Large Time Behavior Of Solutions Research Articles

The Cauchy problem for [formula omitted], large-time behaviour

The nonlinear partial differential equation in the title is typified mathematically as a viscous Hamilton–Jacobi equation. It arises in the study of the growth of surfaces, and in that context is known as the generalized deterministic KPZ equation. Considering the Cauchy problem with initial data that are merely supposed to be bounded and continuous, results on the temporal decay and large-time behaviour of solutions are presented. Corresponding results for the heat equation serve as benchmarks.

Stability analysis for Allen–Cahn type equation associated with the total variation energy

In this paper, we shall discuss about the large-time behavior of solutions of an Allen–Cahn type equation generated by the total variation functional with constraints. In the one-dimensional case, the large time behavior of solutions has been studied in (Nonlinear Anal. 46 (2001) 435; Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195). According to the results, all steady-state patterns are represented as piecewise constant solutions of the equation, and any stable (steady-state) solution takes only values corresponding to pure phases. In the argument, the authors in (Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195) introduced an original concept, named as “local stability”, and discussed the stability of steady-state solutions by means of this concept. The main objective of this paper is to investigate the situation of multi-dimensional solutions. Referring to the results in the one-dimensional case, we target piecewise constant (steady-state) solutions as the object of consideration, and try to extend the theory of local stability to multi-dimensional cases. Consequently, some geometric conditions concerned with the structure of steady-state solutions and the stability will be shown.

Stability analysis for Allen–Cahn type equation associated with the total variation energy

In this paper, we shall discuss about the large-time behavior of solutions of an Allen–Cahn type equation generated by the total variation functional with constraints. In the one-dimensional case, the large time behavior of solutions has been studied in (Nonlinear Anal. 46 (2001) 435; Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195). According to the results, all steady-state patterns are represented as piecewise constant solutions of the equation, and any stable (steady-state) solution takes only values corresponding to pure phases. In the argument, the authors in (Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195) introduced an original concept, named as “local stability”, and discussed the stability of steady-state solutions by means of this concept. The main objective of this paper is to investigate the situation of multi-dimensional solutions. Referring to the results in the one-dimensional case, we target piecewise constant (steady-state) solutions as the object of consideration, and try to extend the theory of local stability to multi-dimensional cases. Consequently, some geometric conditions concerned with the structure of steady-state solutions and the stability will be shown.

Asymptotic profiles of solutions to viscous Hamilton–Jacobi equations

The large time behavior of solutions to the Cauchy problem for the viscous Hamilton–Jacobi equation u t −Δ u+|∇ u| q =0 is classified. If q> q c :=( N+2)/( N+1), it is shown that non-negative solutions corresponding to integrable initial data converge in W 1,p( R N) as t→∞ toward a multiple of the fundamental solution for the heat equation for every p∈[1,∞] (diffusion-dominated case). On the other hand, if 1< q< q c , the large time asymptotics is given by the very singular self-similar solution of the viscous Hamilton–Jacobi equation. For non-positive and integrable solutions, the large time behavior of solutions is more complex. The case q⩾2 corresponds to the diffusion-dominated case. The diffusion profiles in the large time asymptotics appear also for q c < q<2 provided suitable smallness assumptions are imposed on the initial data. Here, however, the most important result asserts that under some conditions on initial data and for 1< q<2, the large time behavior of solutions is given by the self-similar viscosity solutions to the non-viscous Hamilton–Jacobi equation z t +|∇ z| q =0 supplemented with the initial datum z( x,0)=0 if x≠0 and z(0,0)<0.

Time asymptotics for a Becker–Döring-type model of aggregation with collisional fragmentation

The large time behaviour of solutions to a discrete model of cluster coagulation with collisional breakage is studied. The interactions between clusters taken into account are restricted to the collisions of monoclusters with i-clusters, i≥1, resulting in either an ( i+1)-cluster (coalescence) or an ( i−1)-cluster and two monoclusters (fragmentation)—an assumption reminiscent of the Becker–Döring model. In comparison to the Becker–Döring model, the set of steady states exhibits a more complicated structure and we show the convergence of solutions to steady states in both the coagulation-dominated and the fragmentation-dominated cases.

Combining fast, linear and slow diffusion

Although the pioneering studies of G. I. Barenblatt [ On some unsteady motions of a liquid or a gas in a porous medium , Prikl. Mat. Mekh. 16 (1952), 67–68] and A. G. Aronson and L. A. Peletier [ Large time behaviour of solutions of some porous medium equation in bounded domains , J. Differential Equations 39 (1981), 378–412] did result into a huge industry around the porous media equation, none further study analyzed the effect of combining fast, slow, and linear diffusion simultaneously, in a spatially heterogeneous porous medium. Actually, it might be this is the first work where such a problem has been addressed. Our main findings show how the heterogeneous model possesses two different regimes in the presence of a priori bounds. The minimal steady-state of the model exhibits a genuine {\it fast diffusion behavior}, whereas the remaining states are rather reminiscent of the purely {\it slow diffusion model}. The mathematical treatment of these heterogeneous problems should deserve a huge interest from the point of view of its applications in fluid dynamics and population evolution.

A reaction-diffusion system on noncoincident spatial domains modeling the circulation of a disease between two host populations

We study the global existence and long-time behavior of the solutions to a special reaction-diffusion system arising in mathematical population dynamics, with kinetics occurring on distinct spatial domains. First, we give a comprehensive description of the dynamics of the solutions of the underlying system of ordinary differential equations. Next, we analyze a simpler problem where the spatial domain is the same for all the partial differential equations. Last, we prove global existence for the original problem; we offer a conjecture concerning the large-time behavior of solutions and give some hints on its derivation and proof.

STABILITY OF SHEAR BANDS IN AN ELASTOPLASTIC MODEL FOR GRANULAR FLOW: THE ROLE OF DISCRETENESS

Continuum models for granular flow generally give rise to systems of nonlinear partial differential equations that are linearly ill-posed. In this paper we introduce discreteness into an elastoplasticity model for granular flow by approximating spatial derivatives with finite differences. The resulting ordinary differential equations have bounded solutions for all time, a consequence of both discreteness and nonlinearity. We study how the large-time behavior of solutions in this model depends on an elastic shear modulus ℰ. For large and moderate values of ℰ, the model has stable steady-state solutions with uniform shearing except for one shear band; almost all solutions tend to one of these as t→∞. However, when ℰ becomes sufficiently small, the single-shear-band solutions lose stability through a Hopf bifurcation. The value of ℰ at the bifurcation point is proportional to the ratio of the mesh size to the macroscopic length scale. These conclusions are established analytically through a careful estimation of the eigenvalues. In numerical simulations we find that: (i) after stability is lost, time-periodic solutions appear, containing both elastic and plastic waves, and (ii) the bifurcation diagram representing these solutions exhibits bi-stability.

Stress and heat flux stabilization for viscous compressible medium equations with a nonmonotone state function

We study a system of quasilinear equations describing one-dimensional flow of a viscous compressible heat-conducting medium with a nonmonotone state function and mass force. The large-time behavior of solutions is considered for arbitrarily large initial data. In spite of possible nonuniqueness and discontinuity of the stationary solution, we prove L 2-stabilization for the stress and heat flux as t → ∞ along with corresponding global energy estimates for them. The new method of proof utilizes a combination of energy type equalities for the stress and heat flux. Consequently, H 1-stabilization of the velocity and temperature along with global estimates for their derivatives are valid as well.

Large Time Behaviour of Linear Functional Differential Equations

The aim of this paper is to show that the spectral theory for linear autonomous and periodic functional differential equations yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus and yield insight, new proofs and generalizations of results that have recently appeared in the literature.

Stationary Solutions of the KdV Equation

In this article, as a first step to study the large time behavior of solutions to the KdV equation on a half-line, we prove the existence of the stationary solutions to the corresponding problem. The aim is to concentrate on understanding the influence of boundary conditions.

Blow-up and Large Time Behavior of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Blow-up and Large Time Behavior of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

The asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave.

The global dynamics of isothermal chemical systems with critical nonlinearity

In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system u1,t = u1,xx − uα1 uβ2, u2,t = du2,xx + uα1 uβ2 with non-negative initial data, where the exponents α,β satisfy 1<α,β<2, α+β = 3 and the constant d>0 is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as |x|→∞. We show the exact large time behaviour of solutions which is universal.

Large time behavior of fronts governed by eikonal equations

Motivated by a model of solid combustion in heterogeneous media, we investigate the time-asymptotic behaviour of flame fronts evolving with a periodic space-dependent normal velocity; using the so-called \`\`level set approach'' we are led to study the large time behaviour of solutions of eikonal equations. We first provide a general approach which shows that the asymptotic normal velocity of such a flame front depends only on its normal direction and is given by the homogenized Hamiltonian of the eikonal equation. Then we turn to a more precise study of the asymptotic behaviour of the flame front when the initial front is a graph of a periodic function: in this case, the front moves asymptotically with a constant normal velocity and we are able to prove that, in coordinates moving with this constant velocity, the front has a time-periodic asymptotic behaviour in the following two cases: (i) when there is a straight line of maximal speed, and (ii) when the space dimension is 2. These results are obtained by using homogenization, control theory and dynamical systems methods (Aubry--Mather sets).

Asymptotics of solutions of nonlinear parabolic equations

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations: u t= Δu+F(u,D xu,D x 2u), u∈ R n, u(t,x)| t=0=u 0(x), x∈ R N, N⩾1. Under the optimal growth conditions on the smooth nonlinear function F( u, D x u, D x 2 u), we obtain the global existence results to the above Cauchy problem. The influence of the nonlinear function F( u, D x u, D x 2 u) on the large time behaviour of the global smooth solution u( t, x) is also studied.

Large time behaviour of solutions of the Swift–Hohenberg equation

We study the limiting profiles v of solutions of the Swift–Hohenberg equation on a one-dimensional domain (0, L) for different values of L. We identify those values of L for which v=0, and discuss the size and the shape of v when it is nontrivial and a global minimiser of an associated energy functional. To cite this article: L.A. Peletier, V. Rottschäfer, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Stability for a phase field model with the total variation functional as the interfacial energy

In this paper, we treat a one-dimensional solid–liquid phase transition model in which the interface energy is given by the total variation functional of the order parameter. We discuss the large time behavior of solutions of our problem under such a condition on initial data that the initial temperature is uniformly close to the phase transition temperature and the initial state of the material is sufficiently close to a class of steady states.

The Existence and Large Time Behavior of Solutions to a System Related to a Phase Transition Problem

We discuss the existence and the asymptotic behavior of weak solutions to a hyperbolic-elliptic mixed type system related to a phase transition problem. As in the hyperbolic case, the weak solutions are not unique. To select the admissible solution, we need to impose admissibility criteria. One of the criteria we use is the entropy rate admissibility criterion. The question is how it can be applied. We examine two alternatives, and the result is used to study the existence and large time behavior of solutions to the perturbed Riemann problem.

Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau equations in H1

Considering the Cauchy problem for the critical complex Ginzburg-Landau equation in H1(Rn), we shall show the asymptotic behavior for its solutions in C(0, ∖;H1(Rn)) ∩ L2(0, ∖;H1,2n/(n-2)(R2)), n≥3. Analogous results also hold in the case that the nonlinearity has the subcritical power in H1(Rn), n≥1.