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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 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: This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression with singular coefficients in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the th eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems. [Copyright &y& Elsevier]

Abstract: In this paper we study a free boundary problem modeling solid avascular tumor growth. The model is based on the reaction–diffusion dynamics and mass conservation law. The model is considered with time delays in proliferation process. The quasi-steady-state (i.e., ) is studied by Foryś and Bodnar [see U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling 37 (2003) 1201–1209]. In this paper we mainly consider the case . In the case considered by Foryś and Bodnar, the model is reduced to an ordinary differential equation with time delay, but in the case the model cannot be reduced to an ordinary differential equation with time delay. By theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the long time asymptotic behavior of the solutions under some conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we establish several new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which generalize or improve all related existing ones. Then we use the Lyapunov-type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one-dimensional quasilinear elliptic system with the Dirichlet boundary conditions. [Copyright &y& Elsevier]

Abstract: The selective decay phenomena have been observed by physicists for many dynamic flows such as Navier–Stokes flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. In the previous paper (M.-Q. Zhan, 2010 ), the author showed the validity of the selective decay principle for the 2D magnetohydrodynamic (MHD) flows in the case of small magnetic Prandtl number. In this paper, we shall show the validity of the selective decay principle for the 2D magnetohydrodynamic (MHD) flows for any magnetic Prandtl number with periodic boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain , are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that – together with the uniqueness and existence results – makes the problem under consideration to a well-posed problem in the Hadamard sense. [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]

Abstract: This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii''s fixed point theorems in cones. An exhaustive study of the Green''s function related to a linear fourth order problem is done. [Copyright &y& Elsevier]

Abstract: In this paper, the multiplicity of Lagrangian orbits on smooth compact symmetric star-shaped hypersurfaces with respect to the origin in is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via -index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors'' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425–1436]. [Copyright &y& Elsevier]

Abstract: In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial–boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature. [Copyright &y& Elsevier]

Abstract: In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo–Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions. [Copyright &y& Elsevier]

Abstract: We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917–945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss'' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash–Moser type iteration scheme. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the estimates. [Copyright &y& Elsevier]

Abstract: We show in this paper that the -groups governing neutron transport equations with reflecting boundary conditions in slab geometry satisfy a weak spectral mapping theorem in any spaces . The main part of this paper is dealing with collisionless transport -group while the full transport -group is dealt with by stability arguments. [Copyright &y& Elsevier]

Abstract: In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256]. [Copyright &y& Elsevier]

Abstract: Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator , , acting on with appropriate boundary conditions, generates an exponentially bounded -convoluted cosine function, and consequently, an exponentially bounded analytic -convoluted semigroup of angle , for suitable exponentially bounded kernels and . [Copyright &y& Elsevier]

Abstract: In this paper, the initial value problem for a class of fractional differential equations is discussed, which generalizes the existent result to a wide class of fractional differential equations. Also the theoretical result established in the paper ensures the validity of chaos control of fractional differential equations. In particular, feed-back control of chaotic fractional differential equation is theoretically investigated and the fractional Lorenz system as a numerical example is further provided to verify the analytical result. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a degenerate time-dependent drift-diffusion model for semiconductors. The electric conductivity in the system is assumed to be temperate-dependent. And the pressure function we use in this paper is . We present existence results for general nonlinear diffusivities for the degenerate Dirichlet–Neumann mixed boundary value problem. [Copyright &y& Elsevier]

Abstract: The main aim of the paper is to present a technique that allows to infer wellposedness, trace and Gevrey''s regularity of hyperbolic-like PDE''s with non-monotone boundary conditions. The lack of monotonicity prevents applicability of the known semigroup methods. In this paper we show how recently developed tools of microlocal analysis [D. Tataru, A priori estimates of Carleman''s type in domains with boundary, J. Math. Pure Appl. 73 (1994) 353–387] combined with some spectral theory can be used successfully in order to obtain the needed inequalities. The method will be illustrated on a simple example of beam equation with non-monotone boundary conditions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase-field system in , subject to the Neumann boundary condition for θ the dynamical boundary condition for χ and the initial conditions where Ω is a bounded domain in with smooth boundary Γ, ν is the outward normal direction to the boundary and ϕ is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate. [Copyright &y& Elsevier]

Abstract: In this paper we consider a second-order Sturm–Liouville-type boundary value operator of the form on an arbitrary, bounded time-scale , for suitable functions , together with suitable boundary conditions. Operators of this type on time-scales have normally been considered in a setting involving Banach spaces of continuous functions on . In this paper we introduce a space of square-integrable functions on , and Sobolev-type spaces , , consisting of functions with nth-order generalised -type derivatives. We prove some basic functional analytic results for these spaces, and then formulate the operator L in this setting. In particular, we allow , while — this generalises the usual conditions that , . We give some immediate applications of the functional analytic results to L, such as ‘positivity’, injectivity, invertibility and compactness of the inverse. We also construct a Green''s function for L. The analogues of these results on real intervals are well known, and are fundamental to the usual Sturm–Liouville theory on such intervals. [Copyright &y& Elsevier]

Abstract: In this paper, we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case (see Eq. (1) in the paper). Numerical results are presented to confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel''skii fixed point theorem and an example is given to illustrate it. [Copyright &y& Elsevier]

Abstract: An initial boundary value problem is considered for a nonlinear diffusion equation, the diffusivity being a function of the dependent variable. Dirichlet boundary conditions, independent of time, are considered and positive solutions are assumed. This paper is mainly concerned with the rate of convergence, in time, of the unsteady to the steady state. This is done by obtaining an upper estimate for a positive-definite, integral measure of the perturbation (i.e., unsteady–steady state) using differential inequality techniques. A previous result is recalled where the diffusivity (n being a positive constant) appropriate to mass transport, or filtration, in a porous medium. The present paper treats an alternative model, sharing some of the characteristics of the previous one: , τ being non-negative. The paper concludes by considering a “backwards in time” initial boundary value problem for the perturbation (amenable to the same techniques) and establishes that the solution ceases to exist beyond a critical, computable time. [Copyright &y& Elsevier]

In this paper, we systematically study the dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response. The explorations involve the permanence, extinction, global asymptotic stability (general nonautonomous case); the existence, uniqueness and stability of a positive (almost) periodic solution and a boundary (almost) periodic solution for the periodic (almost periodic) case. The paper ends with some interesting numerical simulations that complement our analytical findings. [Copyright &y& Elsevier]

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper. [Copyright &y& Elsevier]

Guo (Approx. Theory Appl. 4 (1988) 9–18) introduced the integral modification of Meyer-Konig and Zeller operators Mˆn and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Be´zier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364–376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106–107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng. [Copyright &y& Elsevier]

In this paper, we establish existence results for positive solutions for the (n−1,1) three-point boundary value problems consisting of the equation u(n)+λa(t)fu(t)=0, t∈(0,1), with one of the following boundary value conditions: u(0)=αu(η), u(1)=βu(η), u(i)(0)=0 for i=1,2,…,n−2, and u(n−2)(0)=αu(n−2)(η), u(n−2)(1)=βu(n−2)(η), u(i)(0)=0 for i=0,1,…,n−3, where η∈(0,1), α&ges;0, β&ges;0, and a :(0,1)→R may change sign and R=(−∞,+∞). f(0)>0, λ>0 is a parameter. Our approach is based on the Leray–Schauder degree theory. This paper is motivated by Eloe and Henderson (Nonlinear Anal. 28 (1997) 1669–1680). [Copyright &y& Elsevier]

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krejcˇi´ and the authors for phase-field systems involving hysteresis operators. [Copyright &y& Elsevier]

In this paper, some new results about the existence of positive solutions for singular semi-positone boundary value problems are obtained. The results of this paper partially improve the former corresponding work. [Copyright &y& Elsevier]

In this paper, we study the free-boundary problem associated with a singular diffusion equation. An entropy equality derived by the authors for BV solutions in an earlier paper is used in its formulation. Existence, uniqueness, and regularity results are obtained. [Copyright &y& Elsevier]

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]