Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

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.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤ^d, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ ^d } is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e ^{s /θ}] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ ^d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w _ρ∥⁻² _ℓ2Σ_{z ∈ℤd} w _ρ(x+z)w _ρ(y+z). In this expression, ρ = θ/κ while w _ρ:ℤ^d→ℝ⁺ is given by w _ρ = (v _ρ)^{⊗ d} with v _ρ: ℤ→ℝ⁺ the unique centered ground state (i.e., the solution in ℓ²(ℤ) with minimal l ²-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ _{x, y} (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure. Received: 5 March 1997 / Revised version: 21 September 1998     

Nonlinear equations with mixed derivatives

The nonlinear hyperbolic equation ∂²u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).     

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.     

The zero dispersion limit of the korteweg-de vries equation for initial potentials with non-trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. U_t ? 6uu_x + ?²u_xxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.     

An approximation technique for nonlinear integral operations

Approximation results for J. S. Mac Nerney's theory of nonlinear integral operations are established. For the nonlinear product integral _xΠ^y (1 + V)P, approximations of the form Π_{i = 1}ⁿ [1 + L_q(x_i?1, x_i)]P are considered, where L₁(u, v)P = ∝_u^vVP and L_q(u, v)P = ∝_u^vV(r, s)[1 + L_q?1(s, v)]P for q = 2, 3,…. Error bounds are obtained for the difference between the product integral and the preceding product.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

Asymptotic behaviour of a Brownian motion on exterior domains

We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ? ^c is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p ^? _t (x, y) behaves, for large t, as u(x)u(y)(t(log t)²)⁻¹ for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?) ^r , such that u(x) ∼ log ∣x∣. Received: 29 January 1999 / Revised version: 11 May 1999     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation