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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]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

Ordinary and partial differential equations with operators containing the p -Laplace operator, the so called ϕ -Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following boundary value problem { − Δ p u + g (u) | ∇ u | p = h (u) f in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, in a domain Ω ⊂ R N (N ≥ 2) , where 1 ≤ p < N , g is a positive and continuous function on [ 0 , ∞) , and h is a continuous function on [ 0 , ∞) (possibly blowing up at the origin). We show how the presence of regularizing terms h and g allows to prove existence of finite energy solutions for nonnegative data f only belonging to L 1 (Ω). [ABSTRACT FROM AUTHOR]

In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. The solution operator of the quasiperiodic boundary value problem yields a well-defined scattering operator, the numerical approximation of which is nothing else but the S-matrix in scattering matrix algorithms like Rigorous Coupled Wave Analysis (RCWA) or Fourier Modal Method (FMM). [ABSTRACT FROM AUTHOR]

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method. [ABSTRACT FROM AUTHOR]

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

The Hamiltonian systems considered in this paper are obtained by weakly coupling two systems having completely different behaviors. The first one satisfies the twist assumptions usually considered for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered lower and upper solutions. In the higher dimensional case, we also treat a coupling situation where the classical Hartman condition is assumed. [ABSTRACT FROM AUTHOR]

This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with L 2 boundary data in a periodically perforated Lipschitz domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the vanishing dissipation limit of solutions to the initial boundary value problem of three-dimensional incompressible viscous magnetohydrodynamic equations in the upper half space where the magnetic field is assumed to be transverse to the boundary. When the tangential component of the magnetic field is imposed the zero Neumann boundary condition, we can establish the uniform energy estimates of the solutions to the initial boundary value problem in the conormal Sobolev spaces even if the velocity field satisfies the no-slip boundary condition. Then the vanishing dissipation limit of solutions to the initial boundary value problem can be achieved in L ∞ sense based on some compactness arguments. Compared with the previous results, our results show that the strong boundary layer can still be prevented by the transverse magnetic field even with the magnetic diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the free boundary value problem of the three-dimensional full primitive equations for the large-scale ocean with heat insulation in a bounded domain. The domain is periodic in the horizontal direction and finite in the vertical direction, and its upper boundary is a free surface driven by the fluid and bottom boundary is fixed. We show that the unique strong solution of the free boundary value problem for the three-dimensional full primitive equations exists globally in time and converges to the equilibrium state with exponential decay rate, when the initial data is a small perturbation of the steady state in regular Sobolev space. [ABSTRACT FROM AUTHOR]

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a diffusive system modeling the interaction of mussel and algae in the water layer overlying the mussel bed, where the algae are the main food source for mussels, and the advection pushes algaes in one direction but not out of the domain. We present a threshold result on the global extinction and persistence of mussels. It shows that the condition for persistence depends on the principal eigenvalue of a scalar boundary value problem, which is related to the diffusion, the speed of the tidal flow, the conversion rate of algae to mussel production, and mussel mortality rate. We further investigate the asymptotic profile of the positive steady state when it exists. [ABSTRACT FROM AUTHOR]

Morse index theory in the classical framework provides an equality between the spectral properties of the second variation of the Lagrangian functional and the oscillation properties of the space of solutions of the associated boundary value problem. For carrying over a similar result in the non-Hamiltonian context which can be useful for investigating the dynamical properties of dissipative systems, we introduce a new topological invariant, the so-called "degree-index" defined in terms of the Brouwer degree of a suitable determinant map of a boundary matrix, which provides one possible substitute of the Maslov index in this non-Hamiltonian framework and finally we prove the equality between the Morse index and the degree-index in this non-selfadjoint setting through a new abstract trace formula. In the last part of the paper we apply our theoretical results to some 1D reaction-diffusion systems. [ABSTRACT FROM AUTHOR]

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L 4 , 5), families of periodic orbits passing through collision, and orbits connecting L 4 to ejection or collision. [ABSTRACT FROM AUTHOR]

The paper is concerned with the stability property under perturbation Q → Q ˜ of different spectral characteristics of a boundary value problem associated in L 2 ([ 0 , 1 ] ; C 2) with the following 2 × 2 Dirac type equation (0.1) L U (Q) y = − i B − 1 y ′ + Q (x) y = λ y , B = ( b 1 0 0 b 2 ) , b 1 < 0 < b 2 , y = col (y 1 , y 2) , with a potential matrix Q ∈ L p ([ 0 , 1 ] ; C 2 × 2) and subject to the regular boundary conditions U y : = { U 1 , U 2 } y = 0. If b 2 = − b 1 = 1 this equation is equivalent to one dimensional Dirac equation. Our approach to the spectral stability relies on the existence of the triangular transformation operators for system (0.1) with Q ∈ L 1 , which was established in our previous works. The starting point of our investigation is the Lipshitz property of the mapping Q → K Q ± , where K Q ± are the kernels of transformation operators for system (0.1). Namely, we prove the following uniform estimate: ‖ K Q ± − K Q ˜ ± ‖ X ∞ , p 2 + ‖ K Q ± − K Q ˜ ± ‖ X 1 , p 2 ⩽ C ⋅ ‖ Q − Q ˜ ‖ p , Q , Q ˜ ∈ U p , r 2 × 2 , p ∈ [ 1 , ∞ ] , on balls U p , r 2 × 2 in L p ([ 0 , 1 ] ; C 2 × 2). It is new even for Q ˜ = 0. Here X ∞ , p 2 , X 1 , p 2 are the special Banach spaces naturally arising in such problems. We also obtained similar estimates for Fourier transforms of K Q ±. Both of these estimates are of independent interest and play a crucial role in the proofs of all spectral stability results discussed in the paper. For instance, as an immediate consequence of these estimates we get the Lipshitz property of the mapping Q → Φ Q (⋅ , λ) , where Φ Q (x , λ) is the fundamental matrix of the system (0.1). Assuming the spectrum Λ Q = { λ Q , n } n ∈ Z of L U (Q) to be asymptotically simple, denote by F Q = { f Q , n } | n | > N a sequence of corresponding normalized eigenvectors, L U (Q) f Q , n = λ Q , n f Q , n. Assuming boundary conditions (BC) to be strictly regular , we show that the mapping Q → Λ Q − Λ 0 sends L p ([ 0 , 1 ] ; C 2 × 2) either into ℓ p ′ or into the weighted ℓ p -space ℓ p ({ (1 + | n |) p − 2 }) ; we also establish its Lipshitz property on compact sets in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. The proof of the second estimate involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. It is also shown that the mapping Q → F Q − F 0 sends L p ([ 0 , 1 ] ; C 2 × 2) into the space ℓ p ′ (Z ; C ([ 0 , 1 ] ; C 2) of sequences of continuous vector-functions, and has the Lipshitz property on compacts sets in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. Certain modifications of these spectral stability results are also proved for balls U p , r 2 × 2 in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. Note also that the proof of the Lipshitz property of the mapping Q → F Q − F 0 involves the deep Carleson-Hunt theorem for maximal Fourier transform, while the proof of this property for the mapping Q → Λ Q − Λ 0 relies on the estimates of the classical Fourier transform and is elementary in character. We apply our previous results to the damped string equation to establish the Riesz basis property and the asymptotic behavior of the eigenvalues of the corresponding dynamic generator under the assumptions d ∈ L 1 [ 0 , ℓ ] , ρ ∈ W 1 , 1 [ 0 , ℓ ] on the damping coefficient and the density of the string, that are weaker than previously treated in the literature. We also establish Lipshitz dependence on d and ρ in ℓ p -spaces of the remainders in the asymptotic formula for the eigenvalues. [ABSTRACT FROM AUTHOR]

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given. [ABSTRACT FROM AUTHOR]

In the paper we established the L 2 estimates for commutators of the Dirichlet-to-Neumann map generated by a divergence form parabolic operator, defined on R + n + 2 , with real, symmetric and t , λ − independent coefficient matrix, or more generally, a small complex L ∞ perturbation of such. The major new challenge, compared to our previous work in elliptic setting, is to handle the first order derivatives with respect to the time variable t which not only fall on solutions to parabolic equations but also on auxiliary functions. [ABSTRACT FROM AUTHOR]

The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in R n (n ≥ 1) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R 3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge. [ABSTRACT FROM AUTHOR]

Dynamics of the macroscopic motion of a compressible, viscous and heat-conducting, one-dimensional monatomic ionized gas can be described by a compressible Navier-Stokes type system. Compared with the corresponding compressible Navier-Stokes type systems modeling a viscous and heat conducting ideal polytropic gas [7,10,13–16,18,19,23,24,26,28,31,32] , the real gases [17] , nonlinear thermoviscoelasticity [5,6] , and a viscous radiative and reactive gas [11,12,20,21,30] , the dependence of the state of an ionized gas on the degree of ionization leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas makes it impossible to deduce the dissipative estimates on the first-order spatial derivative of the bulk velocity and the absolute temperature from the basic energy estimates obtained through the normalized entropy and, as a consequence, makes the construction of its global large solutions and the study of their large time behavior more difficult. By making full use of the intrinsic structure of equations, the existence of global large smooth solutions to two types of initial boundary value problems of one-dimensional, compressible Navier-Stokes system for a viscous and heat-conducting ionized gas is established in [22] for a class of density and/or temperature dependent transport coefficients, but since the estimates obtained there depend on the time variable, the problem on their large time behavior remains unsolved and the main purpose of this paper is to give the precise description of the large time behavior of the global solutions constructed in [22]. The key ingredient in our analysis is to deduce the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation in the half space when the viscosity tends to zero. By means of the asymptotic analysis with multiple scales, we first formally derive the equations for boundary layer profiles. Next, we study the well-posedness of the equations for the boundary layer profiles by using the compactness argument. Moreover, we construct an accurate approximate solution and use the energy method to obtain the convergence results of the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

This paper is concerned with the local well-posedness for the free boundary value problem of smooth solutions to the cylindrical symmetric Euler equations with damping and related models, including the compressible Euler equations and the Euler-Poisson equations. The free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. However, the compressible Euler equations or Euler-Poisson equations with damping become a degenerate system at the moving boundary. By setting a suitable weighted Sobolev space and using Hardy's inequality, we successfully overcome the singularity at the center point and the vacuum occurring on the moving boundary, and obtain the well-posedness of local smooth solutions. We also summarize the recent related results on the free boundary value problem for the Euler equations with damping, compressible Euler equations and Euler-Poisson equations. [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]

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of the energy associated to an initial boundary value problem involving a semilinear wave equation. The domain is an unbounded subset of R 2 with unbounded boundary. On the function responsible by the semilinearity we consider a very general decay rate control. The main tools are the Trudinger–Moser inequality combined with microlocal analysis arguments and the techniques of Lasiecka and Tataru. [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]

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 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]

In this paper, we prove the existence and uniqueness of solitary wave solutions of time-dependent nonlinear Schrödinger equations with behaviors tending to zero at infinity under certain conditions on trapping potentials and 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. Particularly, for constant trapping potentials, we conclude that there is at most one radial ground state under certain conditions on parameters. [ABSTRACT FROM AUTHOR]

This paper discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time. [ABSTRACT FROM AUTHOR]

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity. [ABSTRACT FROM AUTHOR]

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂ t u − Δ e − Δ u + e − Δ u − 1 = 0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6]. Our investigations reveal that we must control the size of both ‖ e − Δ u (x , 0) ‖ W 2 , 2 (Ω) and ‖ e Δ u (x , 0) ‖ ∞ , Ω suitably to achieve our results. Related results in [8,10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity. [ABSTRACT FROM AUTHOR]

