existence of sign changing solutions for nonlocal kirchhoff schrodinger type equations in r3 abstract in this paper we investigate the existence of sign changing solution for a class nonlocal kirchhoff schrodinger type equation a b r 3 u 2 d x u v x u f x u x r 3 where a and b are positive constants with the help of the constraint variational method and via a direct approach we show the existence of sign changing solution and prove that the sign changing solution has precisely two nodal domains this work can be regarded as the complement for some results of the literature

on nonlocal p x laplacian equations abstract this paper deals with the nonlocal anisotropic p x laplacian dirichlet problems with non variational form i 1 n a i u d i d i u p i x 2 d i u b u f x u x in u 0 and with variational form i 1 n a i d i u p i x p i x d x d i d i u p i x 2 d i u b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a i is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained

existence of optimal controls for spde with locally monotone coefficients the aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations spdes with locally monotone coefficients controlled by

existence of ground states for a kirchhoff type problem without 4 superlinear condition abstract we consider the existence of ground state solutions for the kirchhoff type problem a b r n u 2 d x u v x u u p 2 u x r n u h 1 r n where a b 0 n 1 2 3 and 2 p 2 here we are interested in the case that 2 p 4 since the existence of ground state for 4 p 2 is easily obtained by a standard variational argument our method is based on a poho z aev type identity

regularization and error estimate for an initial inverse nonlocal diffusion problem abstract we study the backward problem of determining the initial condition for a system of parabolic diffusion equations which is severely ill posed in the sense of hadamard to stabilize the solution we develop the quasi reversibility qr and fourier truncation methods to construct the regularized solutions we also investigate the error estimates and convergence rates between the regularized solutions and the true solution in l 2 and h 1 norm a numerical scheme is presented

the multiplicity of positive solutions for a class of nonlocal elliptic problem abstract in this paper using the theory of fixed point index we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem

existence and multiplicity of normalized solutions for the nonlinear kirchhoff type problems abstract in this paper we prove the existence and multiplicity results of solutions with prescribed l 2 norm for a class of kirchhoff type problems a b r 3 u 2 d x u u f u in r 3 where a b 0 are constants r and f c r r to obtain such solutions we look into critical points of the energy functional e b u a 2 r 3 u 2 b 4 r 3 u 2 2 r 3 f u constrained on the l 2 spheres s c u h 1 r 3 u 2 2 c here c 0 and f s 0 s f t d t under some mild assumptions on f we show that critical points of e b unbounded from below on s c exist for c 0 in addition we establish the existence of infinitely many radial critical points u n b of e b on s c provided that f is odd finally the asymptotic behavior of u n b as b 0 is analyzed these conclusions extend some known ones in previous papers

a population model with nonlinear diffusion abstract a model is presented for a single species population moving in a limited one dimensional environment the birth death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age the diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation it is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant the interaction of the birth death and diffusion processes is studied and is shown to yield a modified birth death mechanism which is both time and space dependent

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

virtual element method for a nonlocal elliptic problem of kirchhoff type on polygonal meshes abstract we consider the discretization of the nonlocal elliptic problem of kirchhoff type using the virtual element method vem over polygonal meshes the nonlocal diffusion coefficient is approximated by using the l 2 projection operator which is directly computable from the degrees of freedom however the presence of the nonlocal term reduces the sparsity of the jacobian matrix which would increase the computational burden to avoid this and to retain the sparsity of the jacobian the nonlinear system is replaced by an equivalent system inspired by the work on the fem by gudi 2012 the numerical results show that the proposed methodology yields optimal convergence rate in the l 2 norm theoretical estimates are derived for h 1 norms and are verified by solving a numerical example

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

on computation of bounds of the bifurcation parameter for a non local elliptic equation with increasing nonlinearity a novel approach is developed to prove the existence of d the supremum of the spectrum of a non local elliptic problem associated with homogeneous robin and dirichlet boundary conditions analytical upper and lower bounds of d are obtained in closed form the bounds are presented for the slab cylindrical and spherical geometries

a finite difference scheme for some nonlinear diffusion equations in an absorbing medium support splitting phenomena in the porous media equation vt vm xx m 1 it is well known that there appears a finite propagation of the initial support and that if the initial support is connected so is mbox supp v t cdot for t 0 when the effect of the absorption is considered as the additional lower order term cvp c 0 p 0 the possibility that the support will split is expected rosenau and kamin phys d 8 1983 pp 273 283 tried the numerical computations and suggested the support splitting phenomena but the theoretical justification is not discussed in this paper such phenomena are investigated by use of finite difference schemes and the sufficient conditions under which the support begins to split are obtained in the specific case where m p 2 and 0 p 1

perturbed kirchhoff type neumann problems in orlicz sobolev spaces the aim of this paper is to establish the existence of infinitely many solutions for perturbed kirchhoff type non homogeneous neumann problems involving two parameters to be precise we prove that an appropriate oscillating behaviour of the nonlinear term even under small perturbations ensures the existence of infinitely many solutions our approach is based on recent variational methods for smooth functionals defined on orlicz sobolev spaces

extinction in finite time of solutions to degenerate parabolic equations with nonlinear boundary conditions abstract this work concerns a nonlinear diffusion absorption equation with nonlinear boundary flux the main topic of interest is the problem of finite time extinction i e the solutions vanish after a finite time the sufficient and necessary conditions for occurrence of extinction are established it is shown that extinction is caused by either strong absorption in the interior of the domain or fast diffusion combined with strong absorption through the boundary of the domain extinction results are also obtained for the mixed boundary value problem in contrast to the nonlinear neumann problem the absorption on the boundary is no longer important i e the occurrence of extinction in this case is completely determined by the effects of diffusion and interior absorption

on besov spaces of logarithmic smoothness and lipschitz spaces we compare besov spaces b p q 0 b with zero classical smoothness and logarithmic smoothness b defined by using the fourier transform with the corresponding spaces b p q 0 b defined by means of the modulus of smoothness in particular we show that b p q 0 b 1 2 b 2 2 0 b for b 1 2 we also determine the dual of in b p q 0 b with the help of logarithmic lipschitz spaces lip p q 1 alpha finally we show embeddings between spaces lip p q 1 alpha and b p q 1 b which complement and improve embeddings established by haroske 2000

the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations abstract this paper deals with the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations of the form a i u i u f where i c 1 x r x is a reflexive banach space i 0 0 i is even and strictly convex i u u as u i x x is a bounded homeomorphism but is not necessarily homogeneous a 0 r is continuous f x 0 some properties and examples of such a functional i are given some results on the number of solutions of the nonlocal equation are obtained

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

nonlinear heat conduction with absorption space localization and extinction in finite time additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions the statements answer the questions when the support of the solution is bounded in space and when the solution dies in finite time

on 2 microlocal spaces with all exponents variable abstract in this paper we study various key properties for 2 microlocal besov and triebel lizorkin spaces with all exponents variable including the lifting property embeddings and fourier multipliers we also clarify and improve some statements recently published

finite element analysis of nonlocal coupled parabolic problem using newton s method abstract in this article we propose finite element method to approximate the solution of a coupled nonlocal parabolic system an important issue in the numerical solution of nonlocal problems while using the newton s method is related to its structure indeed unlike the local case the jacobian matrix is sparse and banded the nonlocal term makes the jacobian matrix dense as a consequence computations consume more time and space in contrast to local problems to overcome this difficulty we reformulate the discrete problem and then apply the newton s method we discuss the well posedness of the weak formulation at continuous as well as at discrete levels we derive a priori error estimates for both semi discrete and fully discrete formulations results based on usual finite element method are provided to confirm the theoretical estimates

characterizations of logarithmic besov spaces in terms of differences fourier analytical decompositions wavelets and semi groups we work with besov spaces bp q0 b defined by means of differences with zero classical smoothness and logarithmic smoothness with exponent b we characterize bp q0 b by means of fourier analytical decompositions wavelets and semi groups we also compare those results with the well known characterizations for classical besov spaces bp qs

ground state solutions for kirchhoff type equations with asymptotically 4 linear nonlinearity this paper is concerned with the following kirchhoff type equation a b v 2 v f x v in v 0 on assuming that the primitive of f is asymptotically 4 linear as v a homeomorphism between the nehari manifold and a subset of unit sphere is constructed based on some observations and new techniques inspired by recent work of szulkin and weth 2010 ground state solutions for the equation above are obtained as well as infinitely many pairs of nontrivial solutions for case f x v is odd with respect to v

a high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm the elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering firstly we construct a linear finite element scheme for the nonlocal boundary problem and derive the optimal l 2 error estimate then based on the quadratic finite element and the extrapolation linear finite element methods we present a composite scheme and prove that it is convergent order three furthermore we design an upper triangular preconditioning algorithm for the linear finite element discrete system finally numerical results not only validate that the new algorithm is efficient but also show that the new scheme is convergent order three furthermore order four on uniform grids

on a numerical method for a homogeneous nonlinear nonlocal elliptic boundary value problem abstract in this work we develop a numerical method for the equation 0 1 u t d t u x u x 2 n 1 0 x 0 1 u 0 a u 1 b we begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the schauder fixed point theorem from this analysis we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping utilizing the a priori estimates and the brouwer fixed point theorem next we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error h we conclude with some examples of the numerical process

the euler galerkin finite element method for a nonlocal coupled system of reaction diffusion type in this work we study a system of parabolic equations with nonlocal nonlinearity of the following type u t a 1 l 1 u l 2 v u 1 u p 2 u f 1 x t in 0 t v t a 2 l 1 u l 2 v v 2 v p 2 v f 2 x t in 0 t u x t v x t 0 on 0 t u x 0 u 0 x v x 0 v 0 x in where a 1 and a 2 are lipschitz continuous positive functions l 1 and l 2 are continuous linear forms 1 2 0 and p 2 we prove the convergence of a linearized euler galerkin finite element method and obtain the order of convergence in the l 2 norm finally we implement and simulate the presented method in matlab s environment

