electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate we consider a two dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate the magnetic permeability is assumed to be a fixed positive constant in the media the material properties of the media are characterized completely by an index of refraction which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium in this paper we only examine the tm transverse magnetic polarization case a radiation condition is introduced and equivalence with a second kind lippmann schwinger type integral equation is shown with additional assumptions on the index of refraction in the layer uniqueness of solution is proved existence of solution is then established by employing a form of fredholm alternative using a general result on the solvability of integral equations on unbounded domains published earlier by chandler wilde and zhang an approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that if the index of refraction is appropriately chosen the scattered field can grow with distance from the plate

scattering by stripe grating when bismuth is substituted at certain lattice points of rare earth garnet crystals a material is formed which exhibits an almost ideal magnetic stripe domain structure the structure consists of a one dimensional periodic array of regions of nearly constants magnetization so that the magnitudes of the magnetization vectors in two adjacent regions are the same but their directions are opposite this condition produces the alternating sign magnetic permeability tensors shown in section 1 7 3 when a beam of light is passed through such a film it splits into many diffracted beams all sharing a plane defined by the periodicity direction i e the y component of all the wave vectors is the same see figure 1 1 the angles of the diffracted beam directions relative to the incident beam direction are determined by the periodicity length

a posteriori error analysis for the conical diffraction problem abstract maxwell s equations for conical diffraction can be reduced to a system of two helmholtz equations in r 2 coupled via quasi periodic transmission conditions on a set of piecewise smooth interfaces a finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators an a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place as our work in wang et al 2015 a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo differential mapping does not converge to the original pseudo differential mapping in its operator norm the a posteriori error estimate consists of two parts finite element discretization error and the truncation error of the nonlocal boundary operators in particular the truncation error is exponentially decaying with respect to the truncation parameter

an adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures we develop a finite element adaptive strategy with error control for the wave scattering by periodic structures the unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer pml technique which attenuates both the outgoing and evanescent waves in the pml region pml parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

finite gratings at high frequencies in this paper we construct geometric optics solutions to an acoustic scattering problem by a finite periodic dielectric grating placed on the interface between semi infinite dielectric media we rigorously justify that the first order approximation is the solution of the infinite periodic grating the method is extended to maxwell s equations

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

global existence and nonexistence for nonlinear wave equations with damping and source terms we consider an initial boundary value problem for a nonlinear wave equation in one space dimension the nonlinearity features the damping term u m 1 u t and a source term of the form u p 1 u with m p 1 we show that whenever m p then local weak solutions are global on the other hand we prove that whenever p m and the initial energy is negative then local weak solutions cannot be global regardless of the size of the initial data

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

conservation laws and symmetries of radial generalized nonlinear p laplacian evolution equations abstract a class of generalized nonlinear p laplacian evolution equations is studied these equations model radial diffusion reaction processes in n 1 dimensions where the diffusivity depends on the gradient of the flow for this class all local conservation laws of low order and all lie symmetries are derived the physical meaning of the conservation laws is discussed and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases the symmetries are used to derive exact group invariant solutions from solvable three dimensional subgroups of the full symmetry group which yields a direct reduction of the nonlinear equation to a quadrature the physical and analytical properties of these exact solutions are explored some of which describe moving interfaces and green s functions

blow up phenomena for a doubly degenerate equation with positive initial energy abstract in this paper an initial boundary value problem related to the equation u t div u m p 2 u m f u is studied under suitable conditions on f we establish a blow up result for certain solution with positive initial energy and blow up time will be also considered by using the differential inequality technique the upper estimate of the blow up solution near the single blow up point is also obtained

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

acoustic scattering by mildly rough unbounded surfaces in three dimensions for a nonlocally perturbed half space we consider the scattering of time harmonic acoustic waves a second kind boundary integral equation formulation is proposed for the sound soft case based on a standard ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function due to the unboundedness of the surface the integral operators are noncompact in contrast to the two dimensional case the integral operators are also strongly singular due to the slow decay at infinity of the fundamental solution of the three dimensional helmholtz equation in the case when the surface is sufficiently smooth lyapunov we show that the integral operators are nevertheless bounded as operators on l 2 gamma and on l 2 gamma cap bc gamma and that the operators depend continuously in norm on the wave number and on gamma we further show that for emph mild roughness i e a surface gamma which d

boundary control problem and optimality conditions for the cahn hilliard equation with dynamic boundary conditions this paper is concerned with a boundary control problem for the cahn hilliard equation coupled with dynamic boundary conditions in order to handle the control problem we restrict our analysis to

universal bounds for global solutions of a forced porous medium equation we show that in a smooth bounded domain rn n 2 all global nonnegative solutions of ut um up with zero boundary data are uniformly bounded in by a constant depending on p and but not on u0 provided that 1 m p n 1 n 1 m furthermore we prove an a priori bound in l 0 depending on u0 l under the optimal condition 1 m p n 2 n 2 m

on analyticity of linear waves scattered by a layered medium abstract the scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering in this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high order perturbation of surfaces methods for simulating such waves more specifically we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions this result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of operator expansions field expansions and transformed field expansions

unique determination of periodic polyhedral structures by scattered electromagnetic fields this work is concerned with the unique determination of a periodic diffraction grating profile in three dimensions by some scattered electromagnetic fields measured above the grating in general it is well known that global uniqueness may not be true when the measurement is only taken for one incident field our goal is to completely characterize the global uniqueness properties when the periodic structure is of polyhedral type corresponding to each incident plane wave we are able to classify all unidentifiable structures into three classes and show that any periodic polyhedral structure can be uniquely determined by one incident field if and only if it belongs to none of the three classes consequently the minimum number of incident waves required for the unique determination of a periodic polyhedral structure can be easily read

a well posed integral equation formulation for three dimensional rough surface scattering we consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft rough surface recently a brakhage werner type integral equation formulation of this problem has been proposed based on an ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function moreover it has been shown in the three dimensional case that thi si ntegral equation is uniquely solvable in the space l 2 g when the scattering surface g does not differ too much from a plane in this paper we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope moreover we construct explicit bounds on the inverse of the associated boundary integral operator as a function of the wave number the parameter coupling the single and double layer potentials and the maximum surface slope these bounds show that the norm of the inverse operator is bounded uniformly in the wave number k f orko0 if the coupling parameter h is chosen proportional to the wave number in the case when g is a plane we show that the choice hzk 2 is nearly optimal in terms of minimizing the condition number

optimal boundary control of a nonstandard viscous cahn hilliard system with dynamic boundary condition abstract in this paper we study an optimal boundary control problem for a model for phase separation that was introduced by podio guidugli 2006 the model consists of a strongly coupled system of nonlinear parabolic differential equations in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically in contrast to the existing control literature about this pde system we consider here a dynamic boundary condition involving the laplace beltrami operator for the order parameter of the system which models an additional nonconserving phase transition occurring on the surface of the domain we show the frechet differentiability of the associated control to state operator in appropriate banach spaces and derive results on the existence of optimal controls and on first order necessary optimality conditions in terms of a variational inequality and the adjoint state system

linear and nonlinear heat equations in l q_ delta spaces and universal bounds for global solutions we develop a theory of both linear and nonlinear heat equations in the weighted lebesgue spaces l q delta where delta is the distance to the boundary in particular we prove an optimal l q delta l r delta estimate for the heat semigroup and we establish sharp results on local existence uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces this theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems namely under certain assumptions we prove that any global solution is uniformly bounded for t geq tau 0 by a universal constant independent of the initial data in all previous results the bounds for global solutions were depending on the initial data

