new approximate formulations for em scattering by dielectric objects the extended born approximation exborn has been shown an efficient formulation in the electromagnetic em scattering by dielectric objects in both free space and air earth half space problems in most cases exborn is much more accurate than the conventional born approximation at low frequencies when the frequency is high or the contrast of dielectric objects is large however the exborn approximation becomes inaccurate in this paper new approximations are proposed for the em scattering by dielectric objects buried in a lossy earth which are also suitable for the case of free space it has been shown that the zeroth order form of new approximations is completely equivalent to exborn hence high order approximations can be regarded as high order exborn closed form formulations are derived for the new approximations using the fast fourier transform fft these formulations can be implemented efficiently at a cost of cnlogn where n is the number of unknowns and c is a small number numerical simulations show that high order exborn approximations are much more accurate than the exborn approximation

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

three species monomer monomer model a mean field analysis and monte carlo study we study the phase diagram and critical behavior of a one dimensional three species monomer monomer catalytic surface reaction model static monte carlo simulations are used to roughly map out the phase diagram consisting of a reactive steady state bordered by three equivalent unreactive phases where the surface is saturated with one monomer species the transitions from the reactive phase are all continuous while the transitions between poisoned phases are first order of particular interest are the bicritical points where the reactive phase simultaneously meets two poisoned phases a mean field cluster analysis fails to predict all of the qualitative features of the phase diagram unless correlations up to triplets of adjacent sites are included scaling properties of the continuous transitions and the bicritical points are studied using dynamic monte carlo simulations the transition from the reactive to a saturated phase shows directed percolation critical behavior while the universal behavior at the bicritical point is in the even branching annihilating random walk class the crossover from bicritical to critical behavior is also studied

3d magnetotelluric modeling by using finite difference method comparison study of different forward modeling approaches we have investigated different approaches to solve the magnetotelluric forward modeling problems beside classical direct electromagnetic formulation ungauged and gauged lorenz coulomb and axial vector and scalar potential formulations are discretized using finite difference numerical solution technique linear matrix equations obtained from finite difference discretization for each electromagnetic field formulations are solved by using the conjugate gradient iterative solution method we compared finite different solutions of each approaches with respect to accuracy and speed we show that all approaches generate accurate results however ungauged approach gave solutions with less conjugate gradient iteration and accordingly less computer time all stiffness matrices arising from each formulation are examined in terms of their sizes and type finite difference solution of the coulomb gauge and ungauged approaches generates bigger size stiffness matrices than the other approaches direct electromagne

travelling wave solutions for a balance law we study entropy travelling wave solutions for first order hyperbolic balance laws results concerning existence regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros

path integral formulation of stochastic processes for exclusive particle systems we present a systematic formalism to derive a path integral formulation for hard core particle systems far from equilibrium writing the master equation for a stochastic process of the system in terms of the annihilation and creation operators with mixed commutation relations we find the kramers moyal coefficients for the corresponding fokker planck equation fpe and the stochastic differential equation sde is derived by connecting these coefficients in the fpe to those in the sde finally the sde is mapped onto field theory using the path integral giving the field theoretic action which may be analyzed by the renormalization group method we apply this formalism to a two species reaction diffusion system with drift finding a universal decay exponent for the long time behavior of the average concentration of particles in arbitrary dimension

asymptotic form of the approach to equilibrium in reversible recombination reactions the reversible reactions and are investigated from the exact langevin equations describing our model we set up a systematic approximation scheme to compute the approach of the density of c particles to its equilibrium value we show that for a sufficiently long time t this approach takes the form of a power law for any dimension d the amplitude a is also computed exactly but is expected to be model dependent for uncorrelated initial conditions the c density turns out to be a monotonic time function the cases of correlated initial conditions and unequal diffusion constants are investigated as well in the former correlations may break the monotonicity of the density or in some special cases they may change the long time behaviour for the latter the power law remains valid only the amplitude changes even in the extreme case of immobile c particles we also consider the case of segregated initial condition for which a reaction front is observed and confirm that its width is governed by mean field exponent in any dimension

inversion of 3d electromagnetic data in frequency and time domain using an inexact all at once approach we present a general formulation for inverting frequencyor time domain electromagnetic data using an all at once approach in this methodology the forward modeling equations are incorporated as constraints and thus we need to solve a constrained optimization problem where the parameters are the electromagnetic fields the conductivity model and a set of lagrange multipliers this leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought nevertheless experience shows that the constrained problem can be solved faster than the unconstrained one the primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly in this paper we outline the all atonce approach and apply it to electromagnetic problems in both frequency and time domains this is facilitated by a unified representation for forward modeling for these two types of data the optimization problem is solved by finding a stationary point of the lagrangian numerically this leads to a nonlinear system that is solved iteratively using a gauss newton strategy at each iteration a large indefinite matrix is inverted and we discuss how this can be accomplished as a test we invert frequency domain synthetic data from a grounded electrode system that emulates a field csamt survey for the time domain we invert borehole data obtained from a current loop on the surface

field theory for reaction diffusion processes with hard core particles we show how to build up a systematic bosonic field theory for a general reaction diffusion process involving hard core particles in arbitrary dimension we discuss a recent approach proposed by park kim and park phys rev e 62 7642 2000 as a test bench for our method we show how to recover the equivalence between asymmetric diffusion of excluding particles and the noisy burgers equation

nonequilibrium lattice models series analysis of steady states a perturbation theory for steady states of interacting particle systems is developed and applied to several lattice models with nonequilibrium critical points near an absorbing state the expansion is expressed directly in terms of the kinetic parameter creation rate rather than in powers of the interaction an algorithm for generating series expansions for local properties is described order parameter series 16 terms and precise estimates of critical properties are presented for the one dimensional contact process and several related models

universality classification of restricted solid on solid type surface growth models we consider the restricted solid on solid rsos type surface growth models and classify them into dynamic universality classes according to their symmetry and conservation law four groups of rsos type microscopic models asymmetric a asymmetric conserved ac symmetric s and symmetric conserved sc groups are introduced and the corresponding stochastic differential equations sdes are derived analyzing these sdes using dynamic renormalization group theory we confirm the previous results that a rsos ac rsos and s rsos groups belong to the kardar parisi zhang class the villain lai das sarma class and the edwards wilkinson class respectively we also find that sc rsos group belongs to a new universality class featuring the conserved cubic nonlinearity

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

conservation laws with a degenerate source traveling waves large time behavior and zero relaxation limit abstract the large time behavior and zero reaction time limits of solutions of scalar hyperbolic conservation laws with bistable source terms whose zeroes are degenerate are studied it is proved that as t solutions with periodic boundary condition tend to stable and unstable zeroes of the source terms an example is given which shows that there exists initial data for which the solution converges pointwise to the unstable zero of the source term patterns of traveling waves of the systems and the conditions under which they exist are provided it turns out that the structure of traveling waves and the long time behavior are different from the case where the zeroes are non degenerate the recent results of fan jin and teng concerning the zero reaction time limit is generalized to the case of source terms with degenerate zeroes the zero reaction time limits are quite different from the long time limit

quasi analytical approximations and series in electromagnetic modeling the quasi linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background normal field through an electrical reflectivity tensor in the original formulation of the quasi linear approximation was determined by solving a minimization problem based on an integral equation for the scattering currents this approach is much less time consuming than the full integral equation method however it still requires solution of the corresponding system of linear equations in this paper we present a new approach to the approximate solution of the integral equation using through construction of quasi analytical expressions for the anomalous electromagnetic field for 3 d and 2 d models quasi analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures in the last sections of this paper

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

advances in three dimensional magnetotelluric modeling using integral equations recent progress in integral equation modeling of three dimensional magnetotelluric responses includes the ability to simulate 3 d structures which outcrop excluding topography which transect layer interfaces and which extend indefinitely in one or more directions the most important factor in achieving this capability is an accurate treatment of the electric surface charge in particular a previous integro difference formulation for evaluating charges has been abandoned in favor of true surface integrations over the source cells with potential differencing across the field cells in the 3 d body the new procedure constitutes a good approximation to galerkin s method while preserving internal consistency in terms of pulse basis functions to verify outcropping structures juxtaposed conductive and resistive prisms at the surface are simulated and compared to 2 d results an elongate version of the 3 d model shows good agreement with both transverse electric and transverse magnetic modes of the 2 d res

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

critical behavior of a one dimensional fixed energy stochastic sandpile we study a one dimensional fixed energy version that is with no input or loss of particles of manna s stochastic sandpile model the system has a continuous transition to an absorbing state at a critical value of the particle density and exhibits the hallmarks of an absorbing state phase transition including finite size scaling critical exponents are obtained from extensive simulations which treat stationary and transient properties and an associated interface representation these exponents characterize the universality class of an absorbing state phase transition with a static conserved density in one dimension they differ from those expected at a linear interface depinning transition in a medium with point disorder and from those of directed percolation

a finite element based domain decomposition approach for plane wave 3d electromagnetic modeling we developed a novel parallel domain decomposition approach for 3d large scale electromagnetic induction modeling in the earth we used the edge based finite element method and unstructured meshes unstructured meshes were divided into sets of nonoverlapping subdomains we used the curl curl electric field equation to carry out the analysis in each subdomain the electric field was discretized by first order vector shape functions along the edges of tetrahedral elements the tangential components of the magnetic field on the interfaces of the subdomains were defined as a set of lagrange multipliers the unknown lagrange multipliers were solved from an interface problem defined on the interfaces of the subdomains with the availability of the lagrange multipliers the electric field values in each subdomain were solved independently three synthetic examples were evaluated to verify our code excellent agreement with previously published solutions was obtained synthetic examples revealed that our domain decomposition technique is scalable with respect to the number of subdomains and robust with regard to frequency and the heterogeneous distribution of material parameters i e electric conductivity electric permittivity and magnetic permeability

instability of discontinuous traveling waves for hyperbolic balance laws abstract it is known that scalar hyperbolic conservation laws with source term and periodic initial value have a property of poincare bendixson type namely the solutions converge either to a constant state or to a periodic traveling wave which is necessarily discontinuous in this paper we show that generically with respect to the l 1 topology the solutions exhibit a behaviour of the former type we also show that while the rate of convergence to a constant state is exponential the convergence to a traveling wave can be arbitrarily slow

path integral representation for a stochastic sandpile we introduce an operator description for a stochastic sandpile model with a conserved particle density and develop a path integral representation for its evolution the resulting exact expression for the effective action highlights certain interesting features of the model for example it is nominally massless and the dynamics is via cooperative diffusion using the path integral formalism we construct a diagrammatic perturbation theory yielding a series expansion for the activity density in powers of time

electromagnetic modeling of 3 d structures by the method of system iteration using integral equations a new approach for electromagnetic modeling of three dimensional 3 d earth conductivity structures using integral equations is introduced a conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion the influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively this is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method this method reduces computer memory requirements since only one submatrix at a time needs to be stored the diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure the block iterative solution requires much less computation time than direct matrix inversion or conventional point

cluster mean field approximations with the coherent anomaly method analysis for the driven pair contact process with diffusion the cluster mean field approximations are performed up to 13 cluster sizes to study the critical behavior of the driven pair contact process with diffusion dpcpd and its precedent the pcpd in one dimension critical points are estimated by extrapolating our data to the infinite cluster size limit which are in good accordance with recent simulation results within the cluster mean field approximation scheme the pcpd and the dpcpd share the same mean field critical behavior the application of the coherent anomaly method however shows that the two models develop different coherent anomalies which lead to different true critical scaling the values of the critical exponents for the particle density the pair density the correlation length and the relaxation time are fairly well estimated for the dpcpd these results support and complement our recent simulation results for the dpcpd

hyperbolic conservation laws with stiff reaction terms of monostable type in this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type tu 9 f u u l u 6 is studied solutions of cauchy problems of the above equation with initial value 0 0 is found to originate from the behavior of traveling waves of the above system with viscosity regularization

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

on the dynamics of random sequential absorption the dynamics of random sequential absorption on d dimensional lattices is represented by a simple graphical expansion results for the entire filling process and in particular its saturation limit are easily obtained it is shown that the large d limit of the saturation density is ln 2d 2d the expression gives an accurate approximation for d or 3

effect of static disorder and reactant segregation on the a b 0 reaction we derive the long time behavior of the a b to emptyset reaction in two dimensions finding a universal exponent and prefactor in the absence of disorder sufficiently singular disorder leads to a sub diffusion limited reaction and a continuously variable decay exponent pattern matching between the reactant segregation and the disorder is not strong enough to affect the long time decay

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

derivation of continuum stochastic equations for discrete growth models we present a formalism to derive the stochastic differential equations sdes for several solid on solid growth models our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion we then write the master equations for these corresponding particle systems and find the sdes for the particle densities finally by connecting the particle densities with the growth heights we derive the sdes for the height variables applying this formalism to discrete growth models we find the edwards wilkinson equation for the symmetric body centered solid on solid bcsos model the kardar parisi zhang equation for the asymmetric bcsos model and the generalized restricted solid on solid rsos model and the villain lai das sarma equation for the conserved rsos model in addition to the consistent forms of equations for growth models we also obtain the coefficients associated with the sdes

fermionic field theory for directed percolation in 1 1 dimensions we formulate directed percolation in 1 1 dimensions in the language of a reaction diffusion process with exclusion taking place in one space dimension we map the master equation that describes the dynamics of the system onto a quantum spin chain problem from there we build an interacting fermionic field theory of a new type we study the resulting theory using renormalization group techniques this yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low dimensional directed percolation

fock space for fermion like lattices and the linear glauber model abstract the concept of fock space representation is developed to deal with stochastic spin lattices written in terms of fermion operators a density operator is introduced in order to follow in parallel the developments of the case of bosons in the literature some general conceptual quantities for spin lattices are then derived including the notion of generating function and path integral via grassmann variables the formalism is used to derive the liouvillian of the d dimensional linear glauber dynamics in the fock space representation then the time evolution equations for the magnetization and the two point correlation function are derived in terms of the number operator

equivalences between stochastic systems time dependent correlation functions of unstable particles undergoing biased or unbiased diffusion coagulation and annihilation are studied this is achieved by similarity transformations between different stochastic models and between stochastic and soluble nonstochastic models for special cases we obtain exact results which are in good agreement with experiments on one dimensional annihilation coagulation processes

front formation and motion in quasilinear parabolic equations abstract this paper deals with the singular limit for l u u t f u u x x 1 g u 0 where the function f is assumed to be smooth and uniformly elliptic and g is a bistable nonlinearity denoting with u m the unstable zero of g for any initial datum u 0 for which u 0 u m has a finite number of zeroes and u 0 u m changes sign crossing each of them we show the existence of solutions and describe the structure of the limiting function u 0 lim 0 u where u is the solution of a corresponding cauchy problem the analysis is based on the construction of travelling waves connecting the stable zeros of g and on the use of a comparison principle

diffusion limited reactions of hard core particles in one dimension we investigate three different methods to tackle the problem of diffusion limited reactions annihilation of hard core classical particles in one dimension we first extend an approach devised by lushnikov sov phys jetp 64 811 1986 and calculate for a single species the asymptotic long time and or large distance behavior of the two point correlation function based on a work by grynberg and stinchcombe phys rev e 50 957 1994 phys rev lett 74 1242 1995 76 851 1996 which was developed to treat stochastic adsorption desorption models we provide in a second step the exact two point one and two time correlation functions of lushnikov s model we then propose a formulation of the problem in terms of path integrals for pseudo fermions this formalism can be used to advantage in the multispecies case especially when applying perturbative renormalization group techniques

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

critical behaviour of an interacting surface reaction model the authors investigate the kinetics of a generalized monomer monomer surface reaction model in which there is a variable excluded volume interaction between same species particles as and bs for no interaction this system reduces to the monomer monomer model for which there is a first order transition between a and b saturated phases as a function of the excluded volume interaction the first order transition line for weak interaction terminates at a tricritical point where two second order transitions meet these transitions which appear to be in the reggeon field theory universality class separate the a saturated reactive and b saturated phases series expansions and various time dependent monte carlo simulations are used to estimate the critical parameters and exponents associated with these transitions

field theory of branching and annihilating random walks we develop a systematic analytic approach to the problem of branching and annihilating random walks equivalent to the diffusion limited reaction processes 2a and a m 1 a where m 1 starting from the master equation a field theoretic representation of the problem is derived and fluctuation effects are taken into account via diagrammatic and renormalization group methods for d 2 the mean field rate equation which predicts an active phase as soon as the branching process is switched on applies qualitatively for both even and odd m but the behavior in lower dimensions is shown to be quite different for these two cases for even m and d near 2 the active phase still appears immediately but with nontrivial crossover exponents which we compute in an expansion in 2 d and with logarithmic corrections in d 2 however there exists a second critical dimension d c 4 3 below which a nontrivial inactive phase emerges with asymptotic behavior characteristic of the pure annihilation process this is confirmed by an exact calculation in d 1 the subsequent transition to the active phase which represents a new nontrivial dynamic universality class is then investigated within a truncated loop expansion which appears to give a correct qualitative picture the model with m 2 is also generalized to n species of particles which provides yet another universality class and which is exactly solvable in the limit n for odd m we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2 in this case the transition to the active phase is in the directed percolation universality class finally we study the modification when the annihilation reaction is 3a when m 0 mod 3 the system is always in its active phase but with logarithmic crossover corrections for d 1 while the other cases should exhibit a directed percolation transition out of a fluctuation driven inactive phase

zero reaction limit for hyperbolic conservation laws with source terms abstract in this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term t u x f u 1 e u 1 u 2 for the cauchy problem to the above equation we prove that as e 0 its solution converges to piecewise constant 1 solution where the two constants are the two stable local equilibria the constants are separated by either shocks that travel with speed 1 2 f 1 f 1 as determined by the rankine hugoniot jump condition or a non shock discontinuity that moves with speed f 0 where 0 is the unstable equilibrium our analytic tool is the method of generalized characteristics similar results for more general source term 1 e g u having finitely many simple zeros and satisfying ug u

attractors in inhomogeneous conservation laws and parabolic regularizations the asymptotic behavior of inhomogeneous conservation laws is considered the attractor of the equation is characterized the relationship between attractors of the equation and that of its parabolic regularization is studied