In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability

A discrete frame for L2(Rd) is a countable sequence {ej}j∈J in L2(Rd) together with real constants 0

For a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the H1 convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The H1 convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials

For N≥3, by the seminal paper of Brezis and Véron (1980/81), no positive solutions of −Δu+uq=0 in RN∖{0} exist if q≥N/(N−2). In turn, for 11 and θ∈R, we prove that the elliptic problem (⋆) −Δu−λ|x|−2u+|x|θuq=0 in RN∖{0} with u>0 has a C1(RN∖{0}) solution if and only if λ>λ⁎, where λ⁎=Θ(N−2−Θ) with Θ=(θ+2)/(q−1). We show that (a) if λ>(N−2)2/4, then U0(x)=(λ−λ⁎)1/(q−1)|x|−Θ is the only solution of (⋆)

In this paper, we study the initial-boundary value problem for the coupled chemotaxis(-Navier)-Stokes system with indirect signal production{∂tn+u⋅∇n=Δn−∇⋅(n∇c)+rn−μn2,(x,t)∈Ω×(0,∞),∂tc+u⋅∇c=Δc−c+v,(x,t)∈Ω×(0,∞),∂tv+u⋅∇v=Δv−v+n,(x,t)∈Ω×(0,∞),∂tu+κ(u⋅∇)u+∇P=Δu+n∇Φ,∇⋅u=0,(x,t)∈Ω×(0,∞) in a smooth bounded domain Ω⊂Rd(d=2,3), where κ∈{0,1}, r≥0 and μ>0 are given constants. It is shown that when posed with

In this paper, we obtain the following extension of Bernstein inequality for polynomial differential operators: if 1≤p≤∞, K is an arbitrary compact set in R and P(x) is a polynomial, then there exists a constant C such that‖Pm(D)f‖p≤Cmsupx∈K⁡|Pm(x)|‖f‖p for all m∈N,p∈[1,∞] and all f∈Vp,K, where Vp,K={f∈Lp(R):suppfˆ⊂K} and fˆ is the Fourier transform of f. Further, we use Nikolskii's idea to get Bernstein

Starting from a discrete C⁎-dynamical system (A,θ,ωo), we define and study most of the main ergodic properties of the crossed product C⁎-dynamical system (A⋊αZ,Φθ,u,ωo∘E), E:A⋊αZ→A being the canonical conditional expectation of A⋊αZ onto A, provided α∈Aut(A) commute with the ⁎-automorphism θ up to a unitary u∈A. Here, Φθ,u∈Aut(A⋊αZ) can be considered as the fully noncommutative generalisation of the

The initial value problem for a fifth-order nonlinear dispersive partial differential equation describing the curve flow on the sphere is considered. A typical example of the equation arises in a hierarchy of completely integrable systems containing one-dimensional classical Heisenberg ferromagnetic spin model. This paper establishes the local existence and uniqueness of a solution to the initial value

This paper is concerned with the 3D isentropic compressible Euler equations with damping in periodic domain. Inspired by the theory of Markov semigroups, we show the existence of solution semiflow which satisfies the standard semigroup property and minimizes the energy (maximizes the energy dissipation) among all dissipative solutions.

We show the density of smooth Sobolev functions Wk,∞(Ω)∩C∞(Ω) in the Orlicz-Sobolev spaces Lk,Ψ(Ω) for bounded simply connected planar domains Ω and doubling Young functions Ψ.

In this paper we study the phase portraits in the Poincaré disk of the reversible vector fields of type (2;0) having generic bifurcations around a symmetric singular point p. We also prove the nonexistence of any periodic orbit surrounding p. We point out that some numerical computations were necessary in order to control the number of limit cycles.

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer

Financial fraud has great negative impact on capital markets. It misleads investors and the government and will hurt the development of the capital market in China. To detect corporate fraud, an artificial intelligence‐based method is proposed to evaluate the fraud risk. A new model for assessing the fraud risk of listed companies in China is put forward. The proposed approach collects multisource

We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz-type discretization result for these classes.

We study the boundary continuity of solutions to elliptic equations with Orlicz growth. We first formulate the Wiener criterion which characterizes a regular boundary point by a geometric quantity coming from capacities. Secondly, we develop an estimate for the modulus of continuity at a boundary point, under a geometric condition on operators. The proof relies on capturing the local properties of

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some L2 estimates of solutions, which improve those in the previous research [51]. Furthermore, asymptotic profiles of the solution and an approximate relation in

In this paper, we study steady two-dimensional periodic equatorial water waves of small amplitude which propagate over a flat bed and with a specified fixed mean-depth. In particular, we focus on irrotational flows and develop an equivalent formulation using flow force functions. The analysis is based on the local bifurcation theory due to Crandall-Rabinowitz.

This paper is concerned with a Lotka-Volterra model with nonlinear cross diffusion. The global existence of generalized solutions is established under some proper assumptions, and the nonexistence of nonconstant solutions is also investigated when diffusion rate is sufficiently large. At the same time, sufficient conditions ensuring the existence of non-constant solutions are obtained by using Leray-Schauder

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrödinger equation with combined power-type nonlinearities(FNLS)i∂tψ−(−Δ)sψ+a|ψ|p1ψ+|ψ|p2ψ=0. We prove that when p2=4sN and a(p1−4sN)<0, there exist the standing waves of (FNLS), which are orbitally stable. When a=0 and 4sN

This paper deals with the topological entropy for hom Markov shifts TM on d-tree. If M is a reducible adjacency matrix with q irreducible components M1,⋯,Mq, we show that h(TM)=max1≤i≤q⁡h(TMi) fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets {h(TM):M is binary and irreducible} and {h(TX):X is a one-sided shift} are not

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1325-1347, January 2021. This work develops novel rational Krylov methods for updating a large-scale matrix function $f(A)$ when $A$ is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1299-1324, January 2021. In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numerical schemes would generate spurious oscillations when the solution of the hyperbolic conservation laws contains discontinuities

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1273-1298, January 2021. We introduce a residual-based a posteriori error estimator for a novel $hp$-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust

SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2863-2889, January 2021. We propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with $L^p$-penalties applied to an ill-posed operator equation in a Banach space. Our analysis is built on the celebrated Littlewood--Paley theory and the concept of (Rademacher) R-boundedness. With these two analytical

SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2811-2862, January 2021. We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry

We find asymptotics of entries of Jacobi matrices with lacunary spectral data under some additional growth conditions. We also prove the inverse results. In addition, we study connections between Jacobi matrices, canonical systems and de Branges spaces for lacunary spectral data.

Ecologically, the evolutionary advantage of farmers over hunter-gatherers has led to their worldwide spreading. In this paper, we study a free boundary problem of a farmers and hunter-gatherers model featuring a type of prey-predator dynamics. The nonlinear diffusion of farmers is determined by the density of farmers in a way that higher population density accelerates the migration. We establish a

In this paper, we consider the pullback asymptotic behaviors of weak solutions for the 2D non-autonomous Non-Newtonian micropolar fluids. Employing the methods of ℓ-trajectories in [29], we first establish the existence of a finite dimensional pullback attractor and a pullback exponential attractor for the process {L(t,τ)}t≥τ associated with (1.2)-(1.4) in the phase space Xℓ. Then we construct a pullback

In this paper we investigate a class of two-species invasion models with a free boundary and with cross-diffusion and self-diffusion in interval [0,+∞), where the native species occupies the whole environment [0,+∞), and the invasion species invade into the habitat of the native species. The systems under consideration are strongly coupled, and the position of the free boundary is determined by the

Let f:Rn→R, n≥2, be a real analytic function. In this paper we study the stable set of the gradient flow x˙=∇f(x) at a critical point of f. In particular we present simple topological conditions which imply that this set contains an infinite family of trajectories, or has a non-empty interior.

In this paper, we study the following Schrödinger-Born-infeld system with a general nonlinearity{−Δu+u+ϕu=f(u)+μ|u|4uinR3,−div(∇ϕ1−|∇ϕ|2)=u2inR3,u(x)→0,ϕ(x)→0,asx→∞, where μ≥0 and f∈C(R,R) satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity

We develop a discrete counterpart of the De Giorgi–Nash–Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form |$-\nabla \cdot (A\nabla u)=f-\nabla \cdot F$| with |$A\in L^\infty (\varOmega ; {{\mathbb{R}}}^{n\times n})$| a uniformly elliptic matrix-valued function, |$f\in L^{q}(\varOmega

In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra B of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of B and identify several important subspaces in practical terms. This leads to new spectral mapping

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent

In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μM,D on R2 generated by an expanding integer matrix M∈M2(Z) with det⁡(M)∈3Z and a non-collinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t} with α1β2−α2β1∈3Z. We give the sufficient and necessary conditions for μM,D to be a spectral measure, i.e., there exists a countable subset Λ⊂R2 such that E(Λ)={e2πi〈λ

We investigate an isotropic thermoelastic system with dual-phase-lag heat conduction. We consider a homogeneous Dirichlet boundary condition for displacement and temperature. We formulate the variational formulation for the associated fully hyperbolic coupled system. By using a time discretization method, we show the existence of a unique weak solution under lower regularity assumptions on the data

In this paper the finite time stabilization of an infinite network of vibrating strings is studied. We consider a network with N∈N⁎,N≥2 finite edges and one infinite edge. This network is subject to homogeneous Dirichlet conditions at the endpoints of the finite edges. At the internal vertex 0, we put the continuity condition and a balanced damping condition of the form ∑j=0N(∂xuj(0,t)−∂tuj(0,t))=0

We define the Kelvin-Möbius transform of a function harmonic on the unit ball of Rn and determine harmonic function spaces that are invariant under this transform. When n≥3, in the category of Banach spaces, the minimal Kelvin-Möbius-invariant space is the Bergman-Besov space b−(1+n/2)1 and the maximal invariant space is the Bloch space b(n−2)/2∞. There exists a unique strictly Kelvin-Möbius-invariant

In the present note we are concerned with the study of curvature-based comparison theorems on graphs related to main stochastic properties, such as the Feller property and stochastic completeness. We show that, under our main hypothesis, whilst previous results concerning stochastic properties are improved, it is not possible to obtain comparison theorems concerning volume growth. Finally, we prove

In this paper, some Bohr-type inequalities are investigated for several classes of subordination. As a consequence, a series of known results on Bohr inequality can be deduced and unified in some simplified forms from the results.

This paper is concerned with a class of asymmetric information linear-quadratic (LQ) non-zero sum stochastic differential game problems. The dynamic of system is governed by a forward linear mean-field stochastic differential equation (MF-SDE), and the cost functional is quadratic. Motivated by some practical applications, the mean-field term of state process and control process are taken into account

We use an approximation scheme, sub-super solution and Perron's method to show the existence and nonexistence of positive solutions for a boundary value problem with a singular nonlinearity in the equation and on the boundary condition. We also study the asymptotic behavior of the solutions with respect to the parameters.

We study the uniform weighted resolvent estimates of Schrödinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find u∈V such that a(v,u)=l(v) for each v∈L, where V,L are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very

Abstract The action of the fractional integration operator in weighted Lebesgue classes and mixed-norm spaces is studied in the unit ball from \({\mathbb{R}}^{n}\). Some results of Hardy, Littlewood, and Flett are refined and generalized.

Abstract This paper discusses the location of zeros of polynomials in a polynomial sequence \(\{P_{n}(z)\}_{n=1}^{\infty}\) generated by a three-term recurrence relation of the form \(P_{n}(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0\) with \(k>2\) and the standard initial conditions \(P_{0}(z)=1\), \(P_{-1}(z)=P_{-k+1}(z)=0,\) where \(A(z)\) and \(B(z)\) are arbitrary coprime real polynomials. We show that

Abstract In the paper, we have exhaustively studied the uniqueness of meromorphic function sharing two values with its \(k\)th derivative counterpart. We have obtained a series of results each of which will improve and extend a number of existing results relevant to the content of the paper. We have also pointed out some gaps in one theorem in a paper due to Chen et al. (Pure Mathematics, 8(4), 378–382

Abstract We study uniqueness problems in terms of shared values or shared sets for a large class of entire functions representable as Dirichlet series in some right half-plane. In this article, we obtain a result that extends a recent result due to Oswald and Steuding [Ann. Univ. Sci. Budapest., Sect. Comput. 48 (2018), 117–128 (2018)]. Our result is also a variant of a result of Yuan, Li, and Yi [Lith

Abstract In this paper, we consider the entire solutions of nonlinear difference equation \(f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}\), where \(q\) is a polynomial, and \(p_{1},p_{2},\alpha_{1},\) and \(\alpha_{2}\) are nonzero constants with \(\alpha_{1}\neq\alpha_{2}\). It is showed that if \(f\) is a nonconstant entire solution of \(\rho_{2}(f)<1\) to the above equation, then

Abstract The following Artin theorem about alternative linear algebras defined on the commutative, associative ring with unity is well-known: in an alternative linear algebra, if \((a,b,c)=0\), then the subalgebra generated by the elements \(a\), \(b\), and \(c\) is associative. In this paper a wide generalization of this classical result is proposed using the concepts of hyperidentity and coidentity

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1245-1272, January 2021. This paper is the second part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we construct optimal $hp$-approximations for two classes of functions: (1) functions with general Sobolev regularity which lead to algebraic convergence rates using locally quasi-uniform

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1218-1244, January 2021. We study the conforming finite element spaces $\Sigma_{0}^{h,k}(\Omega) \subset H^2_0(\Omega)$, ${V}^{h,k}_0(\Omega) \subset {H}^1_0(\Omega)$, and $Q_{0}^{h,k}(\Omega) \subset L^2_0(\Omega)$ recently proposed by Falk and Neilan [SIAM J. Numer. Anal., 51 (2013), pp. 1308--1326] and establish necessary and sufficient

SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2776-2810, January 2021. In this paper, we are concerned with an epidemic reaction-diffusion system with nonlinear incidence mechanism of the form $S^qI^p\,(p,\,q>0)$. The coefficients of the system are spatially heterogeneous and time dependent (particularly time periodic). We first establish the $L^\infty$-bounds of the solutions of

The Pearcey process is a universal point process in random matrix theory and depends on a parameter ρ∈ℝ. Let N(x) be the random variable that counts the number of points in this process that fall in the interval [−x,x]. In this note, we establish the following global rigidity upper bound: lims→∞ℙsupx>sN(x)−334πx43−3ρ2πx23logx≤423π+𝜖=1, where 𝜖>0 is arbitrary. We also obtain a similar upper bound

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of

Let MΣ be an n-dimensional Thom–Mather stratified space of depth 1. We denote by βM the singular locus and by L the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms

We relate the quantum Steenrod square to Seidel’s equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in ℤ/2-equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant

We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency-dependent power law. First we prove the existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time-stepping scheme based on the finite element method in space