fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

itd double edge diffraction for complex source beam illumination a new high frequency incremental theory of diffraction itd formulation for the double diffraction by metallic wedges when illuminated by complex source points csp is provided the main motivation is the extension of the class of problems that can be studied using asymptotic i e ray based and incremental methods by providing a double diffraction description for csp which are considered because they are efficient to analyze electrically large structures the new formulation provides an accurate asymptotic description of the interaction between two edges in an arbitrary configuration including slope diffraction contributions advantages of the itd formulation for csp illumination include avoiding the typical ray caustic impairments of the gtd utd ray techniques and not requiring ray tracing in complex space numerical results are presented and compared to a method of moments analysis to demonstrate the accuracy of the solution

short time behaviour of a singular solution to the heat equation with absorption on considere l evolution aux temps petits de l interface qui apparait dans le probleme aux limites u t u xx u p u o t 1 t 0 u x o 0 x 0 pour p 1 on construit une solution interieure d ordre zero consistante avec une interface mobile ou u est zero

mathematical analysis of a chemical reaction with lumped temperature and strong absorption abstract we consider a mathematical model for the evolution of a single reactant and of the temperature in an isothermal catalyst the temperature is assumed to be spatially homogeneous and the absorption term is supposed to be nonlipschitzian existence and uniqueness of a regular solution is proved together with some bounds the steady state problem is also investigated and information about the dead core of the reactant are given

large time behavior of the solution of the heat equation with nonlinear strong absorption abstract we give an estimate for the asymptotic behavior of the solution of a semilinear heat equation and of its free boundary

an efficient algorithm for the computation of the utd t transition function we present an efficient algorithm for the numerical calculation of the t canonical transition function which is encountered in the uniform geometrical theory of diffraction utd such a transition function allows the uniform ray field description of various high frequency diffraction mechanisms such as double wedge or vertex diffraction the proposed algorithm is valid for both real and imaginary arguments as required to deal with the general case in utd applications

stochastic theory of edge diffraction an alternative formulation an alternative formulation is presented for the derivation of the results in which allows to rigorously extend the validity range to any incidence observation aspect provided that the standard deviation of the edge roughness sigma is not large compared to the wavelength lambda while the observation distance r is r gt lambda this was heuristically conjectured by franceschetti ieee trans antennas propag vol 56 no 2 pp 437 449 2008 and supported by numerical examples but without a rigorous proof

a singular reaction diffusion system modelling prey predator interactions invasion and co extinction waves abstract we consider a singular reaction diffusion system arising in modelling prey predator interactions in a fragile environment since the underlying odes system exhibits a complex dynamics including possible finite time quenching one first provides a suitable notion of global travelling wave weak solution then our study focusses on the existence of travelling waves solutions for predator invasion in such environments we devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co extinction tail for both species under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere such a property arising for every admissible wave speeds

study of the blow up set by transformation abstract we obtain some new results on blow up for the heat equation with nonlinear sources by transformation into an absorption problem our work is related to that of kawohl and peletier who first developed such a transformation in some special cases here we provide a systematic method for constructing a suitable transformation by using the class of regularly varying functions through regularization we obtain a precise characterization of the blow up set as zeros of a certain steady state

a uniform double diffraction coefficient for a pair of wedges in arbitrary configuration an analytic closed form solution is presented for the double diffraction at a pair of arbitrarily placed wedges that is suitable to be used in a uniform theory of diffraction utd ray description framework here the particular assumption present in all the past literature that the two diffracting edges are coplanar is removed the doubly diffracted dd field at a pair of wedges is constructed via spectral synthesis such a procedure provides a double spectral integral representation of the dd field that is asymptotically evaluated by resorting to transition functions typical in double diffraction problems that can be expressed in terms of generalized fresnel integrals the final expression is arranged in the typical utd fashion and contains a uniform double diffraction dyadic coefficient the dd field behavior is analyzed in its transition regions to show how the dd field smoothly compensates for the total field discontinuities occurring at shadow boundaries where a singly diffracted field from a wedge is shadowed by the other wedge numerical examples are provided to verify such analysis and to test the accuracy of our utd ray description against method of moments full wave results

