regular packing of congruent polygons on the rectangular sheet this paper deals with the problem of searching for the fragment of dense lattice dense double lattice packing of congruent polygons on the rectangular sheet which is optimal by the dense factor in doing so we consider only those packings for which the polygons form rows columns parallel to one of the sheet sides the problem arises in many branches of industry in particular in footwear industry the mathematical model of the problem is built simplification of the mathematical model reducing of the problem dimensionality and its decomposition on a number of independent subproblems is done on the basis of analysis of the properties of the mathematical model the approximate method to solve the problem is suggested the example of solving of the real problem is given

conditional lie backlund symmetry second order differential constraint and direct reduction of diffusion systems abstract the equivalence relation between conditional lie backlund symmetry higher order differential constraint and direct reduction is presented in the case of evolution systems the second order nonlinear conditional lie backlund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional lie backlund symmetry of evolution systems consequently the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems

solving irregular strip packing problems by hybridising simulated annealing and linear programming abstract in this paper a hybrid algorithm to solve irregular strip packing problems is presented the metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process these linear programming models which are used to locally optimise the layouts derive from the application of compaction and separation algorithms computational tests were run using instances that are commonly used as benchmarks in the literature the best results published so far have been improved by this new hybrid packing algorithm

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

on invariant subspaces for nonlinear finite difference operators we study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities in several cases we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction diffusion type with respect to their spatial discretisation the corresponding lower dimensional reductions of the finite difference solutions on the invariant subspaces are constructed

new exact solutions for a free boundary system a method for constructing explicit exact solutions to nonlinear evolution equations is further developed the method is based on consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation the method is then applied to a free boundary problem based on the process of precipitant assisted protein crystal growth

conditional lie backlund symmetries of hamilton jacobi equations abstract the conditional lie backlund symmetry method as a generalization of the conditional symmetry and lie backlund symmetry methods is developed to study hamilton jacobi equations it is shown that two hamilton jacobi equations admit conditional lie backlund symmetries of second order for certain smooth functions as a result a complete description of the structure of solutions to the resulting equations associated with the conditional lie backlund symmetries is performed

a method of optimal lattice packing of congruent oriented polygons in the plane abstract an approach based on an application of the function apparatus is proposed for constructing a mathematical model of the problem of optimal lattice packing of congruent oriented polygons in the plane the feasible solution region is contained in the set of pairs of points of the function 0 level surface of the polygon the region is partitioned into a finite number of subregions corresponding to pairs of linear segments of the function 0 level surface the frontiers of the subregions are generated by the conditions of non overlapping of polygons only four such conditions imply non overlapping of each pair of polygons in a lattice packing the solution can be attained only in a finite number of points on the frontiers of these subregions hence it is possible to use an organized sorting of such points with discarding of subregions whose objective function upper bound is lower than the record

mathematical model and method of searching for a local extremum for the non convex oriented polygons allocation problem abstract this work deals with the problem of optimal allocation of objects of a so called irregular form the objects are allocated on a strip of given width and with defects this problem is insufficiently studied but it is typical for many industries and is also interesting for developing the theory of solving cutting and packing problems an analytical model of the problem using only continuous variables is written in terms of classical mathematical programming and it is constructed on the basis of the original theory of functions and structures of linear inequalities the presented theory allows one to easily describe the conditions of mutual non overlapping of objects and their allocation in the stock region the exact method for searching a local minimum of the problem from any feasible initial point is based on the application of the active set strategy ideas a number of examples of solving practical problems are considered

maximal sign invariants of quasilinear parabolic equations with gradient diffusivity we consider quasilinear parabolic equations with gradient diffusivity ut div u u f u x rn t 0 where 1 is a fixed constant and f u is a given smooth function we also study quasilinear parabolic equations with a gradient dependent coefficient ut h u u f u with a smooth function h p for both classes of equations we derive first order sign invariants i e first order operators preserving their signs on the evolution orbits u t t 0 we give a complete description of maximal sign invariants of prescribed structures as a consequence we construct new exact solutions of some quasilinear equations