Let Ω be a C 2 bounded domain in R N (N ≥ 3), δ (x) = dist (x , ∂ Ω) and C H (Ω) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , (P ρ) where 0 < μ < C H (Ω) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 < p < N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem (P ρ) admits a positive solution if 0 < ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎. In this paper, we go further in the study of the solution set of (P ρ). We show that the problem admits at least two positive solutions if 0 < ρ < ρ ⁎ and a unique positive solution if ρ = ρ ⁎. We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators whose coefficients are signed measures. Our approach is probabilistic, which also gives a representation of the solution. The heat kernel estimates and the theory of additive functionals play a crucial role in our approach. [ABSTRACT FROM AUTHOR]

In this paper we prove an existence result for "solution obtained as limit of approximations" to a class of Dirichlet boundary value problems whose prototype is { − Δ p u = β (1 + | ∇ u |) q + c (x) | u | p − 2 u + f in Ω u = 0 on ∂ Ω , where Ω is a bounded open subset of R N , N ≥ 2 , 1 < p < 2 , Δ p u = div (| ∇ u | p − 2 ∇ u) , p − 1 < q ≤ p − 1 + p N , β is a positive constant, c with c ≥ 0 , c ≠ 0 and f are measurable functions satisfying suitable summability conditions depending on q. We further assume smallness assumptions on β , c and f. Our approach is based on Schauder's fixed point theorem. Similar results can be proved also for 2 ≤ p < N. [ABSTRACT FROM AUTHOR]

The notion of p-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of p -ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of p -ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity , rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when p -ellipticity is considered. In this paper, we define three new notions of p -ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems. These important roles are demonstrated by establishing extrapolation results for solvability of the L p Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization. [ABSTRACT FROM AUTHOR]

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value. [ABSTRACT FROM AUTHOR]

The paper investigates the existence, asymptotic boundary behavior and uniqueness of convex solutions to the Monge-Ampère equation that take infinite values on the boundary of smooth and bounded open convex sets in R n. [ABSTRACT FROM AUTHOR]

In this paper we consider the semilinear damped wave problem of the form { (α (t) u t) t − β (t) Δ u + γ (t) u t + δ (t) u = β (t) f (u) , x ∈ Ω , t > τ , u (x , t) = 0 , x ∈ ∂ Ω , t ⩾ τ , u (x , τ) = u τ (x) , u t (x , τ) = v τ (x) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space. [ABSTRACT FROM AUTHOR]

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given. [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 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]

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 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]

Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using "bulk-surface models" (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface in the models has no thickness. However, in some real world applications, the "surface" is actually a thin membrane, having positive, albeit small thickness δ , as in the cases of thermal barrier coatings for turbine engine blades and cell membranes. In such situations, rigorous derivations for these models seem lacking and desirable. In this paper, we start with two full models each of which contains reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two. Then in the limit of δ → 0 , we obtain two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which we can also recover the surface density of the other substance. Our analysis reveals that to have such effective models, it is crucial that (i) appropriate transmission conditions are satisfied on the interface between the bulk and the thin membrane, and (ii) the diffusion tensor in the thin membrane is "optimally aligned", which means the principal axes of the tensor are either normal or tangential to the interface. [ABSTRACT FROM AUTHOR]

This paper is concerned with an initial and boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients. In the case when the viscosity μ (θ) = θ α and the heat-conductivity κ (θ) = θ β with α , β ∈ 0 , ∞) , we prove the global-in-time existence of strong solutions under some assumptions on the growth exponent α and the initial data. As a byproduct, the nonlinearly exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if α ≥ 0 is small, and the growth exponent β ≥ 0 can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is the Laplacian and nonlinear part is the divergence of a function analytic in the gradient of the solution. The main novelty in terms of the result is that the coefficients of the nonlinearity are allowed to be "anisotropic". As in previous works, the proof reduces to an integral identity involving the tensor product of the gradients of 3 or more harmonic functions. Employing a construction method using Gaussian quasi-modes, we obtain a convenient family of harmonic functions to plug into the integral identity and establish our result. [ABSTRACT FROM AUTHOR]