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Abstract In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ = σ (x) and the inner pressure p = p (x). Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20]. In this paper we study existence of nonradial solutions by using the bifurcation method. Let { γ k } k = 2 ∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k ⁎ ⩾ 2 such that in the two dimension case for any k ⩾ k ⁎ , γ k is a bifurcation point, and in the three dimension case for any even k ⩾ k ⁎ , γ k is also a bifurcation point. [ABSTRACT FROM AUTHOR]

This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

In this paper we investigate boundary blow-up solutions of the problem { − Δ p ( x ) u + f ( x , u ) = ± K ( x ) | ∇ u | m ( x ) in Ω , u ( x ) → + ∞ as d ( x , ∂ Ω ) → 0 , where Δ p ( x ) u = div ( | ∇ u | p ( x ) − 2 ∇ u ) is called the p ( x ) -Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that K ( x ) | ∇ u ( x ) | m ( x ) is a small perturbation, to the case in which ± K ( x ) | ∇ u | m ( x ) is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d ( x , ∂ Ω ) and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since f ( x , ⋅ ) is not assumed to be monotone in this paper. [ABSTRACT FROM AUTHOR]

Suppose G is a compact Lie group, H is a closed subgroup of G , and the homogeneous space G / H is connected. The paper investigates the Ricci flow on a manifold M diffeomorphic to [ 0 , 1 ] × G / H . First, we prove a short-time existence and uniqueness theorem for a G -invariant solution g ( t ) satisfying the boundary condition II ( g ( t ) ) = F ( t , g ∂ M ( t ) ) and the initial condition g ( 0 ) = g ˆ . Here, II ( g ( t ) ) is the second fundamental form of ∂ M , g ∂ M is the metric induced on ∂ M by g ( t ) , F is a smooth map and g ˆ is a metric on M . Second, we study Perelman's F -functional on M . Our results show, roughly speaking, that F is non-decreasing on a G -invariant solution to the modified Ricci flow, provided that this solution satisfies boundary conditions inspired by a paper of Gianniotis. [ABSTRACT FROM AUTHOR]

This paper addresses an investigation of the asymptotic behaviour as ε → 0 of the solution to the boundary value problem associated with the p -Laplace operator in an ε -periodically structured domain with a nonlinear Robin-type condition specified on the boundary of the periodic subdomains. This kind of domains include the so called perforated media as well as the case of isolated particles distributed in a periodic way. This second case arises quite often in Chemical Engineering. Here we consider a non-critical size of the particles. The objective of this paper is twofold. First we study the homogenization of solutions in the case of a continuous nonlinear reaction term on the boundary of the periodic structure. Then, we move to studying the homogenization of the effectiveness factor of the reactor, which is of relevance in Chemical Engineering. [ABSTRACT FROM AUTHOR]

This note concerns the sufficient condition for regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin's condition. H.-O. Bae and H.J. Choe proved in a 1997 paper that, in the whole space R 3 , it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case R + n under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry is not essential, as shown in a forthcoming paper in cylindrical domains, prepared in collaboration with J. Bemelmans and J. Brand. [ABSTRACT FROM AUTHOR]

In this paper by using tools from bifurcation theory, particularly the Crandall–Rabinowitz bifurcation theorem, the Implicit Function Theorem and the persistence results of degenerate solutions, due to E.N. Dancer, a result will be proved for a S -shaped bifurcation curve of a nonlocal boundary value problem. The main results of this paper partially extend the main results in [11,12] to the nonlocal boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a maximization and a minimization problem associated with a Poisson boundary value problem. Optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions. The main contribution of this paper is to determine the optimal solutions. At first, we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime. Secondly, for the high contrast regime, two optimization algorithms are developed. For the minimization problem, we prove that our algorithm converges to the global minimizer regardless of the initializer. The maximization algorithm is capable of deriving all local maximizers including the global one. Numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function. [ABSTRACT FROM AUTHOR]

Let Γ = def { ( z + w , z w ) : | z | ≤ 1 , | w | ≤ 1 } ⊂ C 2 . A Γ -inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary b Γ of Γ. A rational Γ-inner function h induces a continuous map h | T from T to b Γ. The latter set is topologically a Möbius band and so has fundamental group Z . The degree of h is defined to be the topological degree of h | T . In a previous paper the authors showed that if h = ( s , p ) is a rational Γ-inner function of degree n then s 2 − 4 p has exactly n zeros in the closed unit disc D − , counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s 2 − 4 p prescribed. [ABSTRACT FROM AUTHOR]

This paper deals with an optimal control problem related to a phase field system of Caginalp type with a dynamic boundary condition for the temperature. The control placed in the dynamic boundary condition acts on a part of the boundary. The analysis carried out in this paper proves the existence of an optimal control for a general class of potentials, possibly singular. The study includes potentials for which the derivatives may not exist, these being replaced by well-defined subdifferentials. Under some stronger assumptions on the structure parameters and on the potentials (namely for the regular and the logarithmic case having single-valued derivatives), the first order necessary optimality conditions are derived and expressed in terms of the boundary trace of the first adjoint variable. [ABSTRACT FROM AUTHOR]

This paper deals with the higher dimension quasilinear parabolic–parabolic Keller–Segel system involving a source term of logistic type u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + g ( u ) , τ v t = Δ v − v + u in Ω × ( 0 , T ) , subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in R n , n ≥ 2 , ϕ and g are smooth and positive functions satisfying k s p ≤ ϕ when s ≥ s 0 > 1 , g ( s ) ≤ a s − μ s 2 for s > 0 with g ( 0 ) ≥ 0 and constants a ≥ 0 , τ , χ , μ > 0 . It is known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent 2 n . On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients associated. In the present paper it is proved that there is θ 0 > 0 such that the problem admits global bounded classical solutions whenever χ μ < θ 0 , regardless of the size of initial data and diffusion. This shows the substantial effect of the logistic source has on the behavior of solutions. [ABSTRACT FROM AUTHOR]

Let B denote the unit ball in RN with boundary S. For a non-negative C2 subharmonic function f on B and ∈S, we define the Lusin square area integral Sα(,f) by Sα(f)=[Γα(1-|x|)2-NΔf2(x)dx]12,where for α>1, Γα={x∈B:|x-|<α(1-|x|)} is the non-tangential approach region at ∈S, and Δ is the Laplacian in RN. In the paper we will prove the following: Let f be a non-negative subharmonic function such thatfpois subharmonic for somepo>0. Iffpp=sup0po, then for everyα>1,Sα(•,f)p≤Aα,pfpfor some constantAα,pindependent of f. The above result includes the known results for harmonic or holomorphic functions in the Hardy Hp spaces, as well as for a system F=(u1,...,uN) of conjugate harmonic functions for which it is known that |F|p=(Σuj2)p/2 is subharmonic for p≥(N-2)/(N-1),N≥3. We also consider analogues of the functions g and g\* of Littlewood-Paley, and introduce the function gλ\*, λ>1, defined bygλ\*(,f)=[B(1-|y|)Δf2(y)Kλ(y,)dy]12,whereKλ(y,)=(1-|y|)(λ-1)(N-1)|y-|λ(N-1). In the paper we prove that the inequality gλ\*(•,f)p≤Cpfp holds for all λ≥N/(N-1) when p≥2, and for λ>3-p whenever 1(2N-3)/(N-1). [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to investigate a parabolic problem with coupled reaction terms and free boundary, including the existence, uniqueness, regularity and long-time behavior of positive solutions. Firstly, by the contraction mapping theorem, we establish the (local) existence and uniqueness of positive solutions. Then, we prove the solution blows up in finite time with large initial data by comparison principle, and give more details for these large initial data by exhibiting an energy condition. The simultaneous blow-up result of the maximum of u and v is obtained, and the blow-up set of blow-up solutions is a compact subset of [ 0 , h 0 ]. Furthermore, there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small, while there is a global and slow solution provided that the initial datum is suitably large. Finally, for initial data σ (φ (x) , ψ (x)) , we obtain a trichotomy conclusion by considering the size of parameter σ. [ABSTRACT FROM AUTHOR]

This paper deals with an attraction–repulsion chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) , x ∈ Ω , t > 0 , τ 1 v t = Δ v + α u − β v , x ∈ Ω , t > 0 , τ 2 w t = Δ w + γ u − δ w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N ( N ≥ 2 ), where parameters τ i ( i = 1 , 2 ) , χ , ξ , α , β , γ and δ are positive, and diffusion coefficient D ( u ) ∈ C 2 ( 0 , + ∞ ) satisfies D ( u ) > 0 for u ≥ 0 , D ( u ) ≥ d u m − 1 with d > 0 and m ≥ 1 for all u > 0 . It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution for m > 2 − 2 N . In particular in the case τ 1 = τ 2 and χ α = ξ γ , the solution is globally bounded if m > 2 − 2 N − N + 2 N 2 − N + 2 . Therefore, due to the inhibition of repulsion to the attraction, the range of m > 2 − 2 N of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction–repulsion chemotaxis system with nonlinear diffusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the evolution of the periodic boundary value problem and the mixed boundary value problem for a compressible mixture of binary fluids modeled by the Navier–Stokes–Cahn–Hilliard system in one dimensional space. The global existence and the large time behavior of the strong solutions for these two systems are studied. The solutions are proved to be asymptotically stable even for the large initial disturbance of the density and the large velocity data. We show that the average concentration difference for the two components of the initial state determines the long time behavior of the diffusive interface for the two-phase flow. [ABSTRACT FROM AUTHOR]

We discuss a free boundary problem for a reaction–diffusion equation with Dirichlet boundary conditions on both fixed and free boundaries of a one-dimensional interval. The problem was proposed by Du and Lin (2010) to model the spreading of an invasive or new species by putting Neumann boundary condition on the fixed boundary. Asymptotic properties of spreading solutions for such problems have been investigated in detail by Du and Lou (2015) and Du, Matsuzawa and Zhou (2014). The authors (2011) studied a free boundary problem with Dirichlet boundary condition. In this paper we will derive sharp asymptotic properties of spreading solutions to the free boundary problem in the Dirichlet case under general conditions on f . It will be shown that the spreading speed is asymptotically constant and determined by a semi-wave problem and that the solution converges to a semi-wave near the spreading front as t → ∞ provided that the semi-wave problem has a unique solution. [ABSTRACT FROM AUTHOR]

In this paper we solve the problem of the identification of a coefficient which appears in the equation of a distributed system with persistent memory. This system is encountered for example when modelling the interconnection of linear elastic and viscoelastic systems. The additional data used in the identification are subsumed in the input output map from the deformation to the traction on the boundary. We extend a dynamical approach to identification introduced by Belishev in the case of purely elastic (memoryless) bodies and based on a special equation due to Blagoveshchenskiı̌. So, in particular, we extend Blagoveshchenskiı̌ equation to our class of systems with persistent memory. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of nodal solutions for the following fractional Choquard equation ( − △ ) α u + u = ∫ R N | u ( z ) | p | x − z | μ d z | u ( x ) | p − 2 u ( x ) , x ∈ R N , where 0 < μ < 2 α < N and 2 N − μ N − 1 < p < 2 N − μ N − 2 α with α ∈ ( 1 2 , 1 ) . In view of the nonlocality of the fractional Laplacian operator and the so-called Hartree term ∫ R N | u ( z ) | p | x − z | μ d z | u ( x ) | p − 2 u ( x ) , the corresponding variational functional has entirely different properties with respect to the Laplacian case. For any k ∈ N , we prove that the problem possesses at least a radially symmetrical solution which changes sign k times. [ABSTRACT FROM AUTHOR]

This paper is concerned with an evolution problem having an elliptic equation involving the 1-Laplacian operator and a dynamical boundary condition. We apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle. Our main theorem shows that the solution we found is actually a strong solution. We also compare solutions with different data. [ABSTRACT FROM AUTHOR]

We consider a thin-film equation. By exploiting the boundary condition and the variational structure of the equation, we consider the global existence and blow-up of the solutions. For the global solutions, we study the asymptotic behavior, and for the blow-up solutions, we study the upper and lower bounds of the blow-up time. Especially, we have derived the necessary and sufficient conditions for the solution blowing up in finite time when the initial energy J ( u 0 ) ≤ d , where d is the mountain-pass level. For J ( u 0 ) > d , we also study the conditions for the solution exists globally and blows up in finite time, and we also obtain some necessary and sufficient conditions for the solution blowing up in finite time when the initial energy at arbitrary level. Furthermore, we study the decay behavior of both the global solutions and its corresponding energy functional, and the decay ratios are given specially. The results generalize the former studies on this equation, such as the papers: Li et al. (2016) [16] , Dong and J. Zhou (2017) [4] and Sun et al. (2018) [23] . [ABSTRACT FROM AUTHOR]

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising measure-consistent multivariate orthonormal polynomials in dependent random variables. Unlike existing PCEs, whether classical or generalized, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PCE are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, exploiting whitening transformation for generating orthonormal polynomial bases, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE for a general output variable in terms of the expansion coefficients. An example derived from a stochastic boundary-value problem illustrates the generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new global Carleman estimates for the linear stochastic Kuramoto–Sivashinsky equations. The first one is for the backward linear stochastic Kuramoto–Sivashinsky equation. Based on this estimate and the duality argument, we obtain the null controllability of the forward linear stochastic Kuramoto–Sivashinsky equation by three controls, one is an internal control in the diffusion term and the others are boundary controls. The second one is for the forward linear stochastic Kuramoto–Sivashinsky equation. According to this estimate, we obtain a unique continuation property for the forward linear stochastic Kuramoto–Sivashinsky equation. [ABSTRACT FROM AUTHOR]

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolution–supersolution, together with truncation and comparison techniques. [ABSTRACT FROM AUTHOR]

This paper is concerned with an initial–boundary value problem of the incompressible nematic liquid crystal flows with density-dependent viscosity in a smooth bounded domain Ω ⊂ R 3 . The global well-posedness of strong solutions with large oscillations is established in vacuum, provided ‖ ∇ u 0 ‖ L 2 + ‖ Δ d 0 ‖ L 2 is suitably small with arbitrary large initial density, which extended the local strong solution by Gao et al. [12] to be a global one. [ABSTRACT FROM AUTHOR]

This paper studies the nonisotropic chaotic oscillations of the initial-boundary value problem of one-dimensional wave equation with a mixing transport term. It separately considers that the boundary condition at the right-end of the wave equation is a superlinear type and linear perturbation of such type, each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term. For each type of boundary condition, the occurrence of nonisotropic chaotic oscillations is rigorously proved. Numerical examples verify the effectiveness of theoretical prediction. [ABSTRACT FROM AUTHOR]

The inverse spectral problem for a Sturm–Liouville problem which consists of a Sturm–Liouville equation defined on the interval [ 0 , a ] and two separated boundary conditions at the endpoints 0 and a , one of which is involved function f ( λ ) , is considered in this paper. When f ( λ ) is the type of Herglotz functions and is known as a priori, we prove that the potential on [ 0 , a ] and the other boundary condition can be uniquely determined in terms of appropriate partial information on the spectrum and partial information on the set of normalizing constants. [ABSTRACT FROM AUTHOR]

In this paper we analyze the asymptotic behavior of the pullback attractors of a non-autonomous damped wave equation when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter ε goes to zero. We prove a result of regularity of the pullback attractors and that this family of attractors is upper semicontinuous at ε = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the nonexistences of positive solutions for the following two mixed boundary value problems. The first problem is the nonlinear-Neumann mixed boundary value problem { − Δ u = f ( u ) in R + N , ∂ u ∂ ν = g ( u ) on Γ 1 , ∂ u ∂ ν = 0 on Γ 0 and the second is the nonlinear-Dirichlet mixed boundary value problem { − Δ u = f ( u ) in R + N , ∂ u ∂ ν = g ( u ) on Γ 1 , u = 0 on Γ 0 , where N ≥ 3 , R + N = { x ∈ R N : x N > 0 } , Γ 1 = { x ∈ R N : x N = 0 , x 1 < 0 } and Γ 0 = { x ∈ R N : x N = 0 , x 1 > 0 } . We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main method we use is the moving plane method in an integral form. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cubic fourth-order nonlinear Schrödinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in H s ( T ) with − 1 / 3 ≤ s < 0 for the Cauchy problem of the Wick ordered 4NLS. The other one is the non-squeezing property for the flow map of 4NLS in the symplectic phase space L 2 ( T ) . To prove the former we used the ideas introduced in [36] and [27] , and to prove the latter we used the ideas in [8] . [ABSTRACT FROM AUTHOR]

In this paper the linear theory of triple porosity thermoelasticity is considered and the basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, Green's identities are obtained and the Sommerfeld–Kupradze type radiation conditions are established. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are given. The BVPs are reduced to the always solvable singular integral equations for which Fredholm's theorems are valid. Finally, the existence and uniqueness theorems for classical solutions of the basic BVPs of steady vibrations are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dependence on initial data of solutions to the higher dimensional Camassa–Holm equations with periodic boundary condition in Besov spaces. We show that when s > 1 + d 2 ( d ≥ 2 ) and 1 ≤ r ≤ ∞ , the solution map is not uniformly continuous from B 2 , r s ( T d ) into C ( [ 0 , T ] ; B 2 , r s ( T d ) ) for r < ∞ or from B 2 , ∞ s ( T d ) into L ∞ ( [ 0 , T ] ; B 2 , ∞ s ( T d ) ) for r = ∞ . [ABSTRACT FROM AUTHOR]

The Berry heuristic has been a long standing ansatz about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see [7] ). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace–Beltrami operator on the normalized d -dimensional sphere – also known as spherical harmonics). Some applications to non-Gaussian models are also discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with an initial–boundary value problem for screw pinches arisen from plasma physics. A global-in-time, classical solution to this physically very important problem, is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ . The main difficulties in the proof lie in the temperature dependent magnetic resistive coefficients induced by the magnetic field. The initial data can be large if γ is sufficiently close to 1. [ABSTRACT FROM AUTHOR]

The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L 2 as well as in C 1 under small bounded perturbations. To show this for C 1 , we prove a general smoothing result implying that the solutions to the perturbed problems become eventually C 1 -smooth for any L 2 -initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider the 3D heat-conductive Boussinesq equations with a slip boundary condition for the velocity field and Neumann boundary condition for the temperature. We prove that there exists an interval of time which is uniform for the heat conductivity coefficient κ and for which the 3D heat-conductive Boussinesq equations have a strong solution. The solution is uniformly bounded in some spaces with respect to κ . Based on these uniform estimates, we establish the vanishing diffusivity limit for the Boussinesq equations and also obtain the convergence rates for the velocity field and the temperature. [ABSTRACT FROM AUTHOR]

This paper concerns about the two-dimensional direct scattering problem for Schrödinger equation with a class of decaying potential under the tapered wave incidence. We focus on the numerical solution of the problem by using the perfectly matched layer (PML) method. Based on the idea of complex coordinate stretching, the boundary value problem for the PML equation is formulated. Then the variational equation to the boundary value problem is proved to exist a unique solution except possibly for a set of values of wavenumber. The numerical experiments illustrate the influence of the thickness of PML layer, the absorption parameter, the wavenumber, the control bandwidth and even the incidence angle of the tapered wave on the effectiveness of this method. [ABSTRACT FROM AUTHOR]

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed. [ABSTRACT FROM AUTHOR]

In this paper we introduce the hybrid-penalized method, to show the global existence of at least one solution to the Signorini's problem for a Timoshenko Beam. The main advantage of this method is that we can improve all result about asymptotic behaviour to contact problem to any boundary conditions, which is not possible with multiplicative techniques. Moreover this method also allows us to show polynomial stability to partial dissipative models. Finally, we performed numerical experiments to highlight our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for a class of thin-film equations in R n with a p -Laplace term and a nonlocal source term | u | q − 2 u − 1 | Ω | ∫ Ω | u | q − 2 u dx . We prove that there exist weak solutions for the problem with arbitrarily initial energy that blow up in finite time. We also obtain the upper bounds for the blow-up time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Sturm–Liouville operator L generated in L 2 ( R + , H ) by the differential expression L ( Y ) = − Y ″ + Q ( x ) Y , 0 < x < ∞ , with the boundary condition Y ( 0 ) = 0 , where Q ( x ) is a non-selfadjoint, completely continuous operator in H for every x > 0 . Here H is a separable Hilbert space, B ( H ) denotes the space of bounded operators in H and L 2 ( R + , H ) denotes the space of square-integrable, strongly-measurable vector-valued functions defined on ( 0 , ∞ ) . In particular, we find some special solutions of the equation L ( Y ) = λ 2 Y including Jost solution, then investigate the point spectrum of L under certain conditions on Q ( x ) . We obtain the resolvent of L , if Q ( x ) is quasi-selfadjoint i.e. there exists P ∈ B ( H ) such that P − 1 ∈ B ( H ) , P is positive and Q ⁎ ( x ) = P Q ( x ) P − 1 for every x > 0 . We also show that L has a finite number of spectral singularities. [ABSTRACT FROM AUTHOR]

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger–Poisson problem { − ε 2 Δ u + V ( x ) u + Φ u = u p , x ∈ R N , − Δ Φ = u 2 , x ∈ R N , where ε > 0 is a small parameter, N ≥ 3 , and V ( x ) is a potential function. This system describes some physical phenomena such as quantum mechanics models. We construct non-radial positive solutions, whose components may have spikes clustering at the same point as ε → 0 + , but the distance between them divided by ε will go to infinity. [ABSTRACT FROM AUTHOR]

This paper deals with the parabolic–elliptic–elliptic attraction–repulsion chemotaxis system with logistic source { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( χ u ∇ v ) + ∇ ⋅ ( ξ u ∇ w ) + r u − μ u 2 , x ∈ Ω , t > 0 , 0 = Δ v + α u − β v , x ∈ Ω , t > 0 , 0 = Δ w + γ u − δ w , x ∈ Ω , t > 0 , under no-flux boundary conditions in bounded domain with smooth boundary, where χ , ξ , α , β , γ , δ , r and μ are assumed to be positive. When Ω ⊆ R 3 , D ( u ) is assumed to satisfy D ( 0 ) > 0 , D ( u ) ≥ c D u m − 1 with m ≥ 1 and c D > 0 , it is proved that if χ α − ξ γ > 0 and μ = 1 3 ( χ α − ξ γ ) , then for any given u 0 ∈ W 1 , ∞ ( Ω ) , the system possesses a global and bounded classical solution. For the case where D ( u ) ≡ 1 and n ≥ 3 , the convergence rate of the solution is established. When the random motion of the chemotactic species is neglected i.e. ( D ( u ) ≡ 0 ) and Ω ⊂ R n ( n ≥ 2 ) is a convex domain, boundedness and the finite time blow up of the solution are investigated. [ABSTRACT FROM AUTHOR]

In this article, by introducing a new set and a new concept of φ − ( h , e ) -concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ − ( h , e ) -concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems. [ABSTRACT FROM AUTHOR]

In this paper we study a Neumann problem with non-homogeneous boundary conditions for the p ( x ) -Laplacian. In particular we assume that p ( ⋅ ) is a step function defined in a domain Ω and equals to 1 in a subdomain Ω 1 and 2 in its complementary Ω 2 . By considering a suitable sequence p k of variable exponents such that p k → p and replacing p with p k in the original problem, we prove the existence of a solution u k for each of those intermediate ones. We also show, that under a hypothesis concerning the boundary data g , the limit of the sequence ( u k ) is a function u , which belongs to the space of functions of bounded variation and is a solution to the original p ( ⋅ ) -problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of boundary value problems for p -Laplacian equation ( Φ p ( u ′ ) ) ′ + h ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 with integral boundary conditions u ( 0 ) − α u ′ ( 0 ) = ∫ 0 1 g 1 ( s ) u ( s ) d s , u ( 1 ) + β u ′ ( 1 ) = ∫ 0 1 g 2 ( s ) u ( s ) d s . By using the Avery–Peterson fixed point theorem, we obtain the existence of at least three positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-reflecting boundary condition for the time-harmonic Maxwell's equations in homogeneous waveguides with an inhomogeneous inclusion. We analyze a series representation of solutions to the Maxwell's equations satisfying the radiating condition at infinity, from which we develop the so-called electric-to-magnetic operator for the non-reflecting boundary condition. Infinite waveguides are truncated to a finite domain with a fictitious boundary on which the non-reflecting boundary condition based on the electric-to-magnetic operator is imposed. As the main goal, the well-posedness of the reduced problem will be proved. This study is important to develop numerical techniques of accurate absorbing boundary conditions for electromagnetic wave propagation in waveguides. [ABSTRACT FROM AUTHOR]

Recently, Leray's problem of the L 2 -decay of a special weak solution to the Navier–Stokes equations with nonhomogeneous boundary values was studied by the authors, exploiting properties of the approximate solutions converging to this solution. In this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong energy inequality. [ABSTRACT FROM AUTHOR]

In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator L h by using operator theoretic formulation. The self-adjoint dilation T h of L h in the space H is constructed, furthermore, the incoming and outgoing representations of T h and functional model are also constructed, hence in light of Lax–Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved. [ABSTRACT FROM AUTHOR]