morrey spaces and classification of global solutions for a supercritical semilinear heat equation in rn abstract we prove the boundedness of global classical solutions for the semilinear heat equation u t u u p 1 u in the whole space r n with n 3 and supercritical power p n 2 n 2 this is proved without any radial symmetry or sign assumptions unlike in all the previously known results for the cauchy problem and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter examples moreover we show that any global classical solution has to decay in time faster than t 1 p 1 which is also optimal and in contrast with the subcritical case the proof relies on nontrivial modifications of techniques developed by chou du and zheng 8 and by blatt and struwe 3 5 for the case of convex bounded domains they are based on weighted energy estimates of giga kohn type combined with an analysis of the equation in a suitable morrey space this method actually works for any convex bounded or unbounded smooth domain but at the same time captures some of the specific behaviors associated with the case of the whole space r n as a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies and we show that the so called borderline global weak solutions blow up in finite time and then become classical again and decay as t all these results confirm the key role played by the morrey space m 2 4 p 1 in the understanding of the structure of the set of global solutions for p p s

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

global unbounded solutions of the fujita equation in the intermediate range we consider the fujita equation u t vardelta u u p on mathbb r n with n ge 3 we prove the existence of global unbounded solutions for p s p p jl where p s n 2 n 2 and begin aligned p jl left begin array l quad l frac n 2 2 4n 8 sqrt n 1 n 2 n 10 text if n 10 infty text if n le 10 end array right end aligned previously it was known that global unbounded solutions exist for p ge p jl whereas for p p s there are no such solutions

an adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures an edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures the unbounded computational domain is truncated to a bounded one by a perfectly matched layer pml technique the pml parameters such as the thickness of the layer and the medium properties are determined through sharp a posteriori error estimates numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method

on the boundedness of global solutions of abstract semilinear parabolic equations abstract in this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the formdu dt au t f u t in a banach space whereais the infinitesimal generator of an analytic semigroup andfis a nonlinear function such that the initial value problem possesses nonglobal solutions for some initial data we give sufficient conditions to ensure that global solutions on 0 are necessarily bounded in a suitable norm we give several examples of problems to which the abstract result applies

optimal distributed control of the cahn hilliard brinkman system with regular potential abstract the main objective of this paper is to study the optimal distributed control of the three dimensional cahn hilliard brinkman system with regular potential we apply recently proved well posedness and regularity results in order to establish the existence of optimal controls as well as the first order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

weak solutions and blow up for wave equations of p laplacian type with supercritical sources this paper investigates a quasilinear wave equation with kelvin voigt damping utt pu ut f u in a bounded domain 3 and subject to dirichlet boundary conditions the operator p 2 p 3 denotes the classical p laplacian the nonlinear term f u is a source feedback that is allowed to have a supercritical exponent in the sense that the associated nemytskii operator is not locally lipschitz from w01 p into l2 under suitable assumptions on the parameters we prove existence of local weak solutions which can be extended globally provided the damping term dominates the source in an appropriate sense moreover a blow up result is proved for solutions with negative initial total energy

single point blowup for nonlocal parabolic problems abstract the formation of shear bands in materials has important implications to a variety of physical processes these bands are observed in very thin zones and are generally regarded as a precursor to material failure shear band formation is caused by the heat generated in regions with highest strain rate with insufficient time for diffusion of this heat a localized thermal softening of the material occurs which enhances plastic flow in a thin zone this adiabatic strain localization can be modelled as nonlinear thermally activated reaction diffusion equations this leads to a class of nonlocal parabolic problems and their associated time independent steady state counterparts a discussion of the models and an analysis of the problems will be presented

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

boundedness of global positive solutions of a porous medium equation with a moving localized source abstract in this paper a porous medium equation with a moving localized source u t u r u a f u x 0 t t is considered it is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data moreover in one space dimension case it is also shown that all global positive solutions of the above equation are uniformly bounded and this differs from that of a porous medium equation with a local source

optimal control of a semidiscrete cahn hilliard navier stokes system in this paper the optimal boundary control of a time discrete cahn hilliard navier stokes system is studied a general class of free energy potentials is considered which in particular includes the double obstacle potential the latter homogeneous free energy density yields an optimal control problem for a family of coupled systems which result from a time discretization of a variational inequality of fourth order and the navier stokes equation the existence of an optimal solution to the time discrete control problem as well as an approximate version is established the latter approximation is obtained by mollifying the moreau yosida approximation of the double obstacle potential first order optimality conditions for the mollified problems are given and in addition to the convergence of optimal controls of the mollified problems to an optimal control of the original problem first order optimality conditions for the original problem are derived through a limit process the newly derived statio

on the laplace equation with non linear dynamical boundary conditions the main part of the paper deals with local existence and global existence versus blow up for solutions of the laplace equation in bounded domains with a non linear dynamical boundary condition more precisely we study the problem consisting in 1 the laplace equation in 0 infty times omega 2 a homogeneous dirichlet condition 0 infty times gamma 0 3 the dynamical boundary condition frac partial u partial nu u t m 2 u t u p 2 u on 0 infty times gamma 1 4 the initial condition u 0 x u 0 x on partial omega here omega is a regular and bounded domain in mathbb r n with n ge 1 and gamma 0 and gamma 1 endow a measurable partition of partial omega moreover m 1 2 le p r where r 2 n 1 n 2 when n ge 3 r infty when n 1 2 and u 0 in h 1 2 partial omega u 0 0 on gamma 0 the final part of the paper deals with a refinement of a global non existence result by levine park and serrin which is applied to the previous problem

boundedness of global solutions of nonlinear diffusion equations it is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t cc and other choices for which the solutions blow up in finite time if we are interested in other types of behavior we may consider the possibility p u t u exists globally

exponential growth of solution of a strongly nonlinear generalized boussinesq equation an initial boundary value problem for strongly nonlinear generalized boussinesq equation is studied we show the exponential growth of solution with l p norm for negative or positive initial energy by constructing differential inequalities

acoustic scattering by an inhomogeneous layer on a rigid plate the problem of scattering of time harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in r 2 is considered the density is assumed to be unity in the media within the layer the sound speed is assumed to be an arbitrary bounded measurable function the problem is modelled by the reduced wave equation with variable wavenumber in the layer and a neumann condition on the plate to formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced a generalization of the rayleigh expansion condition for diffraction gratings with the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate under additional assumptions on the wavenumber in the layer uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established general results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data

an adaptive finite element method for the diffraction grating problem with transparent boundary condition the diffraction grating problem is modeled by a boundary value problem governed by a helmholtz equation with transparent boundary conditions an a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place to overcome the difficulty caused by the fact that the truncated dirichlet to neumann dtn mapping does not converge to the original dtn mapping in its operator norm a duality argument without assuming more regularity than the weak solution is applied the a posteriori error estimate consists of two parts the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter based on the a posteriori error control a finite element adaptive strategy is established for the diffraction grating problem such that the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization

second harmonic generation in nonlinear optical films second harmonic generation an important phenomenon in nonlinear optics is modeled in this work the model is derived from a nonlinear system of maxwell s equations which overcomes the known shortcomings of some commonly used models in the literature existence and uniqueness of solutions are established by a combination of a variational approach and the contraction mapping principle some numerical results are also presented

uniform decay of solutions for a quasilinear system of viscoelastic equations abstract in this paper we consider a system of two coupled quasilinear viscoelastic equations in canonical form with dirichlet boundary condition we use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay with exponential and polynomial rates of the solutions energy which extends some existing results for a single equation to the case of a coupled system

boundedness of global solutions of a supercritical parabolic equation we study radially symmetric classical solutions of the dirichlet problem for a heat equation with a supercritical nonlinear source we give a sufficient condition under which blow up in infinite time cannot occur this condition involves only the growth rate of the source term at infinity we do not need the homogeneity property which played a key role in previous proofs of similar results we also establish the blow up rate for a class of solutions which blow up in finite time

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

an inverse problem for scattering by a doubly periodic structure consider scattering of electromagnetic waves by a doubly periodic structure s x3 f xl x2 with f xl nial x2 n2a2 f xl x2 for integers nl n2 above the structure the medium is assumed to be homogeneous with a constant dielectric coefficient the medium is a perfect conductor below the structure an inverse problem arises and may be described as follows for a given incident plane wave the tangential electric field is measured away from the structure say at x3 b for some large b to what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor in this paper results on uniqueness and stability are established for the inverse problem a crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain q l au in q v u 0 in q n xu 0 on q

nonlinear damped wave equation in this paper we consider the following nonlinear wave equation with variable exponents uttu aut ut m 2 bu u p 2 where a b are positive constants by using the faedogalerkin method the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p we also prove the finite time blow up of solutions and give a two dimension numerical example to illustrate the blow up result

on the heat flow for the two dimensional gelfand equation abstract in this paper we study the nonnegative solutions of semilinear heat equation u t u e u in 0 t where is a bounded domain in r 2 and show that they are bounded whenever t and the energy functional along a solution tends to as t t whenever the maximal existence time t is finite

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

on nonlinear wave equations with degenerate damping and source terms in this article we focus on the global well posedness of the differential equation u tt u u k j u t u p 1 u in x 0 t where j is a sub differential of a continuous convex function j under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions uniqueness nonexistence and propagation of regularity under nominal assumptions on the parameters we establish the existence of global generalized solutions with further restrictions on the parameters we prove the existence and uniqueness of a global weak solution in addition we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k m and the initial energy is negative we also address the issue of propagation of regularity specifically under some restriction on the parameters we prove that solutions that correspond to any regular initial data such that u 0 h 2 h 1 0 u 1 h 1 0 are indeed strong solutions

infinite time blow up of solutions to a nonlinear parabolic problem abstract we investigate the blow up of solutions of nonuniformly parabolic equations it will be shown that the gradient of the solution tends to infinity as time goes to infinity even though thel norm of the solution may remain uniformly bounded this phenomenon is in contrast with a result for semilinear heat equations where it follows from regularity theory that the solution converges to an equilibrium if it is uniformly bounded in thel norm

blow up for the porous media equation with source term and positive initial energy abstract we study the cauchy dirichlet problem for the porous media equation with nonlinear source term in a bounded subset of r n the problem describes the propagation of thermal perturbations in a medium with a nonlinear heat conduction coefficient and a heat source depending on the temperature the aim of the paper is to extend the unstable set to a part of the positive energy region a phenomenon which was known only for linear conduction

asymptotic limits and optimal control for the cahn hilliard system with convection and dynamic boundary conditions abstract in this paper we study initial boundary value problems for the cahn hilliard system with convection and nonconvex potential where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential while recent works addressed the case of viscous cahn hilliard systems the pure nonviscous case is investigated here in its first part the paper deals with the asymptotic behavior of the solutions as time approaches infinity it is shown that the limit of any trajectory can be characterized in terms of stationary solutions provided the initial data are sufficiently smooth the second part of the paper deals with the optimal control of the system by the fluid velocity results concerning existence and first order necessary optimality conditions are proved here we have to restrict ourselves to the case of everywhere defined smooth potentials in both parts of the paper we start from corresponding known results for the viscous case derive sufficiently strong estimates that are uniform with respect to the positive viscosity parameter and then let the viscosity tend to zero to establish the sought results for the nonviscous case

global nonexistence for a semilinear petrovsky equation abstract in this paper we consider a semilinear petrovsky equation with damping and source terms it is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition moreover for the linear damping case we show that the solution blows up in finite time even for vanishing initial energy this is an important breakthrough since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature

boundedness of global solutions of a porous medium equation with a localized source abstract in this paper a localized porous medium equation u t u r u af u x 0 t is considered it is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded and this differs from that of a porous medium equation with a local source

artificial boundary conditions for axisymmetric eddy current probe problems we study different strategies for the truncation of computational domains in the simulation of eddy current probes of elongated axisymmetric tubes for axial fictitious boundaries an exact dirichlet to neumann map is proposed and mathematically analyzed via a non selfadjoint spectral problem under general assumptions we show convergence of the solution to an eddy current problem involving a truncated dirichlet to neumann map to the solution on the entire unbounded axisymmetric domain as the truncation parameter tends to infinity under stronger assumptions on the physical parameters of the eddy current problem convergence rates are shown we further validate our theoretical results through numerical experiments for a realistic physical setting inspired by eddy current probes of nuclear reactor core tubes

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

heat flow for dirichlet to neumann operator with critical growth abstract in this article we study the heat flow equation for dirichlet to neumann operator with critical growth by assuming that the initial value is lower energy we obtain the existence blowup and regularity on the other hand a concentration phenomenon of the solution when the time goes to infinity is proved

the time harmonic maxwell equations in a doubly periodic structure abstract consider the diffraction of a beam of particles in r 3 when the dielectric coefficient is a constant e1 above a surface s and a different constant e2 below s and the magnetic permeability is constant throughout r 3 s is assumed to be a doubly periodic surface say z f x y with f x ml1 y nl2 f x y for integers m n the existence and uniqueness of a solution satisfying a radiation condition at infinity is reduced to a system of fredholm equations thus for all but a discrete set of e s there exists a unique solution

optimal boundary control of a simplified ericksen leslie system for nematic liquid crystal flows in 2d in this paper we investigate an optimal boundary control problem for a two dimensional simplified ericksen leslie system modelling the incompressible nematic liquid crystal flows the hydrodynamic system consists of the navier stokes equations for the fluid velocity coupled with a convective ginzburg landau type equation for the averaged molecular orientation the fluid velocity is assumed to satisfy a no slip boundary condition while the molecular orientation is subject to a time dependent dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals we first establish the existence of optimal boundary controls then we show that the control to state operator is frechet differentiable between appropriate banach spaces and derive first order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

boundedness of global solutions for a nonlinear degenerate parabolic porous medium system with localized sources abstract this paper deals with nonlinear degenerate parabolic porous medium system with localized sources it is shown that under certain conditions solutions of the equation blow up in finite time for large a and b or large initial data while there exist global positive solutions for small a and b or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded these extend a recent work of chen et al in y chen q liu h gao boundedness of global solutions of a porous medium equation with a localized source nonlinear anal 64 2006 2168 2182 which considered the single equation with localized source

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

numerical analysis of diffraction by periodic structures tm polarization the diffraction of a time harmonic wave in tm polarization by some periodic inhomogeneous material is studied in this paper the diffraction problem is modeled by a generalized helmholtz equation with transparent boundary conditions existence and uniqueness are proved for the model problem uniform error estimates for the finite element approximation of the solution are established error estimates are also obtained when the truncation of the nonlocal boundary operators takes place

stabilized low order finite elements for stokes equations with damping abstract in this paper we consider the stationary stokes equations with damping low order mixed finite element spaces are used to approximate the velocity and the pressure and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf sup condition the stability of this method is proved and the optimal order error estimates are derived by some nonlinear analysis techniques at last two numerical examples are implemented to test the stability and effectiveness of the proposed method

the reaction diffusion equation with lewis function and critical sobolev exponent abstract in this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of reaction diffusion equation with lewis function and critical sobolev exponent

optimal boundary control of a viscous cahn hilliard system with dynamic boundary condition and double obstacle potentials in this paper we investigate optimal boundary control problems for cahn hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the laplace beltrami operator the cost functional is of standard tracking type and box constraints for the controls are prescribed we prove existence of optimal controls and derive first order necessary conditions of optimality the general strategy which follows the lines of the recent approach by colli farshbaf shaker and sprekels appl math optim 71 2015 pp 1 24 to the simpler allen cahn case is the following we use the results that were recently established by colli gilardi and sprekels appl math optim online first doi 10 1007 s00245 015 9299 z 2015 for the case of differentiable logarithmic potentials and perform a so called deep quench limit using compactness and monotonicity arguments it is shown that this strategy leads to the desired first order necessary optimality conditions for t

finite element approximation of time harmonic waves in periodic structures consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material it is shown that the scattering diffraction problem may be modeled by a helmholtz equation with transparent boundary conditions the diffraction problem may be solved by a finite element method in this paper optimal error estimates for the finite element method are established the error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1