classification of systems of nonlinear ordinary differential equations with superposition principles a system of n first order nonlinear ordinary differential equations x t f x t is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants such a system can be associated with the nonlinear action of a lie group g on a space m we show that indecomposable systems of ode s with supersposition principles are obtained if and only if the lie algebras l0 l corresponding to the isotropy group h of a point and the group g respectively define a transitive primitive filtered lie algebra l l0 using known results from the theory of transitive primitive lie algebras we deduce that l0 must be a maximal subalgebra of l and that g must be an affine group a simple lie group or the direct product of two identical simple lie groups affine groups lead to linear equations the other types to nonlinear equations with polynomial or rational nonlinearities equations corresponding to the classical complex lie algebra

the quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux in this paper we are concerned with the quenching behavior of a nonlinear parabolic problem in one dimensional space of which the nonlinearity appears both in source term and in neumann boundary condition first we proved that the solution quenches only on the left boundary in finite time if the given initial datum is monotone then we give the estimate for the quenching rate of the solution near the quenching time

incomplete quenching of heat equations with coupled singular logarithms boundary fluxes abstract this paper deals with quenching phenomena for a heat equations with coupled singular logarithms boundary fluxes under appropriate hypotheses the non simultaneous quenching of the solution for the system is proved and the estimates of quenching rates are given then we give a natural continuation of the solution u v after the quenching time when the equations occurs non simultaneous quenching moreover we identify the heat equations verified by the continuation beyond quenching time i e the equations occurs incomplete quenching

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

the boundary quenching behavior of a semilinear parabolic equation abstract in this paper we consider the boundary quenching behavior of a semilinear parabolic problem in one dimensional space of which the nonlinearity appears both in the source term and in the neumann boundary condition first we proved that the solution quenches at somewhere in some finite time then we assert that the quenching can only occur on the left boundary if the given initial datum is monotone finally we derived the upper and lower bounds for the quenching rate of the solution near the quenching time thus we generalized our former results

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

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

semidiscretization in time of nonlinear parabolic equations with blowup of the solution in this paper the author considers the first boundary value problem for the nonlinear equation u t delta u m alpha u m in omega a smooth bounded domain in mathbb r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be larger than one and alpha positive a scheme for the discretization in time of that problem is proposed it is proven that if the exact solution blows up in a finite time it is the same for the numerical solution estimates of the blow up time are obtained the stability of the method and the convergence for a class of initial conditions is proved

nonlinear equations with superposition formulas and the exceptional group g2 ii classification of the equations nonlinear equations with superposition formulas are obtained corresponding to the action of the complex and real forms of the exceptional lie group g2 on the homogeneous spaces g2 h the isotropy group of the origin h is taken as one of the maximal parabolic subgroups of g2 or as one of the maximal reductive subgroups leaving some vector space v c7 or v r7 invariant the parabolic subgroups as well as the simple subgroups sl 3 c su 3 sl 3 r or su 2 1 lead to equations with quadratic or quartic nonlinearities the semisimple subgroups sl 2 c sl 2 c su 2 su 2 and su 1 1 su 1 1 lead to equations with quadratic nonlinearities and additional nonlinear constraints on the independent variables

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

lie families theory and applications we analyze the families of non autonomous systems of first order ordinary differential equations admitting a common time dependent superposition rule i e a time dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants we next study the relations of these families called lie families with the theory of lie and quasi lie systems and apply our theory to provide common time dependent superposition rules for certain lie families

the blow up rate of solutions of semilinear heat equations on caracterise le comportement asymptotique de la solution de l equation de la chaleur semilineaire u t u exp u dans o t u k sur o t u x 0 x k pour xe k 0 dans un voisinage d un point d explosion quand t approche du temps d explosion fini t

self similar solutions for a quenching problem with spatially dependent nonlinearity abstract this paper is devoted to the study of the self similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro electro mechanical system we first provide a result on the non existence of slow orbit for a certain range of parameters next we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument then we give a detailed analysis of the behavior of global solutions at the origin finally as an application of the above results we prove a uniqueness theorem

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

quenching for a parabolic equation with variable coefficient modeling mems technology abstract this paper deals with a parabolic equation with variable coefficient modeling mems technology and we will prove the solution of the problem quenches in finite time in the case without stationary solutions numerical simulations are presented to illustrate the theoretical results

self similar subsolutions and blow up for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the form u t u f u u we give finite time blow up results for large initial data blow up time estimates are also provided these results rely on a new method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results on reaction diffusion equations on generalized burgers equations and on the equation of chipot and weissler this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

nonlinear equations with superposition formulas and exceptional group g2 iii the superposition formulas superposition formulas are derived expressing the general solution of several different systems of nonlinear ordinary differential equations in terms of a fundamental set of particular solutions the equations as well as the superposition formulas are induced by the action of the exceptional lie group g2 complex or real on a homogeneous space g2 g where g g2 is a maximal subgroup of g2 when g is either parabolic or simple three particular solutions are needed when g is sl 2 c sl 2 c or one of its real forms then two particular solutions suffice

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

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

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

blow up in a system of partial differential equations with conserved first integral part ii problems with convection a reaction diffusion convection equation with a nonlocal term is studied the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves the equations are parameterised by and for 1 the equation arises as a similarity solution of the navier stokes equations and the nonlocal term plays the role of pressure for 0 the equation is a nonlocal reaction diffusion problem the aim of the paper is to determine for which values of the parameter blow up occurs and to study its form in particular interest is focused on the three cases 1 2 and 1 it is observed that for any 0 1 2 nonuniform global blow up occurs if 1 2 1 then the blow up is global and uniform while for 1 the navier stokes equations there are exact solutions with initial data of arbitrarily large l l 2 and h 1 norms that decay to zero furthermore one of these exact solutions is proved to be nonlinearly stable in l 2 for arbitrarily large supremum norm an understanding of this transition from blow up behaviour to decay behaviour is achieved by a combination of analysis asymptotics and numerical techniques

nonlinear equations with superposition formulas and the exceptional group g2 i complex and real forms of g2 and their maximal subalgebras in order to study nonlinear ordinary differential equations with superposition principles related to the exceptional simple lie group g2 the complex and real forms of its lie algebra are examined and their maximal subalgebras are summarized in particular the parabolic subalgebras of the noncompact real form gnc2 r are determined explicit matrix realizations of the fundamental representation d 1 0 are used and studied in connection with invariant subspaces in a seven dimensional complex or real vector space the results are collected in three tables of specific interest for the study of nonlinear differential equations which will be developed in paper ii of this series

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

necessary and sufficient conditions for complete blow up and extinction for one dimensional quasilinear heat equations we characterize the occurrence of complete or incomplete blow up and extinction for a general quasilinear heat equation of the form he text u t phi u xx pm f u text in r times 0 t in terms of the constitutive functions and f we assume that u 0 for u 0 and that f u 0 for the positive sign before f u in he with f u superlinear as u blow up occurs in finite time supxu x t as t t t i e after the singularity occurs if such continuation exists we say that the blow up extinction is incomplete otherwise it is called complete our characterization is based on the qualitative behaviour of the family of travelling wave solutions and a proper use of the intersection comparison argument the analysis applies to other nonlinear models like the equations with gradient dependent diffusivity

the matrix representations of g2 i representations in an so 4 basis irreducible representations of the real compact lie algebra g2 are given in g2 so 4 bases all missing labels are accounted for by the explicit construction of a g2 so 4 basis of vector holomorphic functions the general problem of missing internal labels is also briefly discussed

uniqueness of the solution of a semilinear boundary value problem on considere le probleme aux valeurs limites suivant 2 u f x u 0 x u x 0 x sur pour e 0 il y a un e 0 tel que pour 0 e e ce probleme ait une solution positive unique u dans c 2 c

on quenching for some parabolic problems abstract the universal quenching of a porous medium equation is established we also obtain some universal quenching properties for some semilinear parabolic problems and dichotomy properties for some general one dimensional problems

dynamics for a complex valued heat equation with an inverse nonlinearity abstract we study the cauchy problem for a parabolic system which is derived from a complex valued heat equation with an inverse nonlinearity first we provide some criteria for the global existence of solutions then we consider the case when the initial data are asymptotically constants and obtain that depending on the asymptotic limits the solution quenches at space infinity or exists globally in time

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

quenching rate for a nonlocal problem arising in the micro electro mechanical system abstract in this paper we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro electro mechanical system this question is equivalent to the stabilization of the solution to the transformed problem in self similar variables first some a priori estimates are provided in order to construct a lyapunov function due to the lack of time monotonicity property we then derive some very useful and challenging estimates by a delicate analysis finally with this lyapunov function we prove that the quenching rate is self similar which is the same as the problem without the nonlocal term except the constant limit depends on the solution itself

an adaptive numerical method to handle blow up in a parabolic system we study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval coupled at the boundary in a nonlinear way in certain cases the system develops a blow up singularity in finite time fixed mesh methods are not well suited to approximate the problem near the singularity as an alternative to reproduce the behaviour of the continuous solution we present an adaptive in space procedure the scheme recovers the conditions for blow up and non simultaneous blow up it also gives the correct non simultaneous blow up rate and set moreover the numerical simultaneous blow up rates coincide with the continuous ones in the cases when the latter are known finally we present numerical experiments that illustrate the behaviour of the adaptive method

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

the dirichlet problem for δφ e φ in an infinite sector application to plasma equilibria abstract we consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane we prove the existence of a minimal solution to this equation and study its properties we infer from this analysis some asymptotics for the stationary solution of an equation arising in plasma physics

incomplete quenching in a system of heat equations coupled at the boundary in this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition this system exhibits simultaneous and non simultaneous quenching for non simultaneous quenching our continuation is a solution of a parabolic problem with neumann boundary conditions we also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

on a hyperbolic equation arising in electrostatic mems abstract we consider the initial boundary value problem of a damped wave equation with singular nonlinearity which describes an electrostatic micro electro mechanical system mems device the results of the pull in voltage being the critical threshold for global existence and quenching are obtained if the applied voltage then the equation admits a unique global small solution that exponentially converges to the minimal steady state while large solution may quench in finite time if then any solution quenches in finite time finally in the sense of the viscosity dominated limit the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated also the related error estimates in arbitrary order are derived

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

quenching behavior of solutions to heat equations with coupled boundary singularities abstract this paper deals with finite time quenching for one dimensional heat equations coupled via negative singular boundary sources in the more general product form of two component powers u p 1 v q 1 and u p 2 v q 2 differently from the generally considered there are two additional singular factors namely u p 1 and v q 2 for the boundary sources of u and v respectively we at first propose a criterion to distinguish simultaneous and non simultaneous quenching and then determine all possible blow up rates in the model three for simultaneous quenching and two for non simultaneous quenching under different nominations this shows real contributions of the two additional singular factors to the quenching behavior of solutions

on the critical exponent for reaction diffusion equations in this paper we study the initial boundary value problem for u t u h t u p with homogeneous dirichlet boundary conditions where h t t q for large t let s sup s positive solutions w of u t u such that mathop lim sup limits t to infty t s parallel w cdot t parallel infty infty then for p 1 q 1 s we show if p p there are global positive solutions that decay to zero uniformly for t if 1 p p then all nontrivial solutions blow up in finite time we determine p for some conical domains in r 2 and r 3 a similar result is derived for a bounded domain if h t e t for large t

thermal avalanche for blowup solutions of semilinear heat equations we consider the semilinear heat equation ut u up both in n and in a bounded domain with homogeneous dirichlet boundary conditions with 1 t we consider a natural approximation by reaction problems with global solution and pass to the limit as is well known the limit solution undergoes complete blowup after it blows up at t t the continuation is identically infinite for all t t we perform here a deeper analysis of how complete blowup occurs actually the singularity set of a solution that blows up as t t propagates instantaneously at time t t to cover the whole space producing a catastrophic discontinuity between t t and t t this is called the avalanche we describe its formation as a layer appearing in the limit of the natural approximate problems after a suitable scaling the initial structure of the layer is given by the solution of a limit problem described by a simple ordinary differential equation as t proceeds past t the solutions of the approximate problems have a traveling wave behavior with a speed that we compute the situation in the inner region depends on the type of approximation a conical pattern is formed with time when we approximate the power up by a flat truncation at level n while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern 2003 wiley periodicals inc

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

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

numerical quenching for the semilinear heat equation with a singular absorption in this paper we study the numerical approximation for the heat equation with a singular absorption we prove that the numerical quenching rate coincides with the continuous one we also see that the quenching time and the quenching set converge to the continuous one in fact under some restriction on the initial data the numerical quenching coincides with the continuous one finally we give some numerical results to illustrate our analysis

asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains two approaches abstract in this paper we study the asymptotic behavior as x 1 of solutions of semilinear elliptic equations in quarter or half spaces for which the value at x 1 0 is given we prove the uniqueness and characterize the one dimensional or constant profile of the solutions at infinity to do so we use two different approaches the first one is a pure pde approach and it is based on the maximum principle the sliding method and some new liouville type results for elliptic equations in the half space or in the whole space r n the second one is based on the theory of dynamical systems

symmetry and uniqueness of nonnegative solutions of some problems in the halfspace abstract we derive some 1 d symmetry and uniqueness or non existence results for nonnegative solutions of div a x u u g x in r n u 0 on r n in low dimension under suitable assumptions on a and g our method is based upon a combination of fourier series and liouville theorems

semi discretization in time of a fast diffusion equation abstract in this paper we consider the first boundary value problem for the nonlinear concentration dependent diffusion equation u t um t 0 in a smooth bounded domain in r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be between 0 and 1 then the solution decays to zero in some finite time t depending upon the initial data here we propose a scheme for the discretization in time of that problem we prove that the numerical solution is null after a finite number of time steps and we obtain the convergence of the method and estimates of the numerical extinction time

quenching for a reaction diffusion system with logarithmic singularity abstract in this paper we study the quenching phenomenon for a reaction diffusion system with singular logarithmic source terms and positive dirichlet boundary conditions some sufficient conditions for quenching of the solutions in finite time are obtained and the blow up of time derivatives at the quenching point is verified furthermore under appropriate hypotheses the non simultaneous quenching of the system is proved and the estimates of quenching rate is given

simple subgroups of simple lie groups and nonlinear differential equations with superposition principles nonlinear ordinary differential equations admitting a superposition principle based on the action of the group sl n c on the homogeneous spaces sl n c o n c and sl n c sp n c are derived the superposition formulas are presented explicitly in the o n c case the general solution is expressed in terms of three particular solutions for any n for the sp n c case three solutions are needed for n 2k 8 four solutions for n 6 and five solutions for n 4

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies