Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Perturbation for fractional-order evolution equation

Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for ^CD^a-e₀₊u(t)=A ^CD^d₀₊u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t), u(0)=u _o , α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C _o -semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Asymptotics of the trajectory of an interval that contains the preimage of a periodic point

We consider dynamical systems defined by continuous maps of an interval I of the real axis into itself. We prove that if an interval J in I contains the preimage of a periodic point of period p of a map f ∈ C ⁰(I, I), then the sequence of intervals f ^2pn (J), n= 0, 1, 2,…, is convergent. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 130–133, January–March, 2009.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods

Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ω^ε having cross section with diameter of order ε, and let u ⁽⁰⁾ –Bernoulli–Navier displacement – and u ⁽²⁾ be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u ⁽⁰⁾ + ε² u ⁽²⁾) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the rod. This revised version was published online in August 2006 with corrections to the Cover Date.     

Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

We consider reaction diffusion equations of the prototype form u _t = u _xx + λ u + |u| ^p-1 u on the interval 0 < x < π, with p > 1 and λ > m ². We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u ₁, . . . ,u _m . Since u _k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x ₁, . . . ,x _m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation). Dedicated to Palo Brunovsky on the occasion of his birthday.     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.     

On the Linearization of Semilinear Wave Equations

We consider the class of wave equations u _tt−u _xx=f(u, u _t, u _x). By using the differential invariants, with respect to the equivalence transformation algebra of this class, we characterize subclasses of linearizable equations. Wide classes of general solutions for some nonlinear forms of f(u, u _t, u _x) are found.     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

Uniqueness of Positive Solutions of Δu+f(u)=0 in ℝN, N≦3

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation in ℝ ^N . We assume that the function f has two zeros, the origin and u ₀>0. Above u ₀ the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u ₀, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state. (Accepted July 3, 1996)     

Gearhart–Prüss Theorem and Linear Stability for Riemann Solutions of Conservation Laws

We study the spectral and linear stability of Riemann solutions with multiple Lax shocks for systems of conservation laws. Using a self-similar change of variables, Riemann solutions become stationary solutions for the system u _t + (Df(u) − x I)u _x = 0. In the space of O((1 + |x|)^−η) functions, we show that if , then λ is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, called resonance lines, the determinant can be arbitrarily small but nonzero. A C ⁰ semigroup is constructed. Using the Gearhart–Prüss Theorem, we show that the solutions are O(e ^{γ t} ) if γ is greater than the real parts of the eigenvalues and the coordinates of resonance lines. We study examples where Riemann solutions have two or three Lax-shocks. Dedicated to Professor Pavol Brunovsky on his 70th birthday.     

Another Thin-Film Limit of Micromagnetics

We consider the variational problem of micromagnetics for soft, relatively small thin films with no applied magnetic field. In terms of the film thickness t, the diameter l and the magnetic exchange length w, we study the asymptotic behavior in the small-aspect-ratio limit t/l→0, when either (a) w²/l²≫(t/l)| log (t/l)| or (b) w²/l²∼(t/l)| log (t/l)|. Our analysis builds on prior work by Gioia & James and Carbou. The limiting variational problem is much simpler than 3D micromagnetics; in particular it is two-dimensional and local, with no small parameters. The contribution of shape anisotropy reduces, in this limit, to a constant times the boundary integral of (m·n)².     

Time-Periodic Solutions of the Burgers Equation

We investigate the time periodic solutions to the viscous Burgers equation u_t − μu_xx + uu_x = f for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces.      

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].     

Connectivity and Design of Planar Global Attractors of Sturm Type. III: Small and Platonic Examples

Based on a Morse-Smale structure, we study planar global attractors A_f{{\mathcal A}_f} of the scalar reaction-advection-diffusion equation u _t = u _xx + f (x, u, u _x ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, and hyperbolicity of equilibria. We call A_f{{\mathcal A}_f} Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. The planar Sturm attractor A_f{{\mathcal A}_f} consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph C_f{{\mathcal C}_f} , which we call the connection graph. Its 1-skeleton C¹_f{{\mathcal C}^1_f} is the closure of the unstable manifolds (separatrices) of the index-1 Morse saddles. We summarize and apply two previous results (Fiedler and Rocha, J. Diff. Equ. 244: 1255–1286, 2008, Crelle J. Reine Angew. Math. 26 pp., 2009, doi:) which completely characterize the connection graphs C_f{{\mathcal C}_f} and their 1-skeletons C¹_f{{\mathcal C}^1_f}, in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations. Such orientations are called bipolar in de Fraysseix et al. (Discrete Appl. Math. 56: 157–179, 1995). We describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons of their connection graphs. We present complete lists for the tetrahedron, octahedron, and cube. We provide representative examples for the design of dodecahedral and icosahedral Sturm attractors. Unlike previous examples, and in particular unlike the classification of Sturm attractors with up to nine equilibria, our present results are based on analytic insight rather than mindless computer-based enumeration.     

Global Nonexistence for Nonlinear Kirchhoff Systems

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, u _t ), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped.