Traveling-wave solution to a nonlinear equation in semiconductors with strong spatial dispersion

The third-order nonlinear differential equation (u _xx ? u) _t + u _xxx + uu _x = 0 is analyzed and compared with the Korteweg-de Vries equation u _t + u _xxx ? 6uu _x = 0. Some integrals of motion for this equation are presented. The conditions are established under which a traveling wave is a solution to this equation.     

LIE SYMMETRY ANALYSIS AND PAINLEVE ANALYSIS OF THE NEW （2＋1）-DIMENSIONAL KdV EQUATION

Lie point symmetries associated with the new （2＋1）-dimensional KdV equation ut ＋ 3uxuy ＋ uxxy = 0 are investigated. Some similarity reductions are derived by solving the corresponding characteristic equations. Painlevé analysis for this equation is also presented and the soliton solution is obtained directly from the Bǎcklund transformation.     

On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ ? u + ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ ? u + ā. A calibrated sub-action u gives the possibility to construct, for any point x ε Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.We prove that separating sub-actions are generic among Hölder sub-actions. We also prove that, under certain conditions on Ω(A), any calibrated sub-action is of the form u(x) = u(x _i ) + h _A (x _i , x) for some x _i ∈ Ω(A), where h _A (x, y) denotes the Peierls barrier of A. We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type \((\hat \Sigma , \hat \sigma )\).     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

Existence of Semilinear Differential Equations with Nonlocal Initial Conditions

In this paper we give the existence of mild solutions for semilinear Cauchy problems u′（t） = Au（t） ＋f（t, u（t））, t ∈ I, a.e. with nonlocal initial condition u（O） = g（u） ＋uo when the map g loses compactness in Banach spaces.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Asymptotics of the Gurevich-Pitaevskii universal special solution of the Korteweg-de Vries equation as |<Emphasis Type="Italic">x</Emphasis>| → ∞

A complete asymptotic expansion as x → ±∞ of the Gurevich-Pitaevskii universal special solution of the Korteweg-de Vries equation u _t + u _xxx + u _ux = 0 is constructed and justified. The expansion is infinitely differentiable with respect to the variables t and x and, together with the asymptotic expansions of all its derivatives with respect to independent variables, is uniform on any compact interval of variation of the time t.     

Nonexistence and longtime behaviors of solutions to a class of nonlinear degenerate parabolic equations not in divergence form

In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation ? _t u ⁿ = u ^m div(|?u ^m | ^p?2?u ^m ) + γ|?u ^m | ^p + β u ⁿ with zero boundary condition. Blow-up time is derived when the blow-up does occur.     

Convergence of ground state solutions for nonlinear Schrdinger equations on graphs

We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|~(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|~(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λ_j,φ_j} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u ³, u ², (u - p), u ²(u ² - p), are completed and some conjectures are presented.     

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt^?1u′(t) = Au(t) on the half-line. (Here k ∈ ? is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t ^k u′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.     

Blow-up of critical norms for the 3-D Navier-Stokes equations

Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

Some extensions of Bairstow's method

u _xx +u _yy =u _t Bairstow's method for improving an approximate real quadratic factor (x ²?px?q) of a polynomial with real coefficients which leaves a remainderr(x), is to determine δp and δq to satisfy

$$0 = r\left( x \right) + \frac{{\partial r\left( x \right)}}{{\partial P}}\delta P + \frac{{\partial r\left( x \right)}}{{\partial q}}\delta q$$