Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Time-Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?₁,?₂, we investigate the multilinear mapping from to the localization operator A_a^?₁,?₂ and we give sufficient and necessary conditions for A_a^?₁,?₂ to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.     

Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ωm to Ω=R×(−L,L), where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ωm→Ω as m→+∞. That is, let A and Am be the global attractors of the fluid corresponding to Ω and Ωm, respectively, we establish that for any neighborhood O(A) of A, the global attractor Am enters O(A) if m is large enough.     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Balanced Leonard pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A^∗ : V → V that satisfy the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Blow-up results for a class of first-order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u_t=Δu+a(x)|u|^p+λf(x) in R^N, t>0 include the non-existence results of positive global solutions obtained by Fujita and others when a≡1 and f≡0, Bandle-Levine and Levine-Meier when a≡|x|^m and f≡0, Pinsky when either f≡0 or f?0 and λ>0, Zhang and Bandle-Levine-Zhang when a≡1 and λ=1.     

The Hilbert-Chow morphism and the incidence divisor: Zero-cycles and divisors

For a smooth projective variety X of dimension n, on the product of Chow varieties C_a(X)×C_n−a−1(X) parameterizing pairs (A,B) of an a-cycle A and an (n−a−1)-cycle B in X, Barry Mazur raised the problem of constructing a Cartier divisor supported on the locus of pairs with A∩B≠0?. We introduce a new approach to this problem, and new techniques supporting this approach.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

Effects of reactive gradient term in a multi-nonlinear parabolic problem

This paper deals with parabolic equation ut=Δu+r|∇u|−aepu subject to nonlinear boundary flux ∂u/∂η=equ, where r>1, p,q,a>0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r?2, where the critical r=2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r>2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n, let R₁(A,n), R₂(A,n) and R₃(A,n) denote the number of solutions of a+a^′=n with the additional restriction a,a^′∈A; a,a^′∈A,a<a^′ and a,a^′∈A,a≤a^′ respectively. In this paper, we specially focus on the monotonicity of R₃(A,n). Moreover, we show that there does not exist any set A⊂N such that R₂(A,n) or R₃(A,n) is eventually strictly increasing.     

Well-posedness and weak rotation limit for the Ostrovsky equation

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0?a?−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L². To show this result, we prove a bilinear estimate which is uniform with respect to γ.     

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A∈R^n×n is called a bisymmetric matrix if its elements a_i,j satisfy the properties a_i,j=a_j,i and a_i,j=a_n-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.