stability of fredholm type integro parabolic equations linear time variant integrodifferential equations in a banach space are considered from a particular inner product conditions for solution existence and stability are derived applications to integro parabolic equations of the fredholm type are discussed

multiple nontrivial solutions to a pp kirchhoff equation abstract in this paper morse theory is used to show the existence of nontrivial solutions to the nonlocal p kirchhoff type quasilinear elliptic equation m u p d x p 1 p u f x u in a smooth bounded domain r n with dirichlet boundary condition

existence and nonexistence of positive solutions for nonlocal problems with inhomogeneous strong allee effect in this paper we deal with the nonlocal elliptic problem with inhomogeneous strong allee email protected w1p u email protected email protected x u email protected u 0 email protected email protected where the nonlocal coefficient email protected w1p u pdx is a continuous function of w12 u 2dx by means of variational approach we prove that the problem has at least two positive solutions for large l under suitable hypotheses about nonlinearity we also prove some nonexistence results

the sign changing solutions for nonlinear elliptic problem with carrier type abstract in this paper using the fixed point index method we are concerned with the existence of sign changing solutions of the following problem 0 1 a b u x d x u f u x in u x 0 x on where a 0 b 0 and r n is a bounded open set with smooth boundary and 1

finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption the energy solutions to the equation have finite speed of propagation if l or l if 1 vq t is uniformly bounded for t 0 localisation property and if q sharp upper bounds of the interface or free boundary are obtained we use a weighted energy method the weights being powers of the distance to a variable half space we also study decay rates as t and extinction in finite time for bounded and unbounded domains with null dirichlet boundary conditions our equation includes the porous media equation with absorption analogous results hold if m is replaced by an appropriate quasilinear elliptic operator

a convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross diffusion in this paper we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non coincident spatial domains with nonlocal and cross diffusion the epidemic disease transmission occurring through a contaminated environment the mobility of each class is assumed to be influenced by the gradient of the other classes we address the questions of existence of weak solutions and existence and uniqueness of classical solution by using respectively a regularization method and an interpolation result between banach spaces moreover we propose a finite volume scheme and proved the well posedness nonnegativity and convergence of the discrete solution the convergence proof is based on deriving a series of a priori estimates and by using a general l p compactness criterion finally the numerical scheme is illustrated by some examples

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

two positive solutions for the kirchhoff type elliptic problem with critical nonlinearity in high dimension abstract we investigate the kirchhoff type elliptic problem with critical nonlinearity 1 u 2 d x u u q u 2 1 u 0 in u 0 on where n 5 r n is a bounded domain with smooth boundary 0 r 2 2 n n 2 and q 1 2 1 we prove the existence of two solutions of it via the variational method since n 5 and 0 the uniqueness assertion for the associated limiting problem may fail this causes serious difficulties in controlling concentrating palais smale sequences we overcome these by introducing new techniques for a mountain pass type solution we utilize the limit function of the fibering maps of the concentrating palais smale sequence this tool is based on our careful setting of nehari type sets on the other hand a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution this is the first work which proves the multiplicity of positive solutions of the kirchhoff type critical problem in high dimension

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

on a class of problems involving a nonlocal operator in this paper we prove a result on existence of positive solution for a nonlocal elliptic problem by using a result on fixed point index theory and we establish another existence result which is proved by sub and supersolution and a comparison principle moreover we prove the existence uniqueness and asymptotic behaviour of the solutions for the evolution case

on generalized besov and triebel lizorkin spaces of regular distributions abstract we establish conditions on the parameters which are both necessary and sufficient in order that besov and triebel lizorkin spaces of generalized smoothness contain only regular distributions we also connect this with the possibility of embedding such spaces in some particular lebesgue spaces

finite extinction time for nonlinear parabolic equations with nonlinear mixed boundary data the last theorem in the article i e theorem 4 2 and the subsequent example on pp 10 11 have to be deleted there is an error in the construction of the upper solution u w t h x in the proof of theorem 4 2 since w s 1 m p w w s p w p w m 2 2 condition i in the statement of the theorem implies that w s 0 consequently u is not an upper solution as claimed since this is the very last theorem of the paper the rest of the article from p 1 to the end of theorem 4 1 on p 10 all remain valid without any change

long time behavior of a non autonomous parabolic equation with nonlocal diffusion and sublinear terms abstract this paper is devoted to study the asymptotic behavior of a time dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth namely we extend some previous results from the literature obtaining existence uniqueness and continuity results analyzing the stationary problem and decay of the solutions of the evolutionary problem and finally under more general assumptions ensuring the existence of pullback attractors for the associated dynamical system in both l 2 and h 1 norms relationships among these objects are established using regularizing properties of the equation

nontrivial solutions for kirchhoff type problems with a parameter abstract the paper deals with the following kirchhoff type problems a b r n u 2 d x u v x u f x u in r n u h 1 r n where constants a b 0 1 is a parameter and f x u is either sublinear or superlinear at infinity under relaxed assumptions on v x we establish the existence and multiplicity results without the compactness of embedding of the working space