On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ? ⁿ , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ? ⁿ . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.     

Existence and asymptotic behavior of entire blow-up solutions for quasilinear elliptic equation

In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ _p u ≡ div(|?u| ^p?2?u) = ρ(x)f(u), x ∈ R ^N . No monotonicity condition is assumed upon f(u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques.     

On the Structure of ∞-Harmonic Maps

Let H ∈ C ²(? ^N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E _∞(u, Ω) = ‖H(Du)‖ _{L ^∞(Ω)} defined on maps u: Ω ? ? ⁿ  → ? ^N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? ^N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT

The variational problem in L ^∞ considered is to minimize F(u) = ‖Du‖ _{L ^∞(Ω)} subject to ∈ t _Ω |Du|² dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ_∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ _{L ^∞}  ≤ Λ_∞.     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.     

The negative extended KdV equation with self-consistent sources: soliton, positon and negaton

The negative extended KdV equation with self-consistent sources (eKdV⁻ESCSs) is firstly presented and the associated linear auxiliary equations are derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁻ESCSs such as singular N-soliton solution, N-soliton solution with finite amplitude, N-positon solution and N-negaton solution. The properties of these solutions are analyzed. Moreover, the interactions of two solitons, positon and negaton, positon and soliton, and two positons are discussed.     

Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

Abstract

In this paper we consider a semilinear heat equation (in a bounded domain Ω of ? ^N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ? Ω, that insensitizes the L ² ? norm of the observation of the solution in another open subset 𝒪 ? Ω when ω ∩ 𝒪 ≠ ?, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L ^r -controls (r large enough) starting from insensitizing controls in L ².     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

Spatial Analyticity of Solutions to Integrable Systems. I. The KdV Case

We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles q's which are, in a certain sense, essentially bounded from below and q(x) = O(e^?cx? ), x → + ∞, with some positive c and ?. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if ? > 1/2, (2) meromorphic on a strip around the real line if ? = 1/2, and (3) Gevrey regular if ? <1/2. Note that q's need not have any decay or pattern of behavior at ? ∞.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

On Finite Unions of Submodules

This paper is concerned with finite unions of ideals and modules. The first main result is that, if N ? N ₁ ∪ N ₂ ∪ … ∪ N _s is a covering of a module N by submodules N _i , such that all but two of the N _i are intersections of strongly irreducible modules, then N ? N _k for some k. The special case when N is a multiplication module is considered. The second main result generalizes earlier results on coverings by primary submodules. In the last section unions of cosets is studied.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.