global asymptotical behavior and some new blow up conditions of solutions to a thin film equation abstract we consider a thin film equation by exploiting the boundary condition and the variational structure of the equation we consider the global existence and blow up of the solutions for the global solutions we study the asymptotic behavior and for the blow up solutions we study the upper and lower bounds of the blow up time especially we have derived the necessary and sufficient conditions for the solution blowing up in finite time when the initial energy j u 0 d where d is the mountain pass level for j u 0 d we also study the conditions for the solution exists globally and blows up in finite time and we also obtain some necessary and sufficient conditions for the solution blowing up in finite time when the initial energy at arbitrary level furthermore we study the decay behavior of both the global solutions and its corresponding energy functional and the decay ratios are given specially the results generalize the former studies on this equation such as the papers li et al 2016 16 dong and j zhou 2017 4 and sun et al 2018 23

uniqueness of solutions of degenerate diffusion equations with measures as initial conditions on considere le probleme de cauchy u t u m u p sur r n 0 t u x 0 x sur r n ou m 1 p 1 n 1 0 1 p m 2 n et une mesure finie

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity abstract the cauchy problem p u t u m u p in r n 0 t u x 0 u 0 x in r n is considered where 0 p m 1 and u 0 x 0 is a bounded continuous radially symmetric function in r n we choose p in some open interval p s p p with p s m n 2 n 2 such that a peaking solution incomplete blow up solution of p exists denote the solution of p by u we show that if u 0 x is nonincreasing in large r x and decays slowly u 0 x o x as x 2 p m then u is classified into one of the next three types according to the value as follows there exists 1 0 such that i u blows up completely in finite time if 1 ii u blows up incompletely in finite time and u t l r n o t 1 p 1 as t if 1 iii u does not blow up in finite time and u t l r n o t 1 p 1 as t if 0 1

conditional lie backlund symmetries and invariant subspaces to nonlinear diffusion equations with source abstract this paper considers conditional lie backlund symmetries of the radially symmetric nonlinear diffusion equations with source we obtain a complete list of canonical forms for such equations which admit higher order conditional symmetries as a consequence the solutions of the resulting equations are constructed on the invariant subspaces admitted by the corresponding equations

exact solutions to nonlinear equations with quadratic nonlinearity based on the concepts of the generalized conditional symmetry and of the invariant subspace a constructive way to derive nonlinear equations with quadratic nonlinearity that admit exact solutions as well as many new examples is presented this type of solution found by galaktionov describes the blowup phenomena examples of exact solutions that do not correspond to the invariant subspace are also presented and explained by the nonlinear generalized conditional symmetries

a finite difference scheme for some nonlinear diffusion equations in an absorbing medium support splitting phenomena in the porous media equation vt vm xx m 1 it is well known that there appears a finite propagation of the initial support and that if the initial support is connected so is mbox supp v t cdot for t 0 when the effect of the absorption is considered as the additional lower order term cvp c 0 p 0 the possibility that the support will split is expected rosenau and kamin phys d 8 1983 pp 273 283 tried the numerical computations and suggested the support splitting phenomena but the theoretical justification is not discussed in this paper such phenomena are investigated by use of finite difference schemes and the sufficient conditions under which the support begins to split are obtained in the specific case where m p 2 and 0 p 1

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

classification of coupled systems with two component nonlinear diffusion equations by the invariant subspace method the invariant subspace method is developed to perform classification of systems with two component nonlinear diffusion equations which was carried out with respect to the invariant subspaces defined by linear ordinary differential equations as a result the corresponding exact solutions generated by invariant subspaces to the resulting systems are obtained in most cases two components of these exact solutions belong to different scalar subspaces behaviour to several exact solutions of the systems is described

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are 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

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

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

blow up for a thin film equation with positive initial energy abstract in this paper we consider a thin film equation with nonlocal source which was studied by qu and zhou 2016 11 where the authors derived the conditions for global existence blow up and extinction we consider the case that the initial energy is positive and establish a blow up result for this case furthermore the upper bound of the blow up time is derived

lie symmetry analysis conservation laws and exact solutions of the time fractional generalized hirota satsuma coupled kdv system abstract in the present paper lie point symmetries of the time fractional generalized hirota satsuma coupled kdv hs ckdv system based on the riemann liouville derivative are obtained using the derived lie point symmetries we obtain similarity reductions and conservation laws of the considered system finally some analytic solutions are furnished by means of the invariant subspace method in the caputo sense

conditional lie backlund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source abstract the functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional lie backlund symmetry method the variant forms of the considered equations which admit the linear conditional lie backlund symmetries are identified a number of examples are considered and some exact solutions defined on the polynomial trigonometric and exponential invariant subspaces determined by the linear conditional lie backlund symmetries are constructed

