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Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

Abstract: The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians in with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains , , . Second, we extend Friedlander''s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems. [Copyright &y& Elsevier]

Abstract: We compare the above-mentioned article with the content of a previous publication. [Copyright &y& Elsevier]

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]

The aim of this paper is to study the problem { u t t − Δ u + P ( x , u t ) = f ( x , u ) in ( 0 , ∞ ) × Ω , u = 0 on ( 0 , ∞ ) × Γ 0 , u t t + ∂ ν u − Δ Γ u + Q ( x , u t ) = g ( x , u ) on ( 0 , ∞ ) × Γ 1 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω ‾ , where Ω is a bounded open subset of R N with C 1 boundary ( N ≥ 2 ), Γ = ∂ Ω , Γ 1 is relatively open on Γ, Δ Γ denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear perturbations. In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical. [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 study a free boundary problem modeling the growth of vascularized tumors. The model is a modification to the Byrne–Chaplain tumor model that has been intensively studied during the past two decades. The modification is made by replacing the Dirichlet boundary value condition with the Robin condition, which causes some new difficulties in making rigorous analysis of the model, particularly on existence and uniqueness of a radial stationary solution. In this paper we successfully solve this problem. We prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient, whereas unstable for small surface tension coefficient. Tools used in this analysis are the geometric theory of abstract parabolic differential equations in Banach spaces and spectral analysis of the linearized operator. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses the question of the impact of the boundary on the dynamical behaviour of finite Boolean automata networks on Z2. The evolution over discrete time of such networks is governed by a specific stochastic threshold non-linear transition rule derived from the classical rule of formal neural networks. More precisely, the networks considered in this paper are finite but the study is done for arbitrarily large sizes. Moreover, the boundary impact is viewed as a classical definition of a phase transition in probability theory, characterising in our context the fact that a network admits distinct asymptotic behaviours when different boundary instances are assumed. The main contribution of this paper is the highlight of a formula for a necessary condition for boundary sensitivity, whose sufficiency and necessity are entirely proven with natural constraints on interaction potentials. [ABSTRACT FROM AUTHOR]

It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [13] . The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [10] . In this paper, we confirm that the other two are also area-minimizing. In fact, we show that every Lawson–Osserman cone of type ( n , p , 2 ) constructed in [26] is area-minimizing. [ABSTRACT FROM AUTHOR]

In the paper a numerical approach to the four equation model of water hammer (WH) with fluid–structure interaction (FSI) is presented. An algorithm for numerical solution of that model in time domain based on the method of characteristics (MOC) is proposed. Special attention is paid to modeling of the influence of viscoelastic pipe supports. The boundary condition at the support is formulated as a differential equation of junction motion which is solved numerically concurrently with the MOC compatibility equations. Numerical simulations were done with the use of an own computer program. The tests were conducted for a model of real pipeline built in the lab and fastened with specific, complex, supporting system. The numerical results were compared to the experimental records and after some modeling and calibration quite a good agreement was achieved. Basic behaviors in the pressure records were identified and existed discrepancies were discussed and explained. The other physical model tested and preliminary analyzed in the paper is a straight pipeline fixed to the floor only with viscoelastic supports. Such a design created the possibility of significant displacements of the pipeline on the supports and thus allowing for effective testing of the influence of supports stiffness and damping properties on the WH behaviors. The specific initial conditions for that case were formulated, solved and implemented in a computer code. It was found that proper selection of support parameters may produce significant reduction of WH pressure amplitudes, mainly due to effective energy absorption and dissipation at the supports. [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]

In this paper, we study a free boundary model describing growth of tumors under action of drugs. To our knowledge, in theoretical discussion for free boundary problems, the proliferation rate in tumor models discussed in previous bifurcation results is a linear function of nutrients and inhibitors. Whereas in this paper we consider the net proliferation rate as a nonlinear function depending on both nutrients and drugs. First, the existence and the uniqueness of radially symmetric stationary solutions are obtained. Second, we prove that symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions when the concentration of drug on the boundary of tumor is less than one in the rescaled model. [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 paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f = f ( t , x , u ) , the distributed damping Q = Q ( t , x , u , u t ) , the perturbation term μ | u | p − 2 u and the dissipative term ϱ ( t ) M ( [ u ] s p ) | u t | p − 2 u t , with ϱ ≥ 0 and in L loc 1 ( R 0 + ) , when the initial data are in a special region. Here u = ( u 1 , … , u N ) = u ( t , x ) represents the vectorial displacement, with N ≥ 1 . Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5 . Finally, the results are extended to problems where the fractional p -Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero. [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]

This paper deals with the initial-boundary value problem (P) for the nonlinear diffusion equation ∂ u ∂ t + ( − Δ + 1 ) β ( u ) = g in Ω × ( 0 , T ) in a general domain Ω ⊂ R N with smooth bounded boundary, where N ∈ N , T > 0 and β is a single-valued maximal monotone function on R , e.g., β ( r ) = | r | q − 1 r ( q > 0 ) . The above equation represents a number of well-known models, e.g., porous media equation. Colli and Fukao [6] studied the above problem (P) and the approximate problem ( P ) ε , which consists of the Cahn–Hilliard system, with error estimates when Ω is a bounded domain, N = 2 or 3 and − Δ + 1 is replaced with −Δ. They considered one more approximation ( P ) ε , λ to solve ( P ) ε . They used compactness methods for abstract doubly nonlinear evolution equations to solve the second approximate problem (P) ε , λ . They established the error estimate saying that the solution of ( P ) ε converges to the solution of (P) as ε ↘ 0 . The present work asserts that one can solve the original problem (P) and the approximate problem ( P ) ε individually and directly even if Ω is a general domain. Moreover, this paper gives an error estimate between the solution of ( P ) ε and the solution of (P). [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary value problem for the one-dimensional non-isentropic Euler-Poisson system. The physical subsonic inflow and outflow conditions are prescribed at boundaries. We prove the existence of multi-transonic solutions, i.e., the existence of subsonic-sonic-supersonic-shock-subsonic flows in fact nozzles. Furthermore, the monotonicity between the location of the transonic shock and the density at the exit of the nozzle will be elucidated. [ABSTRACT FROM AUTHOR]

In this paper, we study some overdetermined boundary value problems for anisotropic elliptic PDEs in a bounded domain Ω in a convex cone of R n. By using some integral identities and maximum principle, we prove the corresponding Wulff shape characterizations, which includes the classical Serrin's overdetermined boundary value problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the structure of sonic-supersonic solutions near sonic curves arising from the transonic channel flow problems in gas dynamics. Given two smooth curves, one of which is a sonic curve and the other is a characteristic curve, we construct a local classical supersonic solution for the two-dimensional steady Euler equations in the angular region. A novel set of dependent and independent variables are introduced to transform the Euler equations into a linear system with explicitly singularity-regularity structures. We use the iteration method to establish the existence and uniqueness of smooth solutions for the new system and then express the solution in terms of the original variables. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity properties of the “good” Boussinesq equation on the half line. We obtain local existence, uniqueness, and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [17] , being the first result that constructs solutions for the initial and boundary value problem of the “good” Boussinesq equation below the L 2 space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in [20] and [13] . [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial–boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data ( v 0 , T 0 ) ∈ H 1 , by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data ( v 0 , T 0 ) ∈ H 1 ∩ L ∞ , such that ∂ z v 0 ∈ L m , for some m ∈ ( 2 , ∞ ) , by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in Cao et al. (2016) [10] , where the initial data ( v 0 , T 0 ) was assumed to have H 2 regularity. [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]

This paper is concerned with the following Keller–Segel–Navier–Stokes system (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + κ ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω ⊂ R 3 is a bounded domain with smooth boundary ∂ Ω , κ ∈ R and S denotes a given tensor-valued function fulfilling | S ( x , n , c ) | ≤ C S ( 1 + n ) α with some C S > 0 and α > 0 . As the case κ = 0 has been considered in [25] , it is shown in the present paper that the corresponding initial–boundary problem with κ ≠ 0 admits at least one global weak solution if α ≥ 3 7 . [ABSTRACT FROM AUTHOR]

We consider the boundary value problem ( P λ ) − Δ u = λ c ( x ) u + μ ( x ) | ∇ u | 2 + h ( x ) , u ∈ H 0 1 ( Ω ) ∩ L ∞ ( Ω ) , where Ω ⊂ R N , N ≥ 3 is a bounded domain with smooth boundary. It is assumed that c , h belong to L p ( Ω ) for some p > N with c ≩ 0 as well as μ ∈ L ∞ ( Ω ) and μ ≥ μ 1 > 0 for some μ 1 ∈ R . It is known that when λ ≤ 0 , problem ( P λ ) has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of ( P λ ) assuming that λ > 0 . Our study unveils the rich structure of this problem. We show, in particular, that what happen for λ = 0 influences the set of solutions in all the half-space ] 0 , + ∞ [ × ( H 0 1 ( Ω ) ∩ L ∞ ( Ω ) ) . Most of our results are valid without assuming that h has a sign. If we require h to have a sign, we observe that the set of solutions differs completely for h ≩ 0 and h ≨ 0 . We also show when h has a sign that solutions not having this sign may exists. Some uniqueness results of signed solutions are also derived. The paper ends with a list of open 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]

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , ( x , t ) ∈ Ω × ( 0 , T ) , c t + u ⋅ ∇ c = Δ c − c + n , ( x , t ) ∈ Ω × ( 0 , T ) , u t + ∇ P = Δ u + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , where Ω ⊂ R d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R d × d , and satisfies | S ( x , n , c ) | ≤ C S ( 1 + n ) − α with some C S > 0 and α > 0 . When d = 2 , our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α > 0 , which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3 , we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α > 1 2 , which is slightly stronger than the corresponding subcritical assumption α > 1 3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L p regularity of u from a rather low L p regularity of n , which will be obtained by a new combinational functional. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis system { u t = Δ u − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u ) in Ω × ( 0 , ∞ ) , v t = Δ v − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − v ) in Ω × ( 0 , ∞ ) , w t = d Δ w + h ( u , v , w ) in Ω × ( 0 , ∞ ) , where Ω is a bounded domain in R n with smooth boundary ∂Ω, n ∈ N ; h , χ i are functions satisfying some conditions. Negreanu–Tello (2014, 2015) [12,13] proved global existence and asymptotic behavior of solutions to the above system when 0 ≤ d < 1 . The main purpose of the present paper is to remove the restriction of 0 ≤ d < 1 . [ABSTRACT FROM AUTHOR]

This paper estimates the blow-up time for the heat equation u t = Δ u with a local nonlinear Neumann boundary condition: The normal derivative ∂ u / ∂ n = u q on Γ 1 , one piece of the boundary, while on the rest part of the boundary, ∂ u / ∂ n = 0 . The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We show the finite time blow-up of the solution and estimate both upper and lower bounds of the blow-up time in terms of the area of Γ 1 . In many other work, they need the convexity of the domain Ω and only consider the problem with Γ 1 = ∂ Ω . In this paper, we remove the convexity condition and only require ∂Ω to be C 2 . In addition, we deal with the local nonlinearity, namely Γ 1 can be just part of ∂Ω. [ABSTRACT FROM AUTHOR]

Abstract We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the quasilinear critical problem (P λ) { − Δ p u = λ u q − 1 + u p ⁎ − 1 in Ω u > 0 in Ω u = 0 on ∂ Ω where Ω is a regular bounded domain in R N , N ≥ p 2 , 1 < p < 2 , p ≤ q < p ⁎ , p ⁎ = N p / (N − p) , λ > 0 is a parameter. In spite of the lack of C 2 regularity of the energy functional associated to (P λ) , we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ ⁎ > 0 such that, for each λ ∈ (0 , λ ⁎) , either (P λ) has P 1 (Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (P λ) , each of them having at least P 1 (Ω) distinct solutions. These results complete those obtained in [23] for the case p ≥ 2. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. [ABSTRACT FROM AUTHOR]

Abstract Let α : G ↷ X be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space E G (X) is compact and zero-dimensional, we show that the action α has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space M G (X) of invariant Borel probability measures forms a Bauer simplex. In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is Z -stable. Finally, we discuss some rank properties and provide two classifiability results for crossed products, one of which is based on the work of Elliott and Niu. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation u t t − u x x + M u + ε (V 0 (ω t) u x x + V (ω t , x) u) = 0 , x ∈ R / 2 π Z can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in (t , x) , quasi-periodic in time t with Diophantine frequency ω ∈ R n , and V 0 is finitely smooth and quasi-periodic in time t with Diophantine frequency ω ∈ R n. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent. [ABSTRACT FROM AUTHOR]

Abstract Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) + u (a − μ u r − 1 − λ w) , x ∈ Ω , t > 0 , τ v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0. Here, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary and τ > 0 , r > 1 , λ ≥ 0 , a ∈ R , μ , ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2 , with μ > μ ⁎ = (N − 2) + N (χ + C β) C N 2 + 1 1 N 2 + 1 for some positive constants C β and C N 2 + 1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a μ) 1 r − 1 , (a μ) 1 r − 1 , 0) if μ is large enough. This result improves or extends previous results of several authors. [ABSTRACT FROM AUTHOR]

Abstract This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in [ 0 , M ] and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation v t − max 0 ≤ a ≤ 1 ⁡ (1 2 a 2 σ 2 v x x + a μ v x) + c v − max 0 ≤ l ≤ M ⁡ [ (1 − v x) l ] = 0 under certain boundary conditions on an angular domain Q T = { (x , t) | 0 < x < M t , 0 < t ≤ T }. The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L 2 -framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions. [ABSTRACT FROM AUTHOR]

Abstract A special type of fluid–structure interaction (FSI) problems are problems with periodic boundary conditions like in turbomachinery. The steady state FSI response of these problems is usually calculated with similar techniques as used for transient FSI analyses. This means that, when the fluid and structure problem are not simultaneously solved with a monolithic approach, the problem is partitioned into a fluid and structural part and that each time step coupling iterations are performed to account for strong interactions between the two sub-domains. This paper shows that a time-partitioned FSI computation can be very inefficient to compute the steady state FSI response of periodic problems. A new approach is introduced in which coupling iterations are performed on periodic level instead of per time step. The convergence behaviour can be significantly improved by implementing existing partitioned solution methods as used for time step coupling (TSC) algorithms in the time periodic coupling (TPC) framework. The new algorithm has been evaluated by comparing the convergence behaviour to TSC algorithms. It is shown that the number of fluid–structure evaluations can be considerably reduced when a TPC algorithm is applied instead of a TSC. One of the most appealing advantages of the TPC approach is that the structural problem can be solved in the frequency domain resulting in a very efficient algorithm for computing steady state FSI responses. Highlights • A time periodic coupling (TPC) is introduced for computing the FSI response of periodic problems. • A quasi-Newton inverse least squares (QN-ILS) method is included in the TPC for stabilization. • The QN-ILS TPC method converges for any case, irrespective of the amount of damping and added mass. • Computation time can be significantly reduced with the QN-ILS TPC approach. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates. [ABSTRACT FROM AUTHOR]

Abstract The motion of moderately thick spray can be modeled by a coupled system of equations consisting of the incompressible Navier–Stokes equations and the Vlasov–Boltzmann equation. In this paper, we study the initial value problem for the Navier–Stokes–Vlasov–Boltzmann equations. The existence of global weak solutions is established by the weak convergence method. [ABSTRACT FROM AUTHOR]

Abstract We study the phenomenon of strengthening rate of polynomial approximation near boundary points of a convex domain. The main result of the paper presents this type of approximation near conic boundary points of these domains while Gopengauz's counter-example implies impossibility of such approximation near other points of the boundary. [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]

In this paper, we consider a domain Ω ε ⊂ R N , N ≥ 2 , with a very rough boundary depending on ε . For instance, if N = 3 Ω ε has the form of a brush with an ε -periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε . In Ω ε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε , on the lateral boundary of the teeth. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system. [ABSTRACT FROM AUTHOR]

This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered. [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]