nonlocal diffusion equations with dynamical boundary conditions abstract in this paper we study nonlocal problems that are analogous to the local ones given by the laplacian or the p laplacian with dynamical boundary conditions we deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity

liouville type results for a nonlocal obstacle problem this paper is concerned with qualitative properties of solutions to nonlocal reaction diffusion equations of the form int mathbb r n setminus k j x y big u y u x big d y f u x 0 quad x in r n setminus k set in a perforated open set mathbb r n setminus k where k subset mathbb r n is a bounded compact obstacle and f is a bistable nonlinearity when k is convex we prove some liouville type results for solutions satisfying some asymptotic limiting conditions at infinity we also establish a robustness result assuming slightly relaxed conditions on k

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

a difference method for solving parabolic equations of order 2n in this paper a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined the resulting system is solved by efficient implicit iterations in some numerical examples maple modules are designed for the purpose of testing and using the method

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

high order p stable multistep methods abstract recently numerical integration of the special ivp y f t y whose solutions are of periodic type have been considered of great interest p stability is the appropriate requirement for numerical methods integrating periodic problems as established by lambert and watson 1976 a p stable linear multistep method has order no greater than two in this paper two step multi stage linear methods are proposed for numerical integration of periodic problems and p stable methods of this type obtained in a simple form up to order eight are shown finally those methods are successfully tested

asymptotic behaviour of stokes flow in a thin domain with a moving rough boundary we consider a problem that models fluid flow in a thin domain bounded by two surfaces one of the surfaces is rough and moving whereas the other is flat and stationary the problem involves two small parameters and that describe film thickness and roughness wavelength respectively depending on the ratio three different flow regimes are obtained in the limit as both of them tend to zero time dependent equations of reynolds type are obtained in all three cases stokes roughness reynolds roughness and high frequency roughness regime the derivations of the limiting equations are based on formal expansions in the parameters and

on the principal eigenvalue of some nonlocal diffusion problems abstract in this paper we analyze some properties of the principal eigenvalue 1 of the nonlocal dirichlet problem j u x u x u x in with u x 0 in r n here is a smooth bounded domain of r n and the kernel j is assumed to be a c 1 compactly supported even nonnegative function with unit integral among other properties we show that 1 is continuous or even differentiable with respect to continuous differentiable perturbations of the domain we also provide an explicit formula for the derivative finally we analyze the asymptotic behavior of the decreasing function 1 when the dilatation parameter 0 tends to zero or to infinity

fractional problems in thin domains abstract in this paper we consider nonlocal fractional problems in thin domains given open bounded subsets u r n and v r m we show that the solution u e to x s u e x y y t u e x y f x e 1 y in u e v with u e x y 0 if x u and y e v verifies that u e x y u e x e y u 0 strongly in the natural fractional sobolev space associated to this problem we also identify the limit problem that is satisfied by u 0 and estimate the rate of convergence in the uniform norm here x s u and y t u are the fractional laplacian in the 1st variable x with a dirichlet condition u x 0 if x u and in the 2nd variable y with a neumann condition integrating only inside v respectively that is x s u x y r n u x y u w y x w n 2 s d w and y t u x y v u x y u x z y z m 2 t d z

automorphisms and isomorphisms of real henselian fields let k and l be ordered algebraic extensions of an ordered field f suppose k and l are henselian with archimedean real closed residue class fields then k and l are shown to be f isomorphic as ordered fields if they have the same value group analogues to this result are proved involving orderings of higher level unordered extensions and when k and l are maximal valued fields transcendental extensions as a corollary generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups the results on isomorphisms are applied to the computation of automorphism groups of k and to the study of the fixed fields of groups of automorphisms of k if k is real closed and maximal with respect to its canonical valuation then these fixed fields are shown to be exactly those real closed subfields of k which are topologically closed in k generalizations of this fact are proved an example is given to illustrate several aspects of the problems considered here

an explicit four step phase fitted method for the numerical integration of second order initial value problems abstract an explicit four step method with phase lag of infinite order is developed for the numerical integration of second order initial value problems extensive numerical testing indicates that this new method can be generally more efficient than other four step methods

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

powers of totally positive subgroups of fields abstract for a formally real field f a totally positive co torsion subgroup s of f and m 1 we prove that s m s m this extends a result of becker we prove several refinements of this fact related to higher pythagoras numbers real holomorphy rings and sums with mixed powers

cauchy problem of a nonlocal p laplacian evolution equation with nonlocal convection abstract this paper is concerned with the cauchy problem of a nonlocal equation that takes into account convective and p laplacian diffusive effects u t x t r n j x y u y t u x t p 2 u y t u x t d y g f u f u x t with j radially symmetric and g not necessarily symmetric first we prove the existence and uniqueness of solutions and if the convolution kernels j and g are rescaled appropriately we show that solutions of the nonlocal problem converge to the solution of the usual p laplacian diffusion equation with convection finally as a supplementary result we study the asymptotic behavior of solutions as t and give the decay estimate

existence uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks abstract the author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations he applies fixed point theorems to prove the existence of the traveling waves then he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves

a numerical approach to nonlinear two point boundary value problems for odes in this paper we propose a numerical approach to solve some problems connected with the implementation of the newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two point bvps for odes with mixed linear boundary conditions by using the finite difference method

semilinear elliptic equations in thin domains with reaction terms concentrating on boundary abstract in this paper we analyze the behavior of a family of steady state solutions of a semilinear reaction diffusion equation with homogeneous neumann boundary condition posed in a two dimensional thin domain with reaction terms concentrated in a narrow oscillating neighborhood of the boundary we assume that the domain and therefore the oscillating boundary neighborhood degenerates into an interval as a small parameter goes to zero our main result is that this family of solutions converges to the solution of a one dimensional limit equation capturing the geometry and oscillatory behavior of the open sets where the problem is established

an optimal algorithm for certain boundary value problem abstract the o h 4 finite difference scheme for the second derivative u x leads to a coherent pentadiagonal matrix which is factorized into two tridiagonal matrices this factorization is used to derive an optimal algorithm for solving a linear system of equations with the pentadiagonal matrix as an application a nonlinear system of ordinary differential equations is approximated by an o h 4 convergent finite difference scheme this scheme is solved by the implicit iterative method applying the algorithm at each iteration a mathematica module designed for the purpose of testing and using the method is attached

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

chain signatures and higher level witt rings in many respects becker s theory of orderings of higher level is very much reminiscent of the familiar artin schreier theory of ordered fields a few examples of similarities are fields with higher level orderings have real closures 1 3 orderings of higher level can be used to study sums of powers in fields 3 41 there is a theory of witt rings for fields with higher level preorderings 7 81 however among all these similarities one very obvious difference stands out a field with a higher level ordering has nonisomorphic real closures becker showed that in order to describe the isomorphism classes of real closures a whole sequence of higher level orderings is required 1 p 162 theorem 121 subsequently this has been relined and reformulated through the introduction of chains of higher level orderings 121 and chain signatures 27 see alo 10 11 i in view of this situation one should expect that if results from the usual theory of ordered fields level 1 carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings the purpose of this paper is to look for such results in the theory of higher level witt rings developed recently by becker and rosenberg 7 81 to keep things from getting too complicated this will be done only for the case where all levels are powers of 2 thus throughout the term higher level will be synonymous with level 2 for some n iv w k denotes the witt ring of the field k with respect to the higher level preordering k c k2 nonzero sums of 2 th powers of 9 just as the use of signatures s leads to homomorphisms w 7 chain signatures lead to homomorphisms

a nonlocal dispersal equation arising from a selection migration model in genetics this paper is concerned with the existence uniqueness and asymptotic stability of positive steady states for a nonlocal dispersal equation arising from selection migration models in genetics due to the lack of compactness and regularity of the nonlocal operators many classical methods cannot be used directly to the nonlocal dispersal problems this motivates us to find new techniques we first establish a criterion on the stability and instability of steady states this result is effective to get a necessary condition to guarantee a positive steady state it also gives the uniqueness then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle finally we consider the dynamic behavior of the initial value problem

the dirichlet boundary problem for a nonlocal cahn hilliard equation abstract we study the existence uniqueness and continuous dependence on initial data of the solution for a nonlocal cahn hilliard equation with dirichlet boundary condition on a bounded domain under a nondegeneracy assumption the solutions are classical but when this is relaxed the equation is satisfied in a weak sense also we prove that there exists a global attractor in some metric space

a higher order finite difference method for solving a system of integro differential equations abstract a finite difference method and gauss method of numerical integration are used to approximate the integro differential system of eq 1 with the global error z h o h4 the resulting system of algebraic equations is solved by efficient implicit iterations a mathematica module is designed for the purpose of testing and using the method an example of an integral differential equation with the green s function as the kernel is solved

replacing the finite difference methods for nonlinear two point boundary value problems by successive application of the linear shooting method abstract this paper studies numerical solution of nonlinear two point boundary value problems for second order ordinary differential equations first it establishes a connection between the finite difference method and the quasi linearization method we prove that using finite differences to discretize the sequence of linear differential equations arising from quasi linearization newton method on operator level leads to the usual iteration formula of the newton finite difference method from the provided derivation it can easily be inferred that such a relation holds also for the picard and the constant slope methods based on this result we propose a way of replacing the newton picard and constant slope finite difference methods by respective successive application of the linear shooting method this approach has a number of advantages it removes the necessity of solving systems of algebraic equations hence working with matrices altogether compared to the usual finite difference method with general solver it reduces the number of computational operations from o n 3 where n is the number of mesh points to only o n

the neumann boundary problem for a nonlocal cahn hilliard equation we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal cahn hilliard equation on a bounded domain the equation generates a gradient flow for a free energy functional with nonlocal interaction also we apply a nonlinear poincare inequality to show the existence of an absorbing set in each constant mass affine space

approximate the fokker planck equation by a class of nonlocal dispersal problems abstract this paper is concerned with an inhomogeneous nonlocal dispersal equation we study the limit of the re scaled problem of this nonlocal operator and prove that the solutions of the re scaled equation converge to a solution of the fokker planck equation uniformly we then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero

a higher accuracy finite difference method for an elliptic equation of order four the elliptic equation 1 with boundary value conditions either 2 or 3 is solved by an o h4 convergent finite difference method combined with an implicit iterative method the method is tested by examples of boundary value problems with use of the enclosed mathematica module solve bhpeq the module solve bhpeq gives options either it implements the o h4 convergent nine mesh points scheme or the o h2 convergent standard five points scheme when the finite difference approximation of eq 1 is considered

a nonlocal convection diffusion equation in this paper we study a nonlocal equation that takes into account convective and diffusive effects ut j u u g f u f u in rd with j radially symmetric and g not necessarily symmetric first we prove existence uniqueness and continuous dependence with respect to the initial condition of solutions this problem is the nonlocal analogous to the usual local convection diffusion equation ut u b f u in fact we prove that solutions of the nonlocal equation converge to the solution of the usual convection diffusion equation when we rescale the convolution kernels j and g appropriately finally we study the asymptotic behaviour of solutions as t when f u u q 1u with q 1 we find the decay rate and the first order term in the asymptotic regime

nonlocal problems in thin domains in this paper we consider nonlocal problems in thin domains first we deal with a nonlocal neumann problem that is we study the behavior of the solutions to f x 1 2j x y u y u x dyf x 1 2j x y u y u x dy with j z j z1 z2 j z j z1 z2 and 1 2 rn rn1 rn2 1 2 rn rn1 rn2 a bounded domain we find that there is a limit problem that is we show that u u0u u0 as 0 0 in and this limit function verifies 2f x1 x2 dx2 2 1j x1 y1 0 u0 y1 u0 x1 dy1 with u0 x1 2u0 x1 x2 dx2 in addition we deal with a double limit when we add to this model a rescale in the kernel with a parameter that controls the size of the support of j we show that this double limit exhibits some interesting features we also study a nonlocal dirichlet problem f x rnj x y u y u x dy x x with u x 0u x 0 x rn x rn and deal with similar issues in this case the limit as 0 0 is u0 0u0 0 and the double limit problem commutes and also gives v 0v 0 at the end

linear non local diffusion problems in metric measure spaces the aim of this paper is to provide a comprehensive study of some linear nonlocal diusion problems in metric measure spaces these include for example open subsets in r n graphs manifolds multi structures or some fractal sets for this we study regularity compactness positiveness and the spectrum of the stationary nonlocal operator then we study the solutions of linear evolution nonlocal diusion problems with emphasis in similarities and dierences with the standard heat equation in smooth domains in particular prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods

asymptotic behavior for a nonlocal diffusion equation in domains with holes the paper deals with the asymptotic behavior of solutions to a non local diffusion equation ut j u u lu in an exterior domain which excludes one or several holes and with zero dirichlet data on mathbb r n setminus omega when the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes on the other hand if the solution is scaled according to its decay factor close to the holes it behaves like a function that is l harmonic lu 0 in the exterior domain and vanishes in its complement the height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior the inner and the outer behaviors can be presented in a unified way through a suitable global approximation

lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space abstract we find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form t u r d k x y u y u x d y here we consider a kernel k x y y a x x a y where is a bounded nonnegative function supported in the unit ball and a means a diffeomorphism on r d a simple example being a linear function a x a x the upper and lower bounds that we obtain are given in terms of the jacobian of a and the integral of indeed in the linear case a x a x we obtain an explicit expression for the first eigenvalue in the whole r d and it is positive when the determinant of the matrix a is different from one as an application of our results we observe that when the first eigenvalue is positive there is an exponential decay for the solutions to the associated evolution problem as a tool to obtain the result we also study the behavior of the principal eigenvalue of the nonlocal dirichlet problem in the ball b r and prove that it converges to the first eigenvalue in the whole space as r

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions we present some comparison results for solutions to certain non local elliptic and parabolic problems that involve the fractional laplacian operator and mixed boundary conditions given by a zero dirichlet datum on part of the complementary of the domain and zero neumann data on the rest these results represent a non local generalization of a hopf s lemma for elliptic and parabolic problems with mixed conditions in particular we prove the non local version of the results obtained by davila and davila and dupaigne for the classical case s 1 in 23 24 respectively

blow up results for evolution problems with inhomogeneous nonlocal diffusion abstract we provide a sufficient condition on the existence and nonexistence of global positive solutions to the cauchy problem for an inhomogeneous nonlocal diffusion u t j u u u p f x where j is a nonnegative function p 0 and f 0 0 meanwhile the case of an inhomogeneous nonlocal diffusion system is considered it turns out that the inhomogeneous terms substantially contribute to the blow up exponent which coincides with the classical one when the diffusion is given by the laplacian

o h 4 locally overconvergent semidiscrete scheme for the equation u t u xx f t x u abstract in the equation ut uxx f t x u the second derivative uxx is approximated by the finite difference operator lh which has o h2 local truncation error at the points h and 1 h next to the ends of the interval 0 1 and o h4 at the interior mesh points 2h 3h 1 2h it is proved that the obtained semidiscrete scheme is o h4 globally convergent the semidiscrete scheme is solved by the pure implicit finite difference scheme combined with the method of iteration and the gauss elimination method for tridiagonal matrices the nonstationary liouville s equation ut uxx e u has been solved by the proposed algorithm

neumann conditions for the higher order s fractional laplacian δ su with s 1 abstract in this paper we study a variational neumann problem for the higher order fractional laplacian s s 1 in the process we introduce some nonlocal neumann boundary conditions that appear in a natural way from a gauss like integration formula

explicit two step methods with minimal phase lag for the numerical integration of special second order initial value problems and their application to the one dimensional schro dinger equation abstract two step sixth order methods with phase lag of order eight ten and twelve are developed for the numerical integration of the special second order initial value sproblem an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by chawla rao and neta this journal 1986 1987

asymptotic behavior for nonlocal dispersal equations abstract this paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation by means of super subsolution method and monotone iteration we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation then we apply these results to our equation and show that the nonnegative solution is unique and the behavior of this solution depends on parameter in equation for 1 the solution decays to zero as t while for 1 the solution converges to the unique positive stationary solution as t in addition we show that the solution blows up under some conditions

boundary fluxes for nonlocal diffusion abstract we study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary this problem may be seen as a generalization of the usual neumann problem for the heat equation first we prove existence uniqueness and a comparison principle next we study the behavior of solutions for some prescribed boundary data including blowing up ones finally we look at a nonlinear flux boundary condition

semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries abstract in this work we study the behavior of a family of solutions of a semilinear elliptic equation with homogeneous neumann boundary condition posed in a two dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary our main result is concerned with the upper and lower semicontinuity of the set of solutions we show that the solutions of our perturbed equation can be approximated with one of a one dimensional equation which also captures the effects of all relevant physical processes that take place in the original problem

positive polynomials and tame preorderings from the geometric point of view it is one of the essential merits of theartinschreier theory of ordered fields to have shed new light on certain problems concerning the behaviour of regular functions on an irreducible variety v over r this eventually led to a purely algebraic characterization of those f r v which are positive or nonnegative on v r reg a first example is provided by e artin s positive solution of hilbert s 17 th problem in the seventies classical artin schreier theory was extended to commutative rings and generalizations of artin s result were proven for the purpose of this introduction we consider only a special situation let us fix polynomials f1 fk r x r x1 xn