slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

asymptotics for a parabolic double obstacle problem we consider a parabolic double obstacle problem which is a version of the allen cahn equation u t delta u epsilon 2 psi prime u in omega times 0 infty where omega is a bounded domain epsilon is a small constant and psi is a double well potential here we take psi such that psi u 1 u 2 when u leq 1 and psi u infty when u 1 we study the asymptotic behaviour as epsilon rightarrow 0 of the solution of the double obstacle problem under some natural restrictions on the initial data we show that after a short time of order epsilon 2 ln epsilon the solution takes value 1 in a region omega t and value 1 in omega t where the region omega backslash omega t cup omega t is a thin strip and is contained in either a o epsilon ln epsilon or o epsilon neighbourhood of a hypersurface gamma t which moves with normal velocity equal to its mean curvature we also study the asymptotic behaviour as t rightarrow infty of the solution in the one dimensional case in particular we prove that the omega limit set consists of a singleton

bifurcations and multiple traffic jams in a car following model with reaction time delay we investigate an optimal velocity car following model for n cars on a circular singlelane road where reaction time delay of drivers is taken into account the stability of the uniform ow equilibrium is studied analytically while bifurcating periodic solutions for difierent wave numbers are investigated with numerical continuation techniques this reveals that the periodic solution with the smallest wave number may be stable and all other periodic solutions are unstable as n is increased periodic solutions develop stop and go fronts that correspond to rapid deceleration and acceleration between regions of uniformly owing and stagnant tra c in terms of the positions of all cars on the ring these fronts are associated with tra c jams all tra c jams form a tra c pattern that evolves under time due to slow motion of the fronts the tra c pattern corresponding to the stable periodic motion of cars is the only stable one however we flnd that other periodic orbits may be unstable only so weakly that they give rise to transient tra c jams that may persist for long times eventually such tra c jams either merge with one another or disperse until the stable tra c pattern is reached

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

metastable solutions for the thin interface limit of a phase field model abstract we consider a generalization of the semilinear phase field model from jimenez casas and rodriguez bernal proc roy soc edinburgh 129a 1999 571 600 for a more general enthalphy function we prove the existence of metastable solutions that evolve very slowly in time

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

a mean field equation of motion for the dynamic ising model a mean field type of approximation is used to derive two differential equations one approximately representing the average behavior of the ising model with glauber spin flip stochastic dynamics and the other doing the same for kawasaki spin exchange dynamics the proposed new equations are compared with the cahn allen and cahn hilliard equations representing the same systems and with information about the exact behavior of the microscopic models

evaporation of droplets in the two dimensional ginzburg landau equation abstract we consider the problem of coarsening in two dimensions for the real scalar ginzburg landau equation this equation has exactly two stable stationary solutions the constant functions 1 and 1 we assume most of the initial condition is in the 1 phase with islands of 1 phase we use invariant manifold techniques to prove that the boundary of a circular island moves according to allen cahn curvature motion law we give a criterion for non interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces

almost global existence and transient self similar decay for poiseuille flow at criticality for exponentially long times abstract we consider nonlinear diffusion equations with critical exponent as tu x2u u3 with x r for small initial data in l1 l it is well known that almost all solutions of this system explode in finite time however we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time moreover before explosion the l norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments as an application we consider the long time transient self similar decay to unstable poiseuille flow at criticality for exponentially long times this together with a subcritical bifurcation and short time transient amplification is a principal obstruction in all attempts to measure the critical reynolds number for this experiment more and more precisely

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

a necessary and sufficient condition for robust asymptotic stability of time variant discrete systems a necessary and sufficient condition for the stability of time variant interval matrices is presented this condition allows stability to be tested by checking only products of vertex extreme matrices the implementation of the test in the form of an algorithm and two illustrative examples are provided

on a reaction diffusion equation with robin and free boundary conditions this paper studies the following problem ut uxx f u 0 0 u 0 t bux 0 t t 0 u h t t 0 h t ux h t t t 0 h 0 h0 u x 0 x 0 x h0 where f is an unbalanced bistable nonlinearity b 0 0 and is a compactly supported c2 function we prove that there exists 0 such that vanishing happens when 0 and u t converges as t to 0 uniformly in 0 h t spreading happens when i e h t c t tends to a constant for some c 0 u t converges to a positive stationary solution locally uniformly in 0 and to a traveling semi wave with speed c near x h t in the transition case when u t v t h2 0 h t tends to 0 as t where t is a maximum point of u t and v is the unique even positive solution of v f v 0 subject to v 0 moreover with respect to b and f t pln t q o 1 for some p 0 and q r or t z for some root z of v z bv z

invariant manifolds for metastable patterns in ut e2uxx f u we consider the above equation on the interval 0 x 1 subject to neumann boundary conditions with f u f u where f is a double well energy density function with equal minima our previous work 3 proved the existence and persistence of very slowly evolving patterns metastable states in solutions with two phase initial data here we characterise these metastable states in terms of the global unstable manifolds of equilibria as conjectured by fusco and hale 6

self similarity in a coarsening model in one dimension motivated from asymptotic laws of motion for transition layers in the equation u t 2 u xx u u 2 we consider the following model for coarsening of a fine partition of an interval find the shortest subinterval of the partition and joint it with its neighbours combining three into one making a random order assumption we develop and study an unusual coagulation equation for the distribution of interval lengths we establish the existence of a self similar solution of this equation by using laplace transform techniques simulation data indicate that random order does persist if present initially and the distribution approaches similarity form

solutions to the nonautonomous bistable equation with specified morse index part i existence we investigate the existence of unstable solutions of specified morse index for the equation e 2 u xx f x u 0 on a finite interval and neumann boundary conditions

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

an optimization problem with volume constraint with applications to optimal mass transport in this manuscript we study the following optimization problem with volume constraint min left frac 1 p int omega nabla v pdx int partial omega gv ds colon v in w 1 p left omega right text and v 0 leq alpha right here g is acontinuous function and alpha is a fixed constant such that 0 0 we prove that a minimizer exists and satisfies left begin array ccl delta p u p 0 text in u p 0 cup u p 0 5pt nabla u p p 2 frac partial u p partial nu g text on partial omega cap partial u p 0 cup u p 0 5pt u p 0 alpha end array right next we analyze the limit as p to infty we obtain that any sequence of weak solutions converges up to a subsequence lim limits p j to infty u p j x u infty x uniformly in overline omega and uniform limits u infty are solutions to the maximization problem with volume constraint max left int partial omega gv ds colon v in w 1 infty left omega right nabla v l infty omega leq 1 text and v 0 leq alpha right furthermore we obtain the limit equation that is verified by u infty in the viscosity sense finally it turns out that such a limit variational problem is connected to the monge kantorovich mass transfer problem with the involved measures are supported on partial omega and along the limiting free boundary partial u infty neq 0

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so