Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

A characterization of Q-polynomial distance-regular graphs

Let Γ denote a distance-regular graph with diameter D3. Let θ denote a nontrivial eigenvalue of Γ and let denote the corresponding dual eigenvalue sequence. In this paper we prove that Γ is Q-polynomial with respect to θ if and only if the following (i)–(iii) hold:

(i) There exist such that

(1)

(ii) There exist such that the intersection numbers a_i satisfy

for 0iD, where and are the scalars which satisfy Eq. (1) for i=0, i=D, respectively.

(iii) for 1iD.

Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Cubic convergence of parameter-controlled Newton-secant method for multiple zeros

Let have a multiple zero α with integer multiplicity m≥1 and be analytic in a sufficiently small neighborhood of α. For parameter-controlled Newton-secant method defined by

we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters λ and μ. For various test functions, the numerical method has shown a satisfactory result with high-precision Mathematica programming.     

On a filter for exponentially localized kernels based on Jacobi polynomials

Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c₁, depending only on H, α, and β such that

Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L² space.     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.     

Periodic solutions of sublinear Liénard differential equations

In this paper, we study the existence of periodic solutions of the second order differential equations x^″+f(x)x^′+g(x)=e(t). Using continuation lemma, we obtain the existence of periodic solutions provided that F(x) () is sublinear when x tends to positive infinity and g(x) satisfies a new condition

where M, d are two positive constants.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,

in the unit ball Ω of with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set of initial values in C₀(Ω) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of , which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that is not convex. This contrasts with the case of nonnegative initial values, where the analogous set is known to be convex.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

The maximal range problem for a bounded domain

Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let

where . We relate the maximal polynomial range

to the geometry of Ω.     

On approximation properties of a family of linear operators at critical value of parameter

We introduce the family of linear operators

associated to a certain “admissible bunch” of operators S_t, t>0, acting on , and investigate the approximation properties of this family as α→0⁺. We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations.     

Uniqueness of positive solutions for a boundary blow-up problem

In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem

where Ω is a C² bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/t^p is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x).     

An upper bound on Jacobi polynomials

Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:

where δ_-1<δ₁ are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that

in the region .     

Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients

This paper is concerned with the nonlinear impulsive delay differential equations with positive and negative coefficients

(*)

Sufficient conditions are obtained for every solution of equation (*) tending to a constant as t→∞.     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Homogenization of degenerate nonlinear parabolic equation in divergence form

In this paper the homogenization of degenerate nonlinear parabolic equations

where a(t,y,λ) is periodic in (t,y), is studied via a weighted compensated compactness result.     

Marcinkiewicz–Zygmund strong laws for -statistics of weakly dependent observations

We prove the Marcinkiewicz–Zygmund Strong Law of Large Numbers for U-statistics of strictly stationary, absolutely regular observations (ξ_i)_i≥1. Under suitable moment conditions and conditions on the mixing rate, we show that

for some γ≥0, in the non-degenerate case, and

in the degenerate case.     

A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G) of a graph G is defined by , where is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3 vertices with Randić index R and girth g,

with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture.     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1