global structure of self similar solutions in a semilinear parabolic equation abstract the aim of this paper is to investigate the structure of radial solutions for a semilinear elliptic equation u y u 2 u p 1 u p 1u 0 y rn which is related to forward self similar solutions of a semilinear parabolic equation vt v v p 1v t x 0 rn we study the existence uniqueness and parameter dependency of rapidly decaying solutions with a prescribed number of zeros

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

on the equation y2 x x2 p generators are found for the group of rational points on the title curve for all primes p 5 mod 8 less than 1 000 the rank is always i in accordance with conjectures of selmer and mordell some of the generators are rather large

structure of positive radial solutions to scalar field equations with harmonic potential abstract we study a nonlinear elliptic equation with a harmonic potential u x 2 u u p 1 u 0 x r n e which is related to standing waves for a nonlinear schrodinger equation the existence of positive solution for e in some energy space can be shown by the standard variational method under n 1 n and 1 p n 2 n 2 however the uniqueness is a delicate problem in this paper noting that all positive solutions of e in the energy space are radially symmetric about the origin we study the structure of positive radial solutions to e and show the uniqueness of positive solution with finite energy when n 3

homotopy method for a class of nonconvex brouwer fixed point problems abstract in this paper we present a homotopy method for computing fixed points of twice continuously differentiable self mappings in a class of nonconvex subsets in r n the method need not homeomorphically transform the considered set to a closed unit ball and is numerically implementable

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

a constructive proof of the poincare birkhoff theorem in this paper with the use of the homotopy method a constructive proof of the poincare birkhoff theorem is given this approach provides a global method for finding fixed points of area preserving maps and periodic solutions of duffing equations

a two dimensional ca model for traffic flow with car origin and destination abstract dynamic phase transitions in a two dimensional traffic flow model defined on a decorated square lattice are studied numerically the square lattice point and the decorated site denote intersections and roads respectively in the present model a car has a finite deterministic path between the origin and the destination which is assigned to the car from the beginning in this new model we found a new phase between the free flow phase and the frozen jam phase that is absent from previous models the new model is characterized by the persistence of a macroscopic cluster furthermore the behavior in this macroscopic cluster phase is classified into three regions characterized by the shape of the cluster the boundary of the three regions is phenomenologically estimated when the trip length is short and the car density is high both ends of the belt like cluster connect to each other through the periodic boundary with some probability this type of cluster is classified topologically as a string on a two dimensional torus

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

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

coexisting phases and lattice dependence of a cellular automaton model for traffic flow the biham middleton levine traffic model is perhaps the simplest system exhibiting phase transitions and self organization moreover it is an underpinning to extensive modern studies of traffic flow the general belief is that the system exhibits a sharp phase transition from freely flowing to fully jammed as a function of initial density of cars however we discover intermediate stable phases where jams and freely flowing traffic coexist the geometric structure of such phases is highly regular with bands of free flowing traffic intersecting at jammed wave fronts that propagate smoothly through the space instead of a phase transition as a function of density we see bifurcation points where intermediate phases begin coexisting with the more conventionally known phases we show that the regular geometric structure is in part a consequence of the finite size and aspect ratio of the underlying lattice and that for certain aspect ratios the asymptotic intermediate phase is on a periodic limit cycle the exact microscopic configuration recurs each tau time steps aside from describing these intermediate states which previously were overlooked we derive simple equations to describe the geometric constraints and predict their asymptotic velocities

a continuation method for solving fixed points of self mappings in general nonconvex sets in this paper we present a homotopy continuation method for solving xed points of selfmappings in a class of general nonconvex subsets in r n

on asymptotic behaviors of solutions to parabolic systems modelling chemotaxis abstract this paper deals with large time behaviors of solutions to a keller segel system which possesses self similar solutions by taking into account the invariant properties of the equation with respect to a scaling and translations we show that suitably shifted self similar solutions give more precise asymptotic profiles of general solutions at large time the convergence rate is also computed in details

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

asymptotically self similar global solutions of a general semilinear heat equation we consider the general nonlinear heat equation partial t u delta u a u p 1 1 u g u u 0 varphi on 0 infty times i r n where a in i r p 1 1 2 n and g satisfies certain growth conditions we prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation we also prove that some of those global solutions are asymptotic for large time to self similar solutions of the single power nonlinear heat equation i e with g equiv 0

asymptotically self similar global solutions of a semilinear parabolic equation with a nonlinear gradient term we study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u 0 where a b q 1 p 1 for q 2p p 1 and 1 p 1 1 equivalently q n 2 n 1 we prove the existence of mild global solutions for small initial data with respect to some norm some of those solutions are asymptotically self similar

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 evolution to localized and front solutions in a non lipschitz reaction diffusion cauchy problem with trivial initial data abstract in this paper we establish the existence of spatially inhomogeneous classical self similar solutions to a non lipschitz semi linear parabolic cauchy problem with trivial initial data specifically we consider bounded solutions to an associated two dimensional non lipschitz non autonomous dynamical system for which we establish the existence of a two parameter family of homoclinic connections on the origin and a heteroclinic connection between two equilibrium points additionally we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

a stochastic cellular automaton model for traffic flow with multiple metastable states a new stochastic cellular automaton ca model of traffic flow which includes slow to start effects and a driver s perspective is proposed by extending the burgers ca and the nagel schreckenberg ca model the flow density relation of this model shows multiple metastable branches near the transition density from free to congested traffic which form a wide scattering area in the fundamental diagram the stability of these branches and their velocity distributions are explicitly studied by numerical simulations

a boundary perturbation interior point homotopy method for solving fixed point problems abstract in this paper a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets compared with the previous results by using the newly proposed method initial points can be chosen in the whole space of r n which may improve greatly the computational efficiency of reduced predictor corrector algorithms resulted from that method some numerical examples are given to illustrate the results of this paper

on two signed solutions to a second order semi linear parabolic partial differential equation with non lipschitz nonlinearity abstract in this paper we establish the existence of a 1 parameter family of spatially inhomogeneous radially symmetric classical self similar solutions to a cauchy problem for a semi linear parabolic pde with non lipschitz nonlinearity and trivial initial data specifically we establish well posedness for an associated initial value problem for a singular two dimensional non autonomous dynamical system with non lipschitz nonlinearity additionally we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

on a decay property of solutions to the haraux weissler equation abstract we give a sufficient condition that non radial h 1 solutions to the haraux weissler equation should belong to the weighted sobolev space h 1 r n where is the weight function exp x 2 4 our result provides in some sense a connection between the solutions obtained by ode method and those by variational approach in the space h 1 r n

global existence and blowup for sign changing solutions of the nonlinear heat equation abstract in this paper given 0 2 n we prove the existence of a function with the following properties the solution of the equation u t u u u on r n with the initial condition u 0 is global on the other hand the solution with the initial condition u 0 blows up in finite time if 0 is either sufficiently small or sufficiently large

a semilinear heat equation with singular initial data we first prove existence and uniqueness of non negative solutions of the equation in in the range 1 p n when initial data u x 0 a x 2 p 1 x 0 for a 0 it is proved that the maximal and minimal solutions are self similar with the form where g g a satisfies after uniqueness is proved the asymptotic behaviour of solutions of is studied in particular we show that the case for a 0 is also considered and a sharp decay rate of the above equation is derived in the final we reveal existence of solutions of the first and third equations above which change sign