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

invariant sets and solutions to the two dimensional nonlinear reaction diffusion equations abstract in this paper we utilize the method of invariant set to derive exact solutions of the two dimensional nonlinear reaction diffusion equations with source term it is shown that there exists a class of reaction diffusion equations which are invariant with respect to the sets e 1 u u x v x f t f u u y v y f t f u and e 2 u u x a x f t f u u y b y g t f u with f g as a result we obtain exact solutions of the certain nonlinear reaction diffusion equations these solutions extend the well known self similar solutions and instantaneous source type solutions of the porous medium equation the behavior to some solutions and the corresponding interfaces are also described

support properties of solutions to a degenerate equation with absorption and variable density abstract in this paper we study the properties of solutions to a degenerate parabolic equation with variable density and absorption we first obtain a critical exponent which distinguishes the localization of solutions from the positivity of them when positivity prevails we obtain the other critical exponent with respect to the decay of the variable density which separates the global existence of interfaces from the disappearance of them moreover the long time behavior of interfaces is characterized

separation of variables for scalar evolution equations in one space dimension the flow of an autonomous kth order evolution equation in one space dimension is generated by a kth order ordinary differential operator on the space of fields the characteristic fields are the solutions of the characteristic equation of the operator a solution of the evolution equation is separable if and only if it is a curve in the k c 1 parameter space of characteristic fields the non stationary characteristic fields with separable evolution are those which remain characteristic under small dilation every characteristic field has local separable evolution if and only if the characteristic equation is infinitesimally dilation invariant this is the case when the evolutionary generator is an infinitesimal symmetry of the characteristic equation so that its flow stabilizes the space of characteristic fields the evolution equation is then locally homogeneous if and only if the characteristic value is an invariant of the restricted flow which is a non generic property the separable kth order evolution equations are parametrized by the kth order invariants of the locally homogeneous k c 1 th order ordinary differential equations

a class of self similar solutions to a singular and degenerate diffusion equation abstract in this paper we study self similar solutions to the singular and degenerate diffusion equation u t p u x 2 p u x x x t 0 where 1 2 the existence and uniqueness for the solutions are established in addition the asymptotic behavior is investigated

convergence to equilibrium in degenerate parabolic equations with delay abstract in 11 busenberg huang 1996 showed that small positive equilibria can undergo supercritical hopf bifurcation in a delay logistic reaction diffusion equation with dirichlet boundary conditions consequently stable spatially inhomogeneous time periodic solutions exist previously in 12 badii diaz tesei 1987 considered a similar logistic type delay diffusion equation but differing in two important respects firstly by the inclusion of nonlinear degenerate diffusion of so called porous medium type and secondly by the inclusion of an additional dominating instantaneous negative feedback where terms local in time majorize the delay terms in some sense sufficient conditions were given ensuring convergence of non negative solutions to a unique positive equilibrium a natural question to ask and one which motivated the present work is can one still ensure convergence to equilibrium in delay logistic diffusion equations in the presence of nonlinear degenerate diffusion but in the absence of dominating instantaneous negative feedback the present paper considers this question and provides sufficient conditions to answer in the affirmative in fact the results are much stronger establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay logistic terms to larger classes of nonlinearities furthermore the results obtained are independent of the size of the delay

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

ordinary differential equations which linearize on differentiation in this paper we discuss ordinary differential equations odes which linearize upon one or more differentiations although the subject is fairly elementary equations of this type arise naturally in the context of integrable systems

solution approaches to irregular nesting problems this paper reviews some of the approaches which have been adopted in the solution of problems involving the nesting of irregularly shaped pieces such problems arise in a wide variety of application areas and the purpose of this paper is to bring together problems and solution approaches from a number of diverse domains

global existence and finite time blow up of the solution for a thin film equation with high initial energy abstract this paper is devoted to the study of a thin film equation which was considered extensively in recent years we obtain the conditions for global existence and finite time blow up of solution for the high initial energy case i e j u 0 d while the previous works only considered the case j u 0 d here d is the mountain pass level moreover for the blow up solution with j u 0 d we obtain the lower bound estimate of the blow up time and blow up rate

extended rotation and scaling groups for nonlinear evolution equations a 1 1 dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator v x u u x then the solution satisfies the condition ux x u for equations that do not admit the rotation group we provide an extension of the rotation group the corresponding exact solution can be constructed via the invariant set r0 u ux xf u of a contact first order differential structure where f is a smooth function to be determined the time evolution on r0 is shown to be governed by a first order dynamical system we introduce an extension of the scaling groups characterized by an invariant set tilde s 0 that depends on two constants and n 1 when 0 it reduces to the invariant set s0 introduced by galaktionov we also introduce a generalization of both the scaling and rotation groups which is described by an invariant set e0 with parameters a and b when a 0 or b 0 it respectively reduces to r0 or s0 these approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations

separation of variables and exact solutions to nonlinear diffusion equations with x dependent convection and absorption abstract this paper considers nonlinear diffusion equations with x dependent convection and source terms u t d u u x x q x u u x p x u the functional separation of variables of the equations is studied by using the generalized conditional symmetry approach we formulate conditions for such equations which admit the functionally separable solutions as a consequence some exact solutions to the resulting equations are constructed finally we consider a special case for the equations which admit the functionally separable solutions when the convection and source terms are independent of x

on nonlinear diffusion equations with x dependent convection and absorption abstract for the 1 1 dimensional nonlinear diffusion equations with x dependent convection and source terms u t d u u x x q x u u x p x u we obtain conditions under which the equations admit the second order generalized conditional symmetries x u u xx h u u x 2 g x u u x f x u and the first order sign invariants j x u u t a u u x 2 b x u u x c x u on the solutions u x t several different generalized conditional symmetries and first order sign invariants for equations in which the diffusion term offers different possibilities power law exponential mullin fujita are presented exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first order sign invariants are constructed

propagation profile of support for evolution p laplacian with convection in half space abstract consider the cauchy dirichlet problem in half space for a one dimensional evolution p laplacian with convection for p 2 and pay attention to the interface t sup x u x t 0 it is well known that lim t t in the absence of the convection while the inclusion of the first order term may change the property of finite or infinite speed of propagation in this paper it will be shown that the nonlinear convection plays a very important role to the evolution of t for the convection with promoting diffusion the fast propagation phenomenon occurs i e u x t 0 whenever t 0 if the convection is strong enough otherwise t remains finite and non localized while under the convection with counteracting diffusion if the convection is strong enough localization even shrinking and extinction appears otherwise t keeps non localized in addition it is found that the time related boundary data are significant also to the behavior of solutions the decay or incremental rates of the boundary data affect not only the contraction or expansion of the supports but also the propagation speed of the interface

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

new invariant sets to nonlinear diffusion equations with x dependent convection and absorption abstract in this paper we introduce a new invariant set u x t c d x h x u 0 h u 0 where h is a smooth function of x and u to be determined the invariant sets and exact solutions to nonlinear diffusion equation u t a u u x x q x u u x p x u are discussed the forms of a u q x u and p x u which leads to invariant solutions are determined it is shown that there exist several classes of solutions to the equation that belong to the invariant set another new invariant set is described it is used to some nonlinear evolution equations

non linear reaction diffusion systems with variable diffusivities lie symmetries ansatze and exact solutions abstract this work first considers the classical lie symmetry analysis of a class of systems of two quasilinear reaction diffusion equations having variable diffusivities subsequently non lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems in particular families of exact solutions of a diffusive lotka volterra type system are constructed

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

classification of singular solutions of a nonlinear heat equation on considere toutes les solutions non negatives de l equation de la chaleur non lineaire avec absorption u t u m u p dans q r n 0 t pour n 1 m p 1 avec des conditions initiales u x 0 0 pour x 0

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

a revision on classical solutions to the cauchy boltzmann problem for soft potentials this short note complements the recent paper of the authors alonso gamba in j stat phys 137 5 6 1147 1165 2009 we revisit the results on propagation of regularity and stability using lp estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator we show here the correct range of exponents we require a lebesgue s exponent 1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates as a consequence the lp regularity associated to the cauchy problem of the space inhomogeneous boltzmann equation holds for a finite range of p 1 explicitly determined

non local ansatze for nonlinear heat and wave equations new ansatze reducing nonlinear heat and wave equations to a system of ordinary differential equations are proposed these ansatze can be constructed with the help of symmetry operators of the system corresponding to the equation under study exact solutions of the considered equations are obtained the link between the backlund transformations for a given equation and the conditional symmetry of the corresponding system is discussed

blowup of solutions of semilinear parabolic equations on etudie le comportement d eclatement des solutions de u t uu x u xx f u a 0 u a t 0 t 0 u x 0 x a 2

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

maximal dimension of invariant subspaces admitted by nonlinear vector differential operators in this paper the dimension of invariant subspaces admitted by m component nonlinear vector differential operators is estimated it is shown that if the m component nonlinear vector differential operators of order k preserves the invariant subspace wn11 wnmm where wnqq is the space generated by solutions of linear ordinary differential equations of order nq q 1 m then max n1 nm 2mk 1 to illustrate the approach examples of nonlinear vector differential operators admitting invariant subspace wn11 wn22 with max n1 n2 9 and three component nonlinear vector differential operators admitting invariant subspace wn11 wn22 wn33 with max n1 n2 n3 13 are presented

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

well posedness and scattering for the boltzmann equations soft potential with cut off we prove the global existence of the unique mild solution for the cauchy problem of the cut off boltzmann equation for soft potential model gamma 2 n with initial data small in l n x v where n 2 3 is the dimension the proof relies on the existing inhomogeneous strichartz estimates for the kinetic equation by ovcharov siam j math anal 43 3 1282 1310 2011 and convolution like estimates for the gain term of the boltzmann collision operator by alonso et al commun math phys 298 293 322 2010 the global dynamics of the solution is also characterized by showing that the small global solution scatters with respect to the kinetic transport operator in l n x v also the connection between function spaces and cut off soft potential model n gamma 2 n is characterized in the local well posedness result for the cauchy problem with large initial data

on a time and space discretized approximation of the boltzmann equation in the whole space abstract in this paper convergence results on the solutions of a time and space discrete model approximation of the boltzmann equation for a gas of maxwellian particles in a bounded domain obtained by babovsky and illner 1989 are extended to approximate the solutions of the boltzmann equation in the whole physical space this is done for a class of particle interactions including maxwell and soft cut off potentials in the sense of grad the main result shows that the solutions of the discrete model converge in l 1 to the solutions of the boltzmann equation when the discretization parameters go simultaneously to zero the convergence is uniform with respect to the discretization parameters in addition a sufficient condition for the implementation of the main result is provided the techniques detailed in this paper may be also applied in other contexts

distributional and classical solutions to the cauchy boltzmann problem for soft potentials with integrable angular cross section this paper focuses on the study of existence and uniqueness of distributional and classical solutions to the cauchy boltzmann problem for the soft potential case assuming s n 1 integrability of the angular part of the collision kernel grad cut off assumption for this purpose we revisit the kaniel shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local maxwellian regime with possibly infinite initial mass we study the propagation of regularity using a recent estimate for the positive collision operator given in alonso et al in convolution inequalities for the boltzmann collision operator arxiv 0902 0507 math ap by e carneiro and the authors that allows us to show such propagation without additional conditions on the collision kernel finally an l p stability result with 1 p is presented assuming the aforementioned condition

global existence and blow up of solutions to a nonlocal parabolic equation with singular potential abstract in this paper we study a nonlocal parabolic equation with singular potential on a bounded smooth domain with homogeneous neumann boundary condition firstly we find a threshold of global existence and blow up to the solutions of the problem when the initial data is at the low energy level i e j u 0 d where j u 0 is the initial energy and d is the mountain pass level moreover when j u 0 d the vacuum isolating behavior of the solutions is also discussed secondly we prove that there exist solutions of the problem with arbitrary initial energy that blow up in finite time we also obtain the upper bounds of the blow up time for blow up solutions

new explicit solutions of quasilinear heat equations with general first order sign invariants abstract for the one dimensional quasilinear heat equations ut u uxx f u with smooth functions u 0 and f u we derive conditions under which a general quadratic hamilton jacobi operator h u ut h2 u ux 2 h1 u ux h0 u with the functions h2 h1 h0 to be determined preserves both the signs 0 and 0 on the solutions u x t h u is then called a sign invariant of the given parabolic equation several types of different sign invariants are established which are governed by a linear second order ordinary differential equations new explicit solutions via finite dimensional dynamical systems corresponding to zero preserving of the first order operator are constructed