renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

refined asymptotic profiles for a semilinear heat equation we study the large time behavior of the solutions of the cauchy problem for a semilinear heat equation partial t u delta u f x t u quad rm in bf r n times 0 infty quad u x 0 varphi x quad rm in bf r n quad quad rm p where f in c bf r n times 0 infty times bf r and varphi in l 1 bf r n 1 x k dx with k 0 assume that u is a solution of p satisfying f x t u x t le c 1 t a u x t quad x t in bf r n times 0 infty for some constants c 0 and a 1 then it is well known that the solution u behaves like the heat kernel in this paper we give the k 2 th order asymptotic expansion of the solution u and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term f here k is the integer satisfying k 1 k k

convection diffusion equation with absorption and non decaying initial data in this paper we give the precise description of the large time behavior of the solution u of the cauchy problem tu u a u u in rn 0 u x 0 x 0in rn by using the ordinary differential equation and a linear parabolic equation here n 1 a rn 1 1 0 and is a bounded continuous function such that lp rn for some 1 p furthermore we study the large time behavior of the hot spots for the solution u

integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities abstract we consider the class of polynomial differential equations x x y p n x y y x y q n x y where p n and q n are homogeneous polynomials of degree n these systems have a focus at the origin if 0 and have either a center or a focus if 0 inside this class we identify a new subclass of darbouxian integrable systems having either a focus or a center at the origin additionally under generic conditions such darbouxian integrable systems can have at most one limit cycle and when it exists is algebraic for the case n 2 and 3 we present new classes of darbouxian integrable systems having a focus

monodromy and stability of a class of degenerate planar critical points abstract necessary and also sufficient monodromy conditions for a large class of degenerate singular points of planar differential systems are given for these systems we also find a computable expression of the principal term of the asymptotic expansion of the return map which gives the stability of the point

integrability degenerate centers and limit cycles for a class of polynomial differential systems we consider the class of polynomial differential equations email protected p n x y p n 1 x y p n 2 x y email protected q n x y q n 1 x y q n 2 x y for n 1 and where p i and q i are homogeneous polynomials of degree i these systems have a linearly zero singular point at the origin if n 2 inside this class we identify a new subclass of darboux integrable systems and some of them having a degenerate center i e a center with linear part identically zero moreover under additional conditions such darboux integrable systems can have at most one limit cycle we provide the explicit expression of this limit cycle

deciding context unification in first order term unification variables represent well formed terms over a given signature and we are to solve equations built using function symbols from the signature and such variables this problem is well known to be decidable in linear time in second order term unification the variables take arguments i e other terms and a substitution uses those arguments an arbitrary number of times for instance an equation f x c x c e x f c c has a solution x e b where b is a special symbol denoting the place in which the argument is substituted under this substitution both sides evaluate to f c c there are other solutions for instance x e f b b which evaluates both sides tof f c c f c c in general a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct second order unification is in general undecidable context unification is a natural problem in between first and second order unification we deal with equations over terms the variables take arguments but we restrict the set of solutions the argument is used exactly once formally contexts are terms with exactly one occurrence of the special symbol b and in context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation for instance when the aforementioned equation f x c x c e x f c c is treated as a context unification problem then it has exactly one solution x e b other substitutions that are solutions of it as an instance of the second unification problem say x e f b b are not valid as b is used more than once context unification also generalizes satisfiability of word equations which is decidable in pspace the decidability status of context unification remained unknown for almost two decades in this article we show that context unification is in pspace in exptime when tree regular constraints are also allowed those results are obtained by extending the recently developed recompression technique which was previously defined for strings and used to obtain a new pspace algorithm for satisfiability of word equations in this article the technique is generalized to trees and the corresponding algorithm is generalized from word equations to context unification the idea of recompression is to apply simple compression rules replacing pairs of neighboring function symbols to the solution of the context equation to this end we appropriately modify the equation without the knowledge of the actual solution so compressing the solution can be simulated by compressing parts of the equation it is shown that if the compression operations are appropriately chosen then the size of the instance is polynomial during the whole algorithm thus giving a pspace upper bound

riccati ermakov systems and explicit solutions for variable coefficient reaction diffusion equations abstract we present several families of nonlinear reaction diffusion equations with variable coefficients including generalizations of fisher kpp and burgers type equations special exact solutions such as traveling wave rational triangular wave and n wave type solutions are shown by means of similarity transformations the variable coefficients are conditioned to satisfy riccati or ermakov systems of equations when the riccati system is used conditions are established so that finite time singularities might occur we explore solution dynamics across multi parameters in the supplementary material we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions

radial symmetry of self similar solutions for semilinear heat equations abstract the symmetry properties of positive solutions of the equation u 1 2 x u 1 p 1 u u p 0 in r n where n 2 p n 2 n was studied it was proved that u must be radially symmetric about the origin provided u x o x 2 p 1 as x and that there exist non radial solutions u satisfying lim sup x x 2 p 1 u x 0

on the existence of speed independent circuits abstract a circuit is called speed independent if its nontransient behavior does not depend on the size of the delays in the different components of the circuit we show that the class of speed independent circuits is relatively small in fact very many useful circuits cannot be realized in a speed independent design for example we show that there does not exist any speed independent mod k counter for any k 1 the results are derived using a very general model of a network which is applicable to both gate circuits and more modern mos switch level circuits furthermore the results are also robust with respect to different delay assumptions and definitions of speed independence

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

subcritical nonlinear heat equation abstract we study asymptotic behavior in time of small solutions to nonlinear heat equations in subcritical case we find a new family of self similar solutions which change a sign we show that solutions are stable in the neighborhood of these self similar solutions

asymptotics for nonlinear heat equations abstract we study the cauchy problem for the nonlinear heat equation u t u u 1 0 x r n t 0 u 0 x u 0 x x r n in the sub critical case of 0 2 n in the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper escobedo and kavian 1987 5 u t x t 1 w 0 x t o t 1 as t uniformly with respect to x r n where w 0 is a positive solution of equation w 2 w w 1 1 w which decays rapidly at infinity lim 2 w 0 0

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

subcritical kuramoto sivashinsky type equation on a half line abstract in this paper we are interested in the global existence and large time behavior of solutions to the initial boundary value problem for subcritical kuramoto sivashinsky type equation 0 1 u t n u u x u xx u xxxx 0 x t r r u x 0 u 0 x x r x j 1 u 0 t 0 for j 1 2 where the nonlinear term n u u x depends on the unknown function u and its derivative u x and satisfy the estimate n u v c u v with 0 1 such that 3 2 2 0 the aim of this paper is to prove the global existence of solutions to the initial boundary value problem 0 1 in subcritical case when the nonlinear term has a time decay rate less than that of the linear terms of eq 0 1 also we find the main term of the asymptotic representation of solutions

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

planar polynomial vector fields having first integrals and algebraic limit cycles with the help of abel differential equations we obtain a new class of darboux integrable planar polynomial differential systems which have degenerate infinity moreover such integrable systems may have algebraic limit cycles also we present the explicit expressions of these algebraic limit cycles for quintic systems

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

the undecidability of the second order unification problem abstract it is shown that there is no effective procedure for determining whether or not two terms of the language of second order logic have a common instance

large time behavior for the fast diffusion equation with critical absorption we study the large time behavior of nonnegative solutions to the cauchy problem for a fast diffusion equation with critical zero order absorption partial t u delta u m u q 0 quad quad hbox in 0 infty times real n with m c n 2 n 1

decidability of bounded second order unification it is well known that first order unification is decidable whereas second order and higher order unification is undecidable bounded second order unification is second order unification under the restriction that only a bounded number of bound variables in the instantiating terms for second order variables is permitted however the size of the instantiation is not restricted in this paper a decision algorithm for bounded second order unification is described this is the first nontrivial decidability result for second order unification where the signature is not restricted and there are no restrictions on the occurrences of variables this supports the claim that bounded second order unification is easier than context unification whose decidability is currently an open question

a nonlinear parabolic problem in an exterior domain abstract we consider the problem formula herep 1 n 2 0 is a finite union of disjoint open sets anduo x is a continuous nonnegative and bounded function such that u o x a x as x 0 4 for somea 0 and 0 in this paper we discuss the asymptotic behaviour of solutions to 0 1 0 4 in terms of the various values of the parametersp a n and a common pattern that emerges from our analysis is the existence of an external zone whereu x t uo x and one or several internal regions where the influence of the set as well as that of diffusion and absorption is most strongly felt we present a complete classification of the stabilization profiles in terms of these parameters

the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems abstract in this work we study the centers of planar analytic vector fields which are limit of linear type centers it is proved that all the nilpotent centers are limit of linear type centers and consequently the poincare liapunov method to find linear type centers can be also used to find the nilpotent centers moreover we show that the degenerate centers which are limit of linear type centers are also detectable with the poincare liapunov method

the complexity of monadic second order unification monadic second order unification is second order unification where all function constants occurring in the equations are unary here we prove that the problem of deciding whether a set of monadic equations has a unifier is np complete where we use the technique of compressing solutions using singleton context free grammars we prove that monadic second order matching is also np complete

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles