existence of solution of the logarithmic diffusion equation with bounded above gauss curvature abstract we give a simple proof of an extension of the existence results of ricci flow of giesen and topping 2010 2011 15 20 on incomplete surfaces with bounded above gauss curvature without using the difficult shi s existence theorem of ricci flow on complete non compact surfaces and the pseudolocality theorem of perelman 7 on ricci flow we will also give a simple proof of a special case of the existence theorem of topping 2010 16 without using the existence theorem of shi 1989 9

dynamic patterns in a two dimensional neural field with refractoriness the formation of dynamic patterns such as localized propagating waves is a fascinating self organizing phenomenon that happens in a wide range of spatially extended systems including neural systems in which they might play important functional roles here we derive a type of two dimensional neural field model with refractoriness to study the formation mechanism of localized waves after comparing this model with existing neural field models we show that it is able to generate a variety of localized patterns including stationary bumps localized waves rotating along a circular path and localized waves with longer range propagation we construct explicit bump solutions for the two dimensional neural field and conduct a linear stability analysis on how a stationary bump transitions to a propagating wave under different spatial eigenmode perturbations the neural field model is then partially solved in a comoving frame to obtain localized wave solutions whose spatial profiles are in good agreement with those obtained from simulations we demonstrate that when there are multiple such propagating waves they exhibit rich propagation dynamics including propagation along periodically oscillating and irregular trajectories these propagation dynamics are quantitatively characterized in addition we show that these waves can have repulsive or merging collisions depending on their collision angles and the refractoriness parameter due to its analytical tractability the two dimensional neural field model provides a modeling framework for studying localized propagating waves and their interactions

on l envelopes of banach spaces abstract let a denote any of the following classes of l spaces c k spaces lindenstrauss spaces separably injective spaces universally separably injective spaces lindenstrauss pelczynski spaces or l spaces we show that every banach space x can be isometrically embedded into a space a x a so that every operator x a with a a can be extended to an operator a x a with the same norm

hamilton s gradient estimates for fast diffusion equations under the ricci flow abstract let m be a complete noncompact riemannian manifold of dimension n we derive a hamilton s gradient estimate for positive solutions of the fast diffusion equations u t u m 1 4 n 8 m 1 on m 0 under the ricci flow

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

the mesa problem for neumann boundary value problem abstract in this paper we study the singular limit of the porous medium equation u t u m g x u as m in a bounded domain with neumann boundary condition

convergence of the dirichlet solutions of the very fast diffusion equation abstract for any 1 m 0 positive functions f g and u 0 0 we prove that under some mild conditions on f g and u 0 as r the solution u r of the dirichlet problem u t u m m x x in r r 0 u r t f t m r 1 m u r t g t m r 1 m for all t 0 u x 0 u 0 x in r r converges uniformly on every compact subset of r 0 t to the solution of the equation u t u m m x x in r 0 t u x 0 u 0 x in r which satisfies some mass loss formula on 0 t where t is the maximal time such that the solution u is positive we also prove that the solution constructed is equal to the solution constructed in hui 2007 15 using approximation by solutions of the corresponding neumann problem in bounded cylindrical domains

persistent neural states stationary localized activity patterns in nonlinear continuous n population q dimensional neural networks neural continuum networks are an important aspect of the modeling of macroscopic parts of the cortex two classes of such networks are considered voltage and activity based in both cases our networks contain an arbitrary number n of interacting neuron populations spatial nonsymmetric connectivity functions represent cortico cortical local connections and external inputs represent nonlocal connections sigmoidal nonlinearities model the relationship between average membrane potential and activity departing from most of the previous work in this area we do not assume the nonlinearity to be singular that is represented by the discontinuous heaviside function another important difference from previous work is that we relax the assumption that the domain of definition where we study these networks is infinite that is equal to or we explicitly consider the biologically more relevant case of a bounded subset of a better model of a piece of cortex the time behavior of these networks is described by systems of integro differential equations using methods of functional analysis we study the existence and uniqueness of a stationary i e time independent solution of these equations in the case of a stationary input these solutions can be seen as persistent they are also sometimes called bumps we show that under very mild assumptions on the connectivity functions and because we do not use the heaviside function for the nonlinearities such solutions always exist we also give sufficient conditions on the connectivity functions for the solution to be absolutely stable that is independent of the initial state of the network we then study the sensitivity of the solutions to variations of such parameters as the connectivity functions the sigmoids the external inputs and last but not least the shape of the domain of existence of the neural continuum networks these theoretical results are illustrated and corroborated by a large number of numerical experiments in most of the cases 2 n 3 2 q 3

hamilton s gradient estimates for a nonlinear partial differential equation under the yamabe flow abstract in this paper we derive two hamilton s gradient estimates for positive solutions of the nonlinear partial differential equation t u t u p h u q under the yamabe flow

stability of bumps in piecewise smooth neural fields with nonlinear adaptation abstract we study the linear stability of stationary bumps in piecewise smooth neural fields with local negative feedback in the form of synaptic depression or spike frequency adaptation the continuum dynamics is described in terms of a nonlocal integrodifferential equation in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a heaviside function of local activity discontinuities in the adaptation variable associated with a bump solution means that bump stability cannot be analyzed by constructing the evans function for a network with a sigmoidal gain function and then taking the high gain limit in the case of synaptic depression we show that linear stability can be formulated in terms of solutions to a system of pseudo linear equations we thus establish that sufficiently strong synaptic depression can destabilize a bump that is stable in the absence of depression these instabilities are dominated by shift perturbations that evolve into traveling pulses in the case of spike frequency adaptation we show that for a wide class of perturbations the activity and adaptation variables decouple in the linear regime thus allowing us to explicitly determine stability in terms of the spectrum of a smooth linear operator we find that bumps are always unstable with respect to this class of perturbations and destabilization of a bump can result in either a traveling pulse or a spatially localized breather

two dimensional bumps in piecewise smooth neural fields with synaptic depression we analyze radially symmetric bumps in a two dimensional piecewise smooth neural field model with synaptic depression the continuum dynamics is described in terms of a nonlocal integrodifferential equation in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a heaviside function of local activity synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity we show that in the case of a mexican hat weight distribution sufficiently strong synaptic depression can destabilize a stationary bump solution that would be stable in the absence of depression numerically it is found that the resulting instability leads to the formation of a traveling spot the local stability of a bump is determined by solutions to a system of pseudolinear equations that take into account the sign of perturbations around the circular bump boundary

nonradial solvability structure of super diffusive nonlinear parabolic equations we study the solvability of the cauchy problem for the nonlinear parabolic equation u div u m 1 u at when m 0 then there is no local solution of the initial value problem on the other hand if such an interval does not exist then the initial value problem is locally solvable and the time of existence can be estimated explicitly

effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network we analyze the spatiotemporal dynamics of a system of integro differential equations that describes a one dimensional excitatory neuronal network with synaptic depression and spike frequency adaptation physiologically suggestive forms are used for both types of negative feedback we also consider the effects of employing two different types of firing rate function a heaviside step function and a piecewise linear function we first derive conditions for the existence of traveling fronts and pulses in the case of a heaviside step firing rate and show that adaptation plays a relatively minor role in determining the characteristics of traveling waves we then derive conditions for the existence and stability of stationary pulses or bumps and show that a purely excitatory network with synaptic depression cannot support stable bumps however bumps do not exist in the presence of adaptation finally in the case of a piecewise linear firing rate function we show numerically that the network also supports self sustained oscillations between an up state and a down state in which a spatially localized oscillating core periodically emits pulses at each cycle

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

bifurcations in nonsmooth dynamical systems a review is presented of the one parameter nonsmooth bifurcations that occur in a variety of continuous time piecewise smooth dynamical systems motivated by applications a pragmatic approach is taken to defining a discontinuity induced bifurcation dib as a nontrivial interaction of a limit set with respect to a codimension one discontinuity boundary in phase space only dibs that are local are considered that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows three classes of systems are considered involving either state jumps jumps in the vector field or jumps in some derivative of the vector field a rich array of dynamics are revealed involving the sudden creation or disappearance of attractors jumps to chaos bifurcation diagrams with sharp corners and cascades of period adding for each kind of bifurcation identified where possible a kind of normal form or discontinuity mapping dm is given together with a canonical example and an application the goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators impact oscillators dc dc converters and problems in control theory

singular limit of solutions of the very fast diffusion equation we prove that the distribution solutions of the very fast diffusion equation u t um m u t um m u 0u 0 in rn 0 rn 0 u x 0 u0 x u x 0 u0 x in rnrn where m n 2p n 2 and u0 x e x 2 u0 x e x 2 for any x r1 x r1 where e 0e 0 r1 0r1 0 m0 1 m0 n 2p 1 m0 n 2 we prove that the solution of the above problem will converge uniformly on every compact subset of rn 0 rn 0 to the maximal solution of the equation vt logvvt logv v x 0 u0 x v x 0 u0 x as m 0 m 0 for any smooth bounded domain rn rn m0 1 m0 max 1 n 2 p 1 m0 max 1 n 2 we prove the existence and uniqueness of solutions of the dirichlet problem u t um m u t um m u 0u 0 in 0 0 u u0u u0 in u gu g on 0 0 with either finite or infinite positive boundary value gg we also prove a similar convergence result for the solutions of the above dirichlet problem as m 0m 0

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

aronson benilan estimates for the fast diffusion equation under the ricci flow abstract we study the fast diffusion equation with a linear forcing term 0 1 u t div u p 1 u r u under the ricci flow on a complete manifold m such that m r 2 has bounded curvature and nonnegative isotropic curvature where 0 p 1 and r r x t is the evolving scalar curvature of m at time t we prove aronson benilan and li yau hamilton type differential harnack estimates for positive solutions of 0 1 in addition we use similar method to prove certain li yau hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in cao and hamilton 2009 cao 2008 and kuang and zhang 2008 to the noncompact setting

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

continuation of localised coherent structures in nonlocal neural field equations we study localised activity patterns in neural field equations posed on the euclidean plane such models are commonly used to describe the coarse grained activity of large ensembles of cortical neurons in a spatially continuous way we employ matrix free newton krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation this opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent pde formulation we present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking the regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function the results presented form a basis for the general study of localised cortical activity with inputs and more specifically for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input

from fast to very fast diffusion in the nonlinear heat equation we study the asymptotic behavior of the sign changing solution of the equation u t u u f when the diffusion becomes very fast i e as 1 we prove that a solution u t converges in l 1 uniformly for t in subsets with compact support in 0 t to a solution of u t u 1 u f in contrast with the case of 0 the limiting solutions are either positive or negative or identically equal to 0 in all this causes the limit to be singular in the sense that a boundary layer appears at t 0 when one lets 1