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.     

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.     

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.     

Global Existence for Reaction‐Diffusion Systems Modelling Ignition

The pair of parabolic equations , , with a>0 and b>0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where , which appears in the Frank‐Kamenetskii approximation to Arrhenius‐type reactions. We show here that for a large choice of the nonlinearity f(u,v) in (1), (2)(including the model case (3)), the corresponding initial‐value problem for (1), (2) in the whole space with bounded initial data has a solution which exists for all times. Finite‐time blow‐up might occur, though, for other choices of function f(u,v), and we discuss here a linear example which strongly hints at such behaviour. (Accepted September 16, 1996)     

Contact Transformation Group Classification of Nonlinear Wave Equations

Interest in nonlinear wave equations has been stimulated bynumerous physical applications, such as telecommunication (e.g.nonlinear telegrapher equation), gasdynamics, anisotropic plasticity andnonlinear elasticity, etc. Mathematical models of these phenomena canoften be reduced to particular types of the equation u _tt = f(x, u _x ) u _xx + g(x, u _x ). In this paper, the problem ofclassification of the latter equation with respect to admitted contacttransformation groups is reduced to the investigation of pointtransformation groups of the equivalent system of first-orderquasi-linear equations v _t =a(x, v)w _x , w _t = b(x,v)v _x .     

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.     

An Efficient Derivation of the Aronsson Equation

For 1<p<∞, the equation which characterizes minima of the functional u↦∫ _U |Du| ^p ,dx subject to fixed values of u on ∂U is −Δ _p u=0. Here −Δ _p is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|<sup>2</sup>|| <sub>L∞(U)</sub> . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is −Δ_∞ u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦||f(x,u,Du)|| _L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. (Accepted September 6, 2002) Published online April 14, 2003 Communicated by S. Muller     

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

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.     

Direct displacement method in crack theory (numerical resolution)

Solutions in crack theory can be defined directly by opening displacement v _○=v _○(x) and u _○=u _○(x) for the first and the second mode, respectively. In this case, the boundary conditions are expressed by singular integrals of the second order. Aiming to solve numerically the problems, we apply the finite-part definition for the singular integrals and the discretization procedure. Contributed to the memory of my Mother.     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E _ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E ₀(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E _ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E _ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property. Accepted January 21, 2000?Published online November 24, 2000     

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.     

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

Symmetry, Optima and Bifurcations in Structural Design

Motivated by optimization problems in structural engineering, we study the critical points of symmetric, ‘reflected', one-parameter family of potentials U(p, x) = max (f(p,x), f(p, −x)), yielding modest generalizations of classical bifurcations, predicted by elementary catastrophe theory. One such generalization is the ‘five-branch pitchfork’, where the symmetric optimum persists beyond the critical parameter value. Our theory may help to explain why symmetrical structures are often optimal.     

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.     

An investigation of electromagnetic wave propagation in plasma by shock tube

This paper presents the electromagnetic wave propagation characteristics in plasma and the attenuation coefficients of the microwave in terms of the parameters he, v, w, L, wb. The φ800 mm high temperature shock tube has been used to produce a uniform plasma. In order to get the attenuation of the electromagnetic wave through the plasma behind a shock wave, the microwave transmission has been used to measure the relative change of the wave power. The working frequency is f = (2-35)GHz (ω=2πf, wave length A =15cm-8mm). The electron density in the plasma is ne = (3&#215;10^10-1&#215;10^14) cm^-3. The collision frequency v = (1&#215;10^8-6&#215;10^10) Hz. The thickness of the plasma layer L = (2-80)cm. The electron circular frequency ωb=eBo/me, magnetic flux density B0 = (0-0.84)T. The experimental results show that when the plasma layer is thick (such as L/λ≥10), the correlation between the attenuation coefficients of the electromagnetic waves and the parameters ne,v,ω, L determined from the measurements are in good agreement with the theoretical predictions of electromagnetic wave propagations in the uniform infinite plasma. When the plasma layer is thin (such as when both L and A are of the same order), the theoretical results are only in a qualitative agreement with the experimental observations in the present parameter range, but the formula of the electromagnetic wave propagation theory in an uniform infinite plasma can not be used for quantitative computations of the correlation between the attenuation coefficients and the parameters ne,v,ω, L. In fact, if ω&lt;ωp, v^2&lt;&lt;ω^2, the power attenuations K of the electromagnetic waves obtained from the measurements in the thin-layer plasma are much smaller than those of the theoretical predictions. On the other hand, if ω&gt;ωp, v^2&lt;&lt;ω^2 (just v≈f), the measurements are much larger than the theoretical results. Also, we have measured the electromagnetic wave power attenuation value under the magnetic field and without a magnetic field. The result indicates that the value measured under the magnetic field shows a distinct improvement.     

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].