experimental and theoretical validation for the incremental theory of diffraction experimental and theoretical approaches to verify the validity of the incremental theory of diffraction itd are considered after providing a simple recipe for the application of the itd three geometries are examined for its validation first the itd formulation of the diffraction from a perfect electric conductor pec straight wedge is compared with the uniform theory of diffraction utd and with measurement results second the itd formulation of the diffraction from a pec disc is compared with measurement results and with the exact solution of a boundary value problem involving oblate spheroidal functions third the itd formulation of the diffraction from a hole in a pec plane is compared with the exact solution of a boundary value problem involving oblate spheroidal functions in particular this is the first time that itd results for diffraction from the disc and hole in a plane geometries are validated using exact solutions computed at a caustic in all cases examined very good agreement is found

double tip diffraction modeling 2 d numerical models versus high frequency asymptotics the subject of single and double diffraction phenomena has long been investigated by high frequency asymptotic techniques however integral or differential equation based numerical methods suffer from computational complexity due to electrically large geometries encountered in high frequency problems the main purpose of this paper is to present the finite element fem diffraction modeling of double tip structure and to compare its results with high frequency methods and other numerical models fem is made feasible for modeling of such an infinitely long structure by utilizing the locally conformal perfectly matched layer pml approach which enables the use of finite sized structure matlab codes are developed and various numerical examples are demonstrated in a comparative manner

a utd triple diffraction coefficient for straight wedges in arbitrary configuration a new utd solution is presented for the description of the ray triply diffracted by straight wedges in arbitrary configuration illuminated by a spherical wavefront field the proposed utd coefficient permits to uniformly describe the field at any aspect in particular including the case of overlapping transition regions where the subsequent application of utd single wedge diffraction coefficient fails the triple diffracted ray field is expressed in an analytical form providing physical insight into the triple diffraction mechanism and a new effective engineering tool within the utd framework as required in a modern ray based code

algorithm for the computation of the generalized fresnel integral the generalized fresnel integral appears as a canonical function in the uniform ray field representation of several high frequency diffraction mechanisms such as double wedge or vertex diffraction here we propose an algorithm for its calculation which is valid both for real and imaginary arguments as required for treating the general case in the uniform geometrical theory of diffraction framework

compact solitary waves in linearly elastic chains with non smooth on site potential it was recently observed by saccomandi and sgura that one dimensional chains with nonlinear elastic interaction and regular on site potential can support compact solitary waves i e travelling solitary waves with strictly compact support in this paper we show that the same applies to chains with linear elastic interaction and an on site potential which is continuous but non smooth at minima some different features arise in particular the speed of compact solitary waves is not uniquely fixed by the equation we also discuss several generalizations of our findings

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

uniqueness results for nonlocal hamilton jacobi equations we are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal non monotone velocity for these equations only the existence of global in time weak solutions is available in some particular cases in this paper we propose a new approach for proving uniqueness of the solution when the front is expanding this approach simpli fies and extends existing results for dislocation dynamics it also provides the first uniqueness result for a fitzhugh nagumo system the key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

moving boundaries in reaction diffusion systems with absorption consider the reaction diffusion system t u d 0 xx u ku 1 v 0 0 t u d 1 xx v ku 1 v with boundary and initial conditions u 0 t c 0 u t 0 t 0 x v 0 t 0 v t 0 u x 0 u 0 x v x 0 v 0 x where 0 u 0 x c 0 and 0 v 0 x 1 for x 0

a convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

characterization of the value function of final state constrained control problems with bv trajectories this paper aims to investigate a control problem governed by differential equations with radon measure as data and with final state constraints by using a known reparametrization method by dal maso and rampazzo 18 we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories involving time measurable hamiltonian we study the characterization of the value function of this auxiliary problem and discuss its numerical approximations

dislocation dynamics short time existence and uniqueness of the solution we study a mathematical model describing dislocation dynamics in crystals we consider a single dislocation line moving in its slip plane the normal velocity is given by the peach koehler force created by the dislocation line itself the mathematical model is an eikonal equation with a velocity which is a non local quantity depending on the whole shape of the dislocation line we study the special case where the dislocation line is assumed to be a graph or a closed loop in the framework of discontinuous viscosity solutions for hamilton jacobi equations we prove the existence and uniqueness of a solution for small time we also give physical explanations and a formal derivation of the mathematical model finally we present numerical results based on a level sets formulation of the problem these results illustrate in particular the fact that there is no general inclusion principle for this model

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

a general uniform double wedge diffraction coefficient the uniform theory of diffraction utd is extended to include double diffraction by an arbitrary configuration of two wedges a scalar double diffraction coefficient is defined which is uniform throughout space while reducing to the product of single diffraction coefficients outside overlapping transition regions the result is formulated in terms of double fresnel integrals which can be rapidly evaluated

an efficient ray tracing algorithm for multiple straight wedge diffraction we present an efficient algorithm for the tracing of multiply edge diffracted rays the algorithm assumes a given sequence of infinite edges and complete visibility among them the ray tracing problem is formulated as the minimization of the ray total path length since such a cost function is strictly convex except for coplanar edges in the plane wave far field regime then the problem admits a unique global minimum and allows the use of a modified newton search algorithm which exhibits a very high converging rate we also propose a convenient starting point to effectively initialize the minimization algorithm the proposed algorithm is tested by some numerical examples that show its efficiency and effectiveness

critical exponents and lower bounds of blow up rate for a reaction diffusion system abstract this paper is concerned with a reaction diffusion system with absorption terms under dirichlet boundary conditions modelling the cooperative interaction of two diffusion biological species by constructing blow up sub solutions and bounded super solutions we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions moreover for a special case we derive the lower bound estimates of blow up rate

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

the evolution of travelling waves in fractional order autocatalysis with decay ii the initial boundary value problem in this paper we examine an initial boundary value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination we consider the case where the reaction and termination orders m and n respectively are fractional that is 0 0 k being a parameter which measures the relative strength of the termination step to that of the autocatalysis we show that solutions approach the equilibrium state 1 k for k 1 we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t finally for m n we show that traveling waves with semi infinite support occ

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

approaching an extinction point in one dimensional semilinear heat equations with strong absorption abstract this paper deals with the cauchy problem ut uxx up 0 0 u x 0 u0 x 0 and u x t 0 appear when t is close to t we describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached we also give conditions under which some of these behaviours actually occur

generalization of utd double diffraction to the case of impenetrable wedges with relatively general boundary conditions in the framework of the uniform geometrical theory of diffraction utd a novel high frequency solution is provided which allows the description of the rays doubly diffracted by a pair of impenetrable straight wedges with relatively general boundary conditions under a spherical wave illumination such a utd solution is heuristically derived by imposing the continuity of the total field at the shadow boundaries of the first order i e geometrical optics utd wedge single diffracted ray field utd solution it is the extension to the case of double diffraction of the first order solutions proposed by luebbers and by burnside and burgener it allows a uniform description of the field at any observation point including those aspects where transition regions overlap and the cascading of utd single wedge diffraction coefficients fails the proposed solution does not consider surface wave effects and for the special case of two perfectly conducting wedges it reduces to the formulation present in the literature of which it can be considered a generalization indeed it requires the same special functions involved in utd double diffraction by two perfectly conducting wedges the double diffracted ray field is expressed in analytical closed form providing a deeper understanding of the physics involved in double diffraction and a new powerful tool to be employed in any modern ray based code

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

sublinear elliptic problems with a hardy potential abstract in this paper we study the positive solutions of sub linear elliptic equations with a hardy potential which is singular at the boundary by means of ode techniques a fairly complete picture of the class of radial solutions is given local solutions with a prescribed growth at the boundary are constructed by means of contraction operators some of those radial solutions are then used to construct ordered upper and lower solutions in general domains by standard iteration arguments the existence of positive solutions is proved an important tool is the hardy constant

existence of weak solutions for general nonlocal and nonlinear second order parabolic equations in this article we provide existence results for a general class of nonlocal and nonlinear second order parabolic equations the main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way among applications we present level set equations appearing in dislocation theory and in the study of fitzhugh nagumo systems

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

numerical diffraction of array guided surface waves on finite frequency selective surfaces in this paper an adaptation of hurst s method to take into account the presence of array guided surface waves in the calculation of the scattered field of finite frequency selective surfaces in free space is presented numerical approaches to determine the guided wave s wavenumber needed in this calculation namely prony s method and the discrete fourier transform dft are examined and compared the results show that the scattered fields obtained with the proposed method which is based on numerically extracted diffraction coefficients are in very good agreement with those obtained with the method of moments

convergence of approximation schemes for nonlocal front propagation equations we provide a convergence result for numerical schemes approximating nonlocal front propagation equations our schemes are based on a recently investigated notion of a weak solution for these equations we also give examples of such schemes for a dislocation dynamics equation and for a fitzhugh nagumo type system

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations