universal bounds for global solutions of a diffusion equation with a nonlocal reaction term we consider the nonlinear heat equation with nonlocal reaction term in space ut u up in smoothly bounded domains we prove the existence of a universal bound for all nonnegative global solutions of this equation moreover in contrast with similar recent results for equations with local reaction terms this is shown to hold for all p 1 as an interesting by product of our proof we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction

blowing up of solution for a nonlocal reaction diffusion problem in combustion theory abstract in a nonlocal reaction diffusion model in combustion theory the reaction function involves a physical parameter which is a measure of the strength of the reaction mechanism the purpose of this paper is to show the existence of a critical value such that for a unique global time dependent solution exists and converges to a steady state solution as t and for the solution blows up in finite time a characterization as well as upper and lower bounds of are given

admissible lp norms for local existence and for continuation in semilinear parabolic systems are not the same we say that a banach space e is a continuation space for a given parabolic problem if the e norm of any non global solution has to become unbounded we will prove that for large classes of parabolic systems of two equations the space e l r 1 l r 2 can be a continuation space even though the problem is not locally well posed in e this stands in contrast with classical results for analogous scalar equations

global existence and asymptotic behavior of solutions to a nonlocal fisher kpp type problem abstract in this work we consider a nonlocal fisher kpp reaction diffusion problem with neumann boundary condition and nonnegative initial data in a bounded domain in r n n 1 with reaction term u 1 m t where m t is the total mass at time t with the help of poho aev s identity the non existence of nontrivial stationary solutions with dirichlet boundary conditions is being shown when 1 and the initial mass is greater than or equal to one the problem has nonnegative classical solutions while if the initial mass is less than one then the problem admits global solutions for n 1 2 with any 1 2 or n 3 with any 1 1 2 n moreover the asymptotic convergence to the solution of the heat equation is proved

cross newell equations for hexagons and triangles he cross newell equations for hexagons and triangles are derived for general real gradient systems and are found to be in flux divergence form specific examples of complex governing equations that give rise to hexagons and triangles and which have lyapunov functionals are also considered and explicit forms of the cross newell equations are found in these cases the general nongradient case is also discussed in contrast with the gradient case the equations are not flux divergent in all cases the phase stability boundaries and modes of instability for general distorted hexagons and triangles can be recovered from the cross newell equations

fast computation of a general complete elliptic integral of third kind by half and double argument transformations we developed a novel method to calculate an associate complete elliptic integral of the third kind j n m p n m k m n the key idea is the double argument formula of j n m with respect to n we derived it from the not so popular addition theorem of jacobi s complete elliptic integral of the third kind p 1 a m with respect to a which is a real or pure imaginary argument connected with n and m as n msn 2 a m repeatedly using the half argument transformation fukushima 2010 28 of a new variable y n m or its complement x m n m we reduce y sufficiently small say less than 0 3 or so then we evaluate the integral for the reduced variable by its maclaurin series expansion the coefficients of the series expansion are recursively computed from two other associate complete elliptic integrals b m e m 1 m k m m and d m k m e m m the precise and fast computation of these two integrals is found in our previous work fukushima 2011 17 finally we recover the integral value for the original n by successively applying the double argument formula of j n m the new method is sufficiently precise in the sense that the maximum errors are less than around 10 machine epsilons for the sole computation of j n m the new method runs 1 2 1 5 and 4 7 5 5 times faster than bulirsch s cel and carlson s r j respectively in the simultaneous computation of three associate complete integrals the new method runs 1 6 1 7 and 5 3 8 0 times faster than cel and carlson s r d and r j respectively

asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity we find two convergent series expansions for legendre s first incomplete elliptic integral f l k in terms of recursively computed elementary functions both expansions are valid at every point of the unit square 0 l k 1 truncated expansions yield asymptotic approximations for f l k as l and or k tend to unity including the case when logarithmic singularity l k 1 is approached from any direction explicit error bounds are given at every order of approximation for the reader s convenience we present explicit expressions for low order approximations and numerical examples to illustrate their accuracy our derivation is based on rearrangements of some known double series expansions hypergeometric summation algorithms and inequalities for hypergeometric functions

pattern selection in long wavelength convection abstract the long wavelength of the first instability in rayleigh benard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems the evolution of such an instability is described by the equation t 2 4 2 2 2 2 where is the planform function is the scaled rayleigh number and 1 the quantities represent the effects of finite biot number asymmetry in the boundary conditions at top and bottom and departures from the boussinesq approximation respectively the quantity when the original problem is self adjoint but otherwise planform selection is studied for 1 using equivariant bifurcation theory on the square lattice both rolls and squares can be stable depending on the parameters and the possible secondary bifurcations located near various codimension two singularities are analyzed on the hexagonal lattice the primary bifurcation is always degenerate when 0 of the six primary solution branches possible in this case the hexagon branch is stable when 0 both rolls and hexagons bifurcate supercritically but rolls are stable finally when and or 0 a hysteretic transition to h or h occurs depending on sgn k 2 c if stable h h coexist at larger amplitudes but if k 2 c 0 1 a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls

linear and nonlinear heat equations in l q_ delta spaces and universal bounds for global solutions we develop a theory of both linear and nonlinear heat equations in the weighted lebesgue spaces l q delta where delta is the distance to the boundary in particular we prove an optimal l q delta l r delta estimate for the heat semigroup and we establish sharp results on local existence uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces this theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems namely under certain assumptions we prove that any global solution is uniformly bounded for t geq tau 0 by a universal constant independent of the initial data in all previous results the bounds for global solutions were depending on the initial data

competition in ramped turing structures abstract stationary pattern selection and competition in the uniform brusselator in two 2d and three 3d dimensions are reviewed including reentrant hexagonal and striped zig zag phases influences of linear or chain like profiles of the pool chemicals on this selection are presented in the form of numerical experiments the relation with the recent experimental patterns obtained with the cima reaction is discussed

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

preconditioned steepest descent methods for some nonlinear elliptic equations involving p laplacian terms abstract we describe and analyze preconditioned steepest descent psd solvers for fourth and sixth order nonlinear elliptic equations that include p laplacian terms on periodic domains in 2 and 3 dimensions the highest and lowest order terms of the equations are constant coefficient positive linear operators which suggests a natural preconditioning strategy such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena in particular liquid crystals thin film epitaxial growth and phase transformations the analyses of the schemes involve the characterization of the strictly convex energies associated with the equations we first give a general framework for psd in hilbert spaces based on certain reasonable assumptions of the linear pre conditioner a geometric convergence rate is shown for the nonlinear psd iteration we then apply the general theory to the fourth and sixth order problems of interest making use of sobolev embedding and regularity results to confirm the appropriateness of our pre conditioners for the regularized p lapacian problems our results include a sharper theoretical convergence result for p laplacian systems compared to what may be found in existing works we demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods numerical simulations for some important physical application problems including thin film epitaxy with slope selection and the square phase field crystal model are carried out to verify the efficiency of the scheme

patterns and spatiotemporal chaos in parametrically forced surface waves a systematic survey at large aspect ratio abstract a systematic experimental survey of both the primary patterns and the secondary instabilities of parametrically forced surface waves faraday waves in the large system limit is presented the symmetry of the primary pattern stripes squares or hexagons depends on viscosity and driving frequency f o hexagons are observed at low f o over the whole viscosity range despite the subharmonic symmetry that tends to suppress them possible mechanisms for the occurrence of hexagons for single frequency forcing are discussed boundary induced distortion is absent for the hexagonal and square patterns but present for stripes phase defects occur between hexagonal domains differing in temporal phase by with respect to the forcing patterns of different symmetry coexist in certain parameter ranges the transition to spatiotemporal chaos stc depends on the symmetry of the primary patterns the hexagonal patterns undergo an order disorder transition in which the angular anisotropy in fourier space declines continuously to zero striped patterns at high viscosity become unstable via transverse amplitude modulations in regions of high curvature this instability results in a spatially nonuniform mixed state in which domains of stc coexist with stripes we propose that this phenomenon may be understood in terms of a critical curvature that depends on the acceleration a secondary oscillatory instability is also observed to deform the stripes within the mixed state at intermediate viscosities

stationary dynamical and chaotic states of the two dimensional damped kuramoto sivashinsky equation the long time behavior of the two dimensional damped kuramoto sivashinsky equation is studied numerically in the large aspect ratio limit the equation is shown to lead to three states depending on the magnitude of a damping parameter ensuremath alpha at large intermediate and small values of ensuremath alpha the equation leads respectively to hexagonal breathing hexagonal and disordered states the disordered phase is chaotic in both space and time consistent with spatiotemporal chaos the transitions between these states are examined using standard statistical measures

towards a universal theory for natural patterns abstract our goal is to find a macroscopic description of patterns that both unifies and simplifies classes of externally stressed dissipative pattern forming systems such as convecting fluids liquid crystals wideband lasers that are seemingly unrelated at the microscopic level we construct an order parameter equation which provides a controlled approximation of the original microscopic field in the limit of large aspect ratios it is built from and is a regularization of the cross newell phase diffusion equation obtained by averaging over the local periodicity of the pattern unlike the latter it is valid for all wavenumbers and can correctly capture the nucleation shape and nontrivial properties of the far fields of disclinations dislocations and grain boundaries it reduces to the cross newell equation away from pattern singularities and to the newell whitehead segel equation near onset as a consequence it correctly determines all the long wave instability boundaries zig zag eckhaus skew varicose of the busse balloon far from onset the order parameter is a real variable but its equation involves a functional corresponding to its local amplitude the local amplitude and phase required for the order parameter equation and the reconstruction of the approximation to the original field respectively are extracted from the order parameter field by wavelet analysis numerical comparisons between solutions of the original equation and the regularized equation are carried out we also explore a new class of singular and weak solutions of the cross newell equation which take account of the energetics of defects as well as their topologies these solutions correspond to convex and concave disclinations and their composites including saddles vortices targets dislocations and two new objects handles and bridges finally we show that phase grain boundaries lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation

asymptotics of information entropies of some toda like potentials the spreading of the quantum probability density for the highly excited states of a single particle system with an exponential type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the boltzmann shannon information entropy this problem boils down to the calculation of the asymptotics of the entropy like integrals of the modified bessel function of the second kind also called the mcdonald function or basset function the dependence of the two physical entropies on the large quantum number n is given in detail it is shown that the semiclassical wkb position space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single particle systems with any power type potential of the form v x x2k x r and k n the momentum space entropy calculated with a method based on the properties of the mcdonald function is rigorously found to have a behavior of the form ln ln n in strong contrast with the

uniform blow up profiles for nonlinear and nonlocal reaction diffusion equations abstract in this paper we investigate the blow up properties of positive solutions to the following integro parabolic equations u t u e u 1 e u d x where 0 and 1 and x r n x r for the radially symmetric and non increasing initial data we give a complete classification in terms of global and single point blow up according to the parameters and moreover the blow up rates are also determined in each case particularly for the special case and n 2 it will be proved that the blow up rate at x 0 is faster than that at the other point x 0 this seems to be a new phenomenon for this kind of problem

precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations this is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind fukushima 2010 2011 21 23 we developed a method to compute an associate incomplete elliptic integral of the third kind j f n m p f n m f f m n by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral the relative errors of j f n m computed by the new method are sufficiently small as less than 20 machine epsilons meanwhile the simplicity of the adopted algorithm makes the new method run 1 5 to 3 7 times faster than carlson s duplication method the combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind b f m e f m 1 m f f m m and d f m f f m e f m m which we established recently 23 enables a precise and fast computation of arbitrary linear combination of legendre s incomplete elliptic integrals of all three kinds f f m e f m and p f n m these new procedures share the same device the half argument transformations while the double argument transformation of j f n m includes those of b f m and d f m as its sub component as a result the simultaneous computation of the three associate integrals is significantly faster than computing them separately in fact our combined procedure is 2 7 to 5 9 times faster than the combination of carlson s duplication method to compute r d and r j

nonlocal nonlinear reaction preventing blow up in supercritical case of chemotaxis system abstract this paper is devoted to the analysis of non negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain the qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction when the growth factor is stronger than the dampening effect with the help of the nonlocal nonlinear term in the reaction for appropriately chosen exponents and arbitrary initial data the model admits a classical solution which is uniformly bounded moreover when the growth factor has the same order with the dampening effect the nonlocal nonlinear exponents can prevent the chemotactic collapse

series expansions for the third incomplete elliptic integral via partial fraction decompositions we find convergent double series expansions for legendre s third incomplete elliptic integral valid in overlapping subdomains of the unit square truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity 1 1 if one of the variables approaches this point faster than the other each approximation is accompanied by an error bound

stability of turing patterns in the brusselator model the selection and competition of turing patterns in the brusselator model are reviewed the stability of stripes and hexagons towards spatial perturbations is studied using the amplitude equation formalism for hexagonal patterns these equations include both linear and nonpotential spatial terms enabling distorted solutions the latter modify substantially the stability diagrams and select patterns with wave numbers quite different from the critical value the analytical results from the amplitude formalism agree with direct simulations of the model moreover we show that slightly squeezed hexagons are locally stable in a full range of distortion angles the stability regions resulting from the phase equation are similar to those obtained numerically by other authors and to those observed in experiments

disclinations in square and hexagonal patterns we report the observation of defects with fractional topological charges disclinations in square and hexagonal patterns as numerical solutions of several generic equations describing many pattern forming systems swift hohenberg equation damped kuramoto sivashinsky equation as well as nonlinear evolution equations describing large scale rayleigh benard and marangoni convection in systems with thermally nearly insulated boundaries it is found that disclinations in square and hexagonal patterns can be stable when nucleated from special initial conditions the structure of the disclinations is analyzed by means of generalized cross newell equations

periodic aperiodic and transient patterns in vibrated granular layers abstract experiments on vertically vibrated granular layers in evacuated containers reveal a variety of patterns for acceleration amplitudes above a critical value 2 5 g stripes squares hexagons spirals triangles and targets as well as particle like localized excitations oscillons and fronts kinks between regions with different vibrational phase are observed as the layer depth and the container oscillation frequency and amplitude are varied a zig zag instability unstable hexagons phase disordered patterns and two phase squares are also observed with a few noteworthy exceptions the patterns are essentially independent of the lateral boundary conditions

hexagonal structure of large scale marangoni convection abstract three dimensional surface tension driven convection in a horizontal liquid layer confined between poorly conducting boundaries is considered in this case the horizontal scale of motion is much larger than the layer depth in such circumstances the dynamics of the system is effectively described by a two dimensional equation for the convection planform it is shown that the preferred pattern of the planform is a hexagonal cell tesselation as occurs when the condition of good thermal insulation is not met the possible effects of drift flow on convection structure are also discussed

asymptotic behaviour of nonlocal reaction diffusion equations abstract the existence of a global attractor in l 2 is established for a reaction diffusion equation on a bounded domain in r d with dirichlet boundary conditions where the reaction term contains an operator f l 2 l 2 which is nonlocal and possibly nonlinear existence of weak solutions is established but uniqueness is not required compactness of the multivalued flow is obtained via estimates obtained from limits of galerkin approximations in contrast with the usual situation these limits apply for all and not just for almost all time instants

fast computation of incomplete elliptic integral of first kind by half argument transformation we developed a new method to calculate the incomplete elliptic integral of the first kind f varphi m by using the half argument formulas of jacobian elliptic functions the method reduces the magnitude of varphi by repeated usage of the formulas while fixing m the method is sufficiently precise in the sense that the maximum relative error is 3 5 machine epsilons at most thanks to the simplicity of the half argument formulas the new procedure is significantly faster than the existing procedures for example it runs 20 60 faster than bulirsch function el1 and 1 9 2 2 times faster than the method using carlson s function r f

wavefronts for a nonlinear nonlocal bistable reaction diffusion equation in population dynamics abstract the wavefronts of a nonlinear nonlocal bistable reaction diffusion equation u t 2 u x 2 u 2 1 j u d u t x 0 r with j x 1 j x and r j x d x 1 are investigated in this article it is proven that there exists a c such that for all c c a monotone wavefront c can be connected by the two positive equilibrium points on the other hand there exists a c such that the model admits a semi wavefront c with 0 furthermore it is shown that for sufficiently small the semi wavefronts are in fact wavefronts connecting 0 to the largest equilibrium in addition the wavefronts converge to those of the local problem as 0