution u x t of a scalar linear parabolic equation it is well known that in one space dimension the number of zeros of u t is nonincreasing with t this zero number is an example of an order structure and the zero number property is very useful in unfolding the structure of the global attractor of the semiflow generated by a scalar parabolic equation we shall prove that in one space dimension the zero number is the only order structure preserved by linear parabolic equations in two dimensions the only order structure preserved by linear parabolic equations is the order structure induced by the comparison principle for second order equations consequently in two dimensions there does not exist a fine decomposition into equivalence classes as in the one dimensional case