on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

effects of distributed transmission speeds on propagating activity in neural populations motivated by experimental evidence a distribution of axonal transmission speeds is introduced into a standard field model of neural populations the resulting field dynamics is analytically studied by a systematic investigation of the stability and bifurcations of equilibrium solutions using a perturbation approach the effect of distributed speeds on bifurcations of equilibria are determined for general connectivity and speed distributions in addition a nonlinear analysis of traveling fronts is given it is shown that the variance of the speed distribution affects the frequency of bifurcating periodic solutions and the phase speed of traveling waves moreover a new effect is discovered where the introduction of axonal speed distributions leads to the maximization of the traveling front speed doi 10 1103 physreve 73 021906 pacs numbers 87 18 hf 02 60 nm 87 19 la the measured activity of the neural system indicates that the spatiotemporal dynamics of the brain is intimately related to the various brain functions and cognition as examples we mention the space time instabilities during several types of hallucinations 1 and epileptic seizures 2 hence the study of the dynamics of neural models is important for understanding neural and cognitive processes in recent years several models of coupled neurons have attracted much attention including networks involving spatial structures 3 8 describing a continuous synaptically coupled neural field extended in space the dynamics of these fields is governed by

two bump solutions of amari type models of neuronal pattern formation abstract we study a partial integro differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks for a one dimensional domain we develop criteria for the existence and stability of equal width 2 bump solutions under the assumption that the firing rate function is the heaviside function we apply these criteria to an example for which the connectivity is of lateral inhibition type i e the coupling function has one positive zero and find that families of 2 bump solutions exist but none of the solutions are stable extensive numerical searches suggest that this is true for all coupling functions of this form however for a large class of coupling functions which have three positive zeros we find the coexistence of both stable and unstable 2 bump solutions we also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1 bump and 2 bump solutions our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity

pitchfork hopf bifurcations in 1d neural field models with transmission delays recently local bifurcation theory for delayed neural fields was developed in this paper we show how symmetry arguments and residue calculus can be used to simplify the computation of the spectrum in special cases and the evaluation of the normal form coefficients respectively this is done hand in hand with an extensive study of two pitchfork hopf bifurcations for a 1d neural field model with wizard hat type connectivity

dynamic instabilities in scalar neural field equations with space dependent delays in this paper we consider a class of scalar integral equations with a form of space dependent delay these non local models arise naturally when modelling neural tissue with active axons and passive dendrites such systems are known to support a dynamic oscillatory turing instability of the homogeneous steady state in this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability the appropriate amplitude equations are found to be the coupled mean field ginzburg landau equations describing a turing hopf bifurcation with modulation group velocity of o 1 importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio temporal kernels defining the neural field equation of interest indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio temporal interactions we illustrate the predictive power of this form of analysis with comparison against direct numerical simulations paying particular attention to the competition between standing and travelling waves and the onset of benjamin feir instabilities

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

existence results for schrodinger p laplace equations involving critical growth in rn abstract we establish some existence results for schrodinger p laplace equations in r n with various potentials and critical growth of nonlinearity that may occur on some nonempty set although not necessarily the whole space r n the proofs are mainly based on concentration compactness principles in a suitable weighted variable exponent sobolev space and its imbeddings

existence of radial solutions for p x laplacian equations in rn abstract in this paper we obtain sufficient conditions of the existence of radial solutions for p x laplacian equations in r n

a compact embedding result for anisotropic sobolev spaces associated to a strip like domain and some applications abstract let m 1 and d 2 be integers and consider a strip like domain o r d where o r m is a bounded euclidean domain with smooth boundary furthermore let p o r d r be a uniformly continuous and cylindrically symmetric function we prove that the subspace of w 1 p x y o r d consisting of the cylindrically symmetric functions is compactly embedded into l o r d provided that m d p inf x y o r d p x y p sup x y o r d p x y as an application we study a neumann problem involving the p x y laplacian operator and an oscillating nonlinearity proving the existence of infinitely many cylindrically symmetric weak solutions our approach is based on variational and topological methods in addition to the principle of symmetric criticality

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

how effective delays shape oscillatory dynamics in neuronal networks abstract synaptic dendritic and single cell kinetics generate significant time delays that shape the dynamics of large networks of spiking neurons previous work has shown that such effective delays can be taken into account with a rate model through the addition of an explicit fixed delay roxin et al 2005 2006 29 30 here we extend this work to account for arbitrary symmetric patterns of synaptic connectivity and generic nonlinear transfer functions specifically we conduct a weakly nonlinear analysis of the dynamical states arising via primary instabilities of the asynchronous state in this way we determine analytically how the nature and stability of these states depend on the choice of transfer function and connectivity we arrive at two general observations of physiological relevance that could not be explained in previous work these are 1 fast oscillations are always supercritical for realistic transfer functions and 2 traveling waves are preferred over standing waves given plausible patterns of local connectivity we finally demonstrate that these results show good agreement with those obtained performing numerical simulations of a network of hodgkin huxley neurons

compactness and existence results for the p laplace equation abstract given 1 p n and two measurable functions v r 0 and k r 0 r 0 we define the weighted spaces w u d 1 p r n r n v x u p d x l k q l q r n k x d x and study the compact embeddings of the radial subspace of w into l k q 1 l k q 2 and thus into l k q l k q l k q as a particular case we consider exponents q 1 q 2 q that can be greater or smaller than p our results do not require any compatibility between how the potentials v and k behave at the origin and at infinity and essentially rely on power type estimates of their relative growth not of the potentials separately we then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p laplace equations of the form p u v x u p 2 u g x u in r n 1 p n where v and g u with u fixed may be vanishing or unbounded at zero or at infinity both the cases of g super and sub p linear in u are studied and in the sub p linear case nonlinearities with g 0 0 are also considered

solutions for neumann boundary value problems involving p x laplace operators abstract in this paper we study the nonlinear neumann boundary value problem of the following form p div u p x 2 u u p x 2 u f x u in u p x 2 u g x u on using the variational method under appropriate assumptions on f and g we obtain a number of results on existence and multiplicity of solutions also what is also interesting is that we obtain some results which can be considered as extensions of the classical result named combined effects of concave and convex nonlinearities moreover the positive solution of the problem is considered

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

calderon zygmund singular operators in extrapolation spaces abstract we study the boundedness of the hardy littlewood maximal operator in abstract extrapolation banach function lattices and their kothe dual spaces the extrapolation spaces are generated by compatible families of banach function lattices on quasi metric measure spaces with doubling measure these results combined with a variant of the integral coifman fefferman inequality imply that every calderon zygmund singular operator is bounded in considered extrapolation spaces we apply these results to extrapolation spaces determined by compatible families of calderon lozanovskii spaces in particular to compatible families of orlicz spaces that are interpolation of weighted l p spaces 1 p with a p weights defined on spaces of homogeneous type

some theoretical and numerical results for delayed neural field equations in this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons nonlinear delayed integro differential equations describe the spatio temporal behavior of these fields using methods from the theory of delay differential equations we show the existence and uniqueness of a solution of these equations a lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable we also present a fairly detailed study of the numerical computation of these solutions this is to our knowledge the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed another original contribution of ours is the definition of a lyapunov functional and the result of stability it implies we illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience

compact imbedding theorems with symmetry of strauss lions type for the space w1 p x ω abstract in this paper we extend some compact imbedding theorems of strauss lions type to the space w 1 p x when the domain has some symmetric properties and p x satisfies some conditions

sharp reverse holder property for a weights on spaces of homogeneous type abstract in this article we present a new proof of a sharp reverse holder inequality for a weights then we derive two applications a precise open property of muckenhoupt classes and as a consequence of this last result we obtain a simple proof of a sharp weighted bound for the hardy littlewood maximal function involving a constants m l p w c 1 p 1 w a p a 1 p where 1 p w 1 1 p and c is a dimensional constant our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak reverse holder inequality which is still sufficient to prove the open property for a p classes and the l p boundedness of the maximal function in this latter case the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure and the geometric constant of the quasimetric

existence and blow up rate of large solutions of p x laplacian equations with gradient terms abstract in this paper we investigate boundary blow up solutions of the problem p x u f x u k x u m x in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian our results extend the previous work 25 of y liang q h zhang and c s zhao from the radial case to the non radial setting and 46 due to q h zhang and d motreanu from the assumption that k x u x m x is a small perturbation to the case in which k x u m x is a large perturbation we provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d x and in terms of the growth of the exponents furthermore the comparison principle is no longer applicable in our context since f x is not assumed to be monotone in this paper

compact imbeddings between variable exponent spaces with unbounded underlying domain abstract in this paper we have established some compact imbedding theorems for some subspaces of w 1 p x u when the underlying domain u is unbounded the domain we consider is mainly of type r n n 2 or r l l 2 where r m is a bounded domain with smooth boundary

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega