Exponential growth for the wave equation with compact time‐periodic positive potential

We prove the existence of smooth positive potentials V(t, x), periodic in time and with compact support in x, for which the Cauchy problem for the wave equation u_tt ? Δ_xu + V(t, x)u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z ∈ ?, |z| > 1, associated to V(t, x). © 2008 Wiley Periodicals, Inc.     

On Landis’ Conjecture in the Plane

In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W(z) be a measurable real vector-valued function and V(z) ≥0 be a real measurable scalar function, satisfying ‖W‖ _{L ^∞(R ²)} ≤ 1 and ‖V‖ _{L ^∞(R ²)} ≤ 1. Let u be a real solution of Δu ? ?(Wu) ? Vu = 0 in R ². Assume that u(0) = 1 and |u(z)| ≤exp (C ₀|z|). Then u satisfies inf _{|z 0| =R}  sup _{|z?z 0| <1}|u(z)| ≥exp (?CRlog R), where C depends on C ₀. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

We consider the problem Δ²u = V(x)u^{p + ?} in with u,Δu→0 as |x|→ + ∞, where , N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

An Asymptotic Analysis of Localized Solutions for Some Reaction-Diffusion Models in Multidimensional Domains

In the limit ? → 0, a spike-layer solution is constructed for the reaction-diffusion equation where b > 0 and D is a bounded convex domain. Here Q(u) is such that there exists a unique radially symmetric function u_c(?^?1 r) satisfying ?²Δu_c + Q(u_c) = 0 in all of ?^N, with u_c(ρ) decaying exponentially at infinity. The spike-layer solution has the form u ~ u_c [?|x ? x₀|], where the spike-layer location x₀ ? D is to be determined subject to the condition that dist(x₀, ?D) as ? → D. The determination of x₀ is shown to be exponentially ill conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x₀. For a strictly convex domain, it is shown that there is an x₀ that is located at an O(?) distance away from the point in D that is furthest from ?D. Finally, hot-spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

On global transformations of ordinary differential equations of the second order

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form

Non-homogeneous non-linear damped wave equations in unbounded domains

We present a global existence theorem for solutions of u^tt ? ?_ia_ik (x)?_ku + u_t = ?(t, x, u, u_t, ?u, ?u_t, ?²u), u(t = 0) = u⁰, u(=0)=u¹, u(t, x), t ? 0, x?Ω.Ω equals ?³ or Ω is an exterior domain in ?³ with smoothly bounded star-shaped complement. In the latter case the boundary condition u|_?Ω = 0 will be studied. The main theorem is obtained for small data (u⁰, u¹) under certain conditions on the coefficients a_ik. The L^p - L^q decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.     

On the Terai-Jésmanowicz conjecture

In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）.
Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.     

The stability of rarefaction wave for Navier–Stokes equations in the half‐line

This paper studies the stability of the rarefaction wave for Navier–Stokes equations in the half‐line without any smallness condition. When the boundary value is given for velocity u∥_{x = 0} = u_? and the initial data have the state (v₊, u₊) at x→ + ∞, if u_?<u₊, it is excepted that there exists a solution of Navier–Stokes equations in the half‐line, which behaves as a 2‐rarefaction wave as t→ + ∞. Matsumura–Nishihara have proved it for barotropic viscous flow (Quart. Appl. Math. 2000; 58:69–83). Here, we generalize it to the isentropic flow with more general pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

On the behavior of the solution of the Burgers equation as the viscosity goes to zero

The solution of the initial boundary-value problem u_?′ ? ?D²u_? + u_?Du_? = f on (a, b) x(0, T), u_?(a, t) = u_?(b, t) = 0 and u_?(x, 0) = 0 on (a, b), is shown to converge to the solution of the limiting equation as the viscosity tends to zero. Estimates on the rate of convergence are given.     

Asymptotic behaviour of three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem −?(uxx+uyy)+ut+v₁ux+v₂uy=0 in an octant, (ii) an elliptic problem −?(uxx+uyy+uzz)+v₁ux+v₂uy+v₃uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ?→⁰⁺. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ?− behaviour of the solution and its derivatives in the boundary layers or the internal layers.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

For N = 1,2, we consider singularly perturbed elliptic equations ?²Δ u ? V(x) u + f(u)= 0, u(x)> 0 on R ^N , lim_|x|→∞ u(x)= 0. For small ? > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007 Byeon , J. , Oshita , Y. ( 2004 ). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations . Comm. PDE 29 : 1877 – 1904 . [Google Scholar]).     

Oscillation of partial difference equations with continuous variables

In this paper, we consider the partial difference equation with continuous variables of the form P₁z(x + a, y + b) + p₂z (x + a, y) + p₃z (x, y + b) − p₄z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where P ϵ C(R⁺ × R⁺, R⁺ − {0}), a, b, τ, σ are real numbers and p_i (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.