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This paper is concerned with the multiplicity of positive solutions for the critical Kirchhoff type problems involving indefinite weight functions (0.1) { − M ( ∫ Ω | ∇ u | 2 d x ) Δ u = Q λ ( x ) | u | q − 2 u + K ( x ) | u | 2 ⁎ − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N ( N ≥ 3 ) , 1 q 2 , M ( s ) = a + b s β with β > 0 , a > 0 and b > 0 , the weight functions Q λ and K are continuous and changing-sign. Using the Nehari manifold, fibering maps and Ljusternik-Schnirelmann category, we prove that at least two positive solutions for (0.1) exist provided that β = 1 and 2 ⁎ ≥ 4 . Furthermore, by the mountain pass theorem and Ekeland's variational principle, it is shown that (0.1) possesses at least three positive solutions whenever β > 2 N − 2 , including the case that β = 1 and 2 ⁎ 4 . Our results generalize some recent results in the literature.

A Banach space X is said to have the fixed point property if for every closed convex bounded subset C ⊂ X and for every T : C → C nonexpansive, there is a fixed point. A Banach space is said to have the dual-fixed point property if its dual space X ⁎ verifies the fixed point property. It is a fact that ( C 0 ( K ) , ‖ ⋅ ‖ ∞ ) fails the dual-fixed point property for every locally compact Hausdorff set K. In this article we study the fulfilment of the dual-fixed point property under equivalent norms. We prove that C 0 ( K ) can be renormed to have the dual-fixed point property if and only if K is a countable set. In this case we prove that this renorming can be achieved as close to the standard norm on C 0 ( K ) as we like for the Banach-Mazur distance. We further obtain the corresponding weak⁎-fixed-point-property stability coefficient regarding the duality σ ( C 0 ( K ) ⁎ , C 0 ( K ) ) . It will be displayed that the value of this stability coefficient strongly depends on the distribution of the accumulation points of K. Finally, we prove that the dual-fixed point property is completely ‘‘unstable” for the Banach-Mazur distance in most of nonreflexive Banach spaces.

We reprove the existence of extremals for the critical Trudinger-Moser inequality on a smooth bounded domain of R 2 via the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [11] , Mancini-Martinazzi [12] and Mancini-Thizy [13] . For this purpose, unlike [12] , [13] , it suffices to calculate local energy of maximizers for subcritical Trudinger-Moser functionals near the blow-up point.

We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in its spectrum provided a certain special function written as a series satisfies some lower bound. We show that this is indeed the case if the ratio of the period and the width of strip is less than a certain explicit number, which is approximately equal to 0.10121. We also find explicitly the point in the spectrum, above which there is no internal gaps. We then study the case of a sufficiently small period and we prove that in such case the considered operator has no internal gaps in the spectrum. The conditions ensuring the absence are written as certain explicit inequalities.

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R 2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

A topological space X is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space X has a resolving subset that can be written as an at most countably infinite union of subsets, in such a way that a given vector lattice of (not necessarily bounded) continuous functions on X separates every point outside the resolving subset from each of its constituents, then the order continuous dual of this lattice is trivial. In order to apply this result in specific cases, we show that several spaces have resolving subsets that can be written as at most countably infinite unions of closed nowhere dense subsets. An appeal to the main result then yields, for example, that, under appropriate conditions, vector lattices of continuous functions on separable spaces, metric spaces, and topological vector spaces have trivial order continuous duals if they separate points and closed nowhere dense subsets. Our results in this direction extend known results in the literature. We also show that, under reasonably mild separation conditions, vector lattices of continuous functions on locally connected T 1 Baire spaces without isolated points have trivial order continuous duals. A discussion of the relation between our results and the non-existence of non-zero normal measures is included.

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system ( ◇ ) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) in Ω × ( 0 , t ⁎ ) , 0 = Δ v + α u − β v in Ω × ( 0 , t ⁎ ) , 0 = Δ w + γ u − δ w in Ω × ( 0 , t ⁎ ) , where Ω is a smooth and bounded domain of R 2 , t ⁎ is the blow–up time and α , β , γ , δ , χ , ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u ( x , 0 ) = u 0 ∈ C 0 ( Ω ¯ ) , system (◇) has a unique classical solution which becomes unbounded as t ↗ t ⁎ . The main result of this investigation is to provide an explicit lower bound for t ⁎ estimated in terms of ∫ Ω u 0 2 d x and attained by means of well–established techniques based on ordinary differential inequalities.

Let c k l ∈ W 2 , ∞ ( R d , C ) for all k , l ∈ { 1 , … , d } . We consider the divergence form operator A = − ∑ k , l = 1 d ∂ l ( c k l ∂ k ) in L 2 ( R d ) when the coefficient matrix satisfies ( C ( x ) ξ , ξ ) ∈ Σ θ for all x ∈ R d and ξ ∈ C d , where Σ θ be the sector with vertex 0 and semi-angle θ in the complex plane. We show that for all p in a suitable interval the contraction semigroup generated by −A extends consistently to a contraction semigroup on L p ( R d ) . For those values of p we present a condition on the coefficients such that the space C c ∞ ( R d ) of test functions is a core for the generator on L p ( R d ) . We also examine the operator A separately in the more special Hilbert space L 2 ( R d ) setting and provide more sufficient conditions such that C c ∞ ( R d ) is a core.

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

In the paper, the initial observability of abstract second-order infinite-dimensional dynamical systems is considered. Two cases of observation operator are explored. First, the observation operator is assumed to be bounded. Then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation. Using the frequency-domain method it is proved, that initial observability of second-order system under discussion can be verified by the initial observability conditions for the corresponding simplified first-order system. General results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures. Some special cases are also discussed and practical comments and remarks are given. The paper extends earlier results on initial observability for a class of infinite-dimensional second-order abstract dynamical systems.

This paper considers the initial-boundary value problem for the nonlocal Cahn–Hilliard equation ∂ t φ + ( − Δ + 1 ) ( a ( ⋅ ) φ − J ⁎ φ + G ′ ( φ ) ) = 0 in ( 0 , T ) × Ω in an unbounded domain Ω ⊂ R N with smooth bounded boundary, where N ∈ N , T > 0 , and a ( ⋅ ) , J , G are given functions. In the case that Ω is a bounded domain and − Δ + 1 is replaced with −Δ, this problem has been studied by using a Faedo–Galerkin approximation scheme considering the compactness of the Neumann operator − Δ + 1 (cf. [8] , [16] ). However, the compactness of the Neumann operator − Δ + 1 breaks down when Ω is an unbounded domain. The present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain.

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal vector calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of R m with m ≥ 3 . It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems.

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\subset F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$., 32 pages, LaTeX, uses elsarticle

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane., Comment: 15pages, 4figures

In this paper, we discuss the extinction and non-extinction properties of solutions for the following fractional p-Kirchhoff problem { u t + M ( [ u ] s , p p ) ( − Δ ) p s u = λ | u | r − 2 u − μ | u | q − 2 u ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where [ u ] s , p is the Gagliardo seminorm of u, Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p s is the fractional p-Laplacian with 0 s 1 p 2 , M : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, 1 q ≤ 2 , r > 1 and λ , μ > 0 . Under suitable assumptions, we obtain the extinction of solutions. To get more precisely decay estimates of solutions, we develop the Gagliardo-Nirenberg inequality. Moreover, the non-extinction property of solutions is also investigated.

We prove the Holder continuity of the unique solution to quaternionic Monge-Ampere equation with densities in L p , p > 2 , on a bounded strictly pseudoconvex domains.

We show that a bounded quasinilpotent operator T acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator F and a scalar α ∈ C , α ≠ 0 , α ≠ 1 , such that T + F and T + α F are also quasinilpotent. We also prove that for any fixed rank-one operator F, almost all perturbations T + α F have invariant subspaces of infinite dimension and codimension.

We study the reaction-diffusion system with coupled exponential nonlinearities { ∂ t u = Δ u + e p 1 u + p 2 v in Ω × ( 0 , T ) , ∂ t v = Δ v + e q 1 u + q 2 v in Ω × ( 0 , T ) , u ( x , t ) = v ( x , t ) = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in Ω, where T > 0 , p i ≥ 0 and q i ≥ 0 ( i = 1 , 2 ) with ( p 1 , p 2 ) ≠ ( 0 , 0 ) and ( q 1 , q 2 ) ≠ ( 0 , 0 ) . The domain Ω is R N ( N ≥ 1 ) or a bounded domain in R N with C 2 boundary and the initial functions u 0 and v 0 are nonnegative and measurable. For each ( p 1 , p 2 , q 1 , q 2 ) , we obtain integrability conditions of ( u 0 , v 0 ) which explicitly determine the existence/nonexistence of a local in time nonnegative classical solution. For the existence result, we can take a wider class of initial functions than previously suggested. Our analysis can be applied to other nonlinearities including superexponential ones.

We show that a bounded, supercyclic C 0 -semigroup is already stable. The supercyclic Bourdon and Feldman Theorem for C 0 -semigroups is also proven.

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κ C . In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L 1 -norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ κ C . Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.

We study the complexity of stochastic integration with respect to an isonormal process defined on a bounded Lipschitz domain Q ⊂ R d . We consider integration of functions from Sobolev spaces W p r ( Q ) and analyze the complexity in the deterministic and randomized setting. Matching upper and lower bounds for the n-th minimal error are established, this way determining the complexity of the problem. It turns out that the stochastic integration problem is closely related to approximation of the embedding of W p r ( Q ) into L 2 ( Q ) .

We are concerned with positive solutions of the fractional elliptic system ( E ) { ( − Δ ) s u = f ( v ) in Ω , ( − Δ ) s v = g ( u ) in Ω , where Ω is an arbitrary domain in R N ( N > 2 s ), s ∈ ( 1 2 , 1 ) and f , g ∈ C l o c β ( R ) , for some β ∈ ( 0 , 1 ) . We establish universal a priori estimate for positive solutions of (E) . Then for C 2 bounded domain Ω, we prove the existence of positive solutions of (E) with prescribed boundary value u = μ and v = ν where μ , ν are positive bounded measure on ∂Ω and discuss regularity property of the solutions.

In this paper, we study the asymptotic stability of the following density-suppressed motility system ( ⁎ ) { u t = Δ ( γ ( v ) u ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v + u − v , x ∈ Ω , t > 0 , in a bounded domain with smooth boundary, where the motility function γ ( v ) and μ are positive. Under the conditions that the motility function γ ( v ) has the lower-upper bounded and that μ is large enough, we derive that system (⁎) possesses a unique global solution in three-dimensional space. Moreover, we show that if the global classical solution exists of system (⁎) in any dimensional space, then the solution will converge to the equilibrium ( 1 , 1 ) exponentially as t → + ∞ when μ > K 0 16 with K 0 = max 0 ≤ v ≤ ∞ ⁡ | γ ′ ( v ) | 2 γ ( v ) .

The C ⁎ -subalgebra B of all bounded linear operators on the space L 2 ( R ) , which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on R , is studied. A faithful representation of the quotient C ⁎ -algebra B π = B / K in a Hilbert space, where K is the ideal of compact operators on the space L 2 ( R ) , is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method and the Fredholm symbol calculus for the C ⁎ -algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the C ⁎ -algebra B and a Fredholm criterion for the operators B ∈ B .

We study the question of when two weighted variable exponent Bergman spaces or Hardy spaces are equivalent. As an application, we show that variable exponent Hardy spaces have a close relation to classical Hardy spaces when the exponent is log-Holder continuous and has bounded harmonic conjugate (when extended from its boundary values to be harmonic in the disc). We use this to characterize Carleson measures for these variable exponent Hardy spaces. We also prove under certain conditions an analogue of Littlewood subordination and a result on the boundedness of composition operators.

In this paper, we study a reaction-diffusion system with prey taxis which models the dynamics of a two-predator-one-prey ecosystem in which the predators collaboratively take advantage of the prey's strategy. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and small prey-taxis sensitivity coefficients are proved. It is also shown that such prey-taxis qualitatively affect the stability of coexistence steady state solutions in some cases, and the emergence of forceful prey-taxis will enhance the spatial patterns.

This paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume-filling effect over a bounded rectangular domain. By using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns. We derive the Stuart-Landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and Fredholm theory. In particular, we find asymptotic expressions for a wide range of patterns sustained by the system. These patterns include mixed-mode, square, hexagonal, and roll stationary configurations. Our analytical results are corroborated by direct simulations of the underlying chemotaxis system.

The solutions of the stationary Navier-Stokes equations in R n for n ≥ 3 in the scaling invariant Besov spaces are investigated. It is proved that a sequence of bounded smooth external forces whose B ˙ ∞ , 1 − 3 norms converges to zero can produce a sequence of bounded smooth solutions whose B ˙ − 1 ∞ , ∞ norms never converges to zero. Such norm inflation phenomena are shown by constructing the sequence of external forces, as similar to those of initial data proposed by Bourgain-Pavlovic in the non-stationary problem.

In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ + γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional φ : A × A → C satisfying φ ( a , b ) = 0 whenever a b = b a is of the form φ ( a , b ) = ω ( a b − b a ) for some ω ∈ A ⁎ . We prove that A has this property provided that any of the following three conditions holds: (i) A is a weakly amenable Banach algebra with property B and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from A into A ⁎ is an inner derivation, (iii) A is the algebra of all n × n matrices, where n ≥ 2 , over a cyclically amenable Banach algebra with a bounded approximate identity.

We consider the Banach function space consisting of real functions defined on the half-axis and having increments tempered by a given modulus of continuity. Next, we formulate and prove a sufficient condition for relative compactness of a bounded subset of the mentioned Banach space. On the basis of that condition we construct a measure of noncompactness in the function space in question. Using that measure we prove a theorem on the existence of solutions of a quadratic integral equation in the space of functions satisfying the Holder condition on the real half-axis.

Group representations by bounded Lamperti operators in the spaces L α ( 1 ≤ α ∞ ) form a wide class of representations, including representations by bounded positive operators and (when α ≠ 2 ) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesaro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in L α ( 1 α ∞ ) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989–5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to “weighted” averages of such representations and discuss various conditions on the “weights” under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups R m and Z m .

This work is concerned with a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production: u t = ∇ ⋅ ( ( 1 + u ) − α ∇ u ) − ∇ ⋅ ( u ( 1 + u ) β − 1 ∇ v ) + f ( u ) , v t = Δ v − v + u γ , with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain, where α , β ∈ R and γ > 0 . The logistic type source term f ( u ) satisfies that either f ( u ) ≡ 0 or f ( u ) = r u − μ u k with r ∈ R , μ > 0 and k > 1 . The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameters conditions, which improves the known results.

In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small.

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them., 13 pages. This paper is a substantially reworked version of our paper arXiv:1510.03807, where we now deduce the result of that paper on the global boundedness for Fourier integral operators from a much more general principle for integral operators

We prove the existence of local and global in time weak solutions for the equations of nonhomogeneous incompressible asymmetric fluids in a bounded domain Ω ⊊ R 3 , when the initial vacuum is allowed, i.e., the initial density ρ 0 is not necessarily strictly positive. Our solutions here are more regular than previous results and so satisfy the initial conditions in a stronger sense. The global in time solutions are obtained under smallness assumptions on the initial data and forcing terms.

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction–diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray–Schauder topological degree. A bounding function (i.e. Lyapunov-like function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.

We consider an inverse problem for a system of equations modeling quasi-static conductive and radiative heat transfer. The problem consists in finding the right-hand side of the heat transfer equation, which is a linear combination of given functionals. The prescribed data are the values of these functionals evaluated on the solution. The solvability of the problem is proven without any smallness assumptions on the model parameters. In the class of bounded temperature fields, the uniqueness of the solution of the inverse problem is shown. Further, we study the Tikhonov regularization in the framework of a PDE constrained optimization problem and show that the approximating sequence contains a convergent subsequence. The analytical results depend crucially on new and refined a priori estimates for the solutions.

We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable operator algebras, and we provide an affirmative answer for representations whose range is residually finite-dimensional. Furthermore, we show that weak-⁎ closed, amenable, residually finite-dimensional operator algebras are similar to C ⁎ -algebras, and in particular have the property that all their bounded representations are completely bounded. We prove our results for operator algebras having the so-called total reduction property, which is known to be weaker than amenability.

We study operator algebras arising from monomial ideals in the ring of polynomials in noncommuting variables, through the apparatus of subproduct systems and C*-correspondences. We provide a full comparison amongst the related operator algebras. For our analysis we isolate a partially defined dynamical system, to which we refer as the quantised dynamics of the monomial ideal. In addition we revisit several previously considered constructions. These include Matsumoto's subshift C*-algebras, as well as the tensor and the Pimsner algebras associated with dynamical systems or graphs. We sort out the various relations by giving concrete conditions and counterexamples that orientate the operator algebras of our context. It appears that the boundary C*-algebras do not arise as the quotient with the compact operators unconditionally. We establish a dichotomy to this effect by examining the resulting tensor algebras. We identify their boundary representations, we analyse their C*-envelopes, and we give criteria for hyperrigidity. Moreover we completely classify them in terms of the data provided by the monomial ideals. For tensor algebras of C*-correspondences and bounded isomorphisms this is achieved up to the level of local conjugacy (in the sense of Davidson and Roydor) for the quantised dynamics. For tensor algebras of subproduct systems and algebraic isomorphisms this is achieved up to the level of equality of monomial ideals modulo permutations of the variables. In the process we accomplish more in different directions. Most notably we show that tensor algebras form a complete invariant for isomorphic (resp. similar) subproduct systems of homogeneous ideals up to isometric (resp. bounded) isomorphisms. The results on local conjugacy are obtained via an alternative proof of the breakthrough result of Davidson and Katsoulis on piecewise conjugate systems. For our purposes we use appropriate compressions of the Fock representation. We then apply this alternative proof locally for the partially defined quantised dynamics. In this way we avoid the topological graphs machinery and pave the way for further applications. These include operator algebras of dynamical systems over commuting contractions or over row commuting contractions.

This paper deals with convergence of a solution for the parabolic–parabolic Keller–Segel system { ( u λ ) t = Δ u λ − χ ∇ ⋅ ( u λ ∇ v λ ) in Ω × ( 0 , ∞ ) , λ ( v λ ) t = Δ v λ − v λ + u λ in Ω × ( 0 , ∞ ) , where Ω is a bounded domain in R n ( n ≥ 2 ) with smooth boundary, χ , λ > 0 are constants, to that for the parabolic–elliptic Keller–Segel system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) in Ω × ( 0 , ∞ ) , 0 = Δ v − v + u in Ω × ( 0 , ∞ ) as λ ↘ 0 .

In this paper, we study the spectrum of Toeplitz operators on the Dirichlet space D with bounded conjugate analytic symbols and give a characterization of the kernels of Toeplitz operators with harmonic symbols in P ‾ + M ( D ) . Using this characterization, we show that the spectrum of a Toeplitz operator with symbol, the sum of an analytic polynomial with degree at most two and the conjugate of linear polynomial is connected, but there are Toeplitz operators with general harmonic polynomial symbols having disconnected spectrum.

We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - u - f(u) w + \kappa, \\ v_t = \Delta v - v + f(u) w, \\ w_t = \Delta w - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $\Omega \subset \mathbb R^n, n \in \mathbb N,$ for a parameter $\kappa \ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^\alpha$ for all $s \ge 1$, some $K_f \gt 0$ and $\alpha \in \mathbb R$. We prove that whenever \begin{align*} \alpha \lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case $\alpha \gt 1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac2n$ could be essentially optimal., Comment: 12 pages

The Cauchy problem of the system of compressible adiabatic flow through porous media in R 3 is considered. We obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in H 3 -norm. This is a continuing work of [31] , where due to non-dissipative property of the entropy s, an additional assumption that the initial data is bounded in L 1 -norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution. In this paper, to remove this assumption, we first decompose the solution U into high frequency part U h and low frequency part U l . Then by constructing delicate energy functionals for U h and optimal decay rates for U l , we can get the desired energy estimates to close the a priori assumption. Furthermore, we get the optimal L 2 − L 2 decay rate of the solution and show that the velocity decays faster than the density.

In this paper we classify all bounded travelling wave solutions for the integrable Dullin–Gottwald–Holm equation. It is shown that it decomposes in two known cases: the Camassa–Holm and the Korteweg–de Vries equation. For the former, the classification is similar to the one presented in (2005) [12] , while for the latter it is only possible to obtain smooth solutions.

Let Ω be a smooth bounded domain in R N , N ≥ 1 , let m , n be two nonnegative functions defined in Ω, and let ϕ : R N → R N be a continuous and strictly monotone mapping. We consider the existence and nonexistence of positive solutions for nonlinear problems involving the ϕ-Laplacian, of the form { − div ϕ ( ∇ u ) = λ m ( x ) f ( u ) − n ( x ) g ( u ) in Ω , u = 0 on ∂ Ω , where λ > 0 is a real parameter, and f , g : [ 0 , ∞ ) → [ 0 , ∞ ) are continuous functions modeled by f ( t ) = t q and g ( t ) = t r with 0 q r . The subdiffusive, equidiffusive and superdiffusive cases are all treated, and some qualitative properties of the solutions are also given. As a byproduct, we obtain results in the case of a (roughly speaking) sublinear nonlinearity f and n ≡ 0 . We point out that some of our results are new even for the usual p-Laplacian. Our main tool is the sub-supersolutions method combined with some estimates on related nonlinear problems.

In this paper we study the stability and moment boundedness of the solutions to the linear age-structured model with randomly-varying immigration or harvesting. For this model, we study stability and the asymptotic behavior of the first moment and the results are identical to that of the corresponding deterministic age-structured model and the linear age-structured model with the white noise. However, the stability and boundedness of the second moment are complicated and depend on the randomly-varying immigration or harvesting. For the linear age-structured model with randomly-varying immigration or harvesting, we directly prove that the second moment M ( t , a ) is bounded for 0 ≤ t a . When t > a ≥ 0 , using the Laplace transform in the framework of Ito–Doob integral, we give the explicit expressions of the second moments M ( t , 0 ) and M ( t , a ) and then establish the sufficient conditions for the second moments to be bounded and unbounded, respectively, through the supreme of the real parts of all characteristic roots. We also study the asymptotic behaviors of the second moments M ( t , 0 ) and M ( t , a ) and give the sufficient condition for stochastically ultimately boundedness of the model.

In this paper, we consider stochastic second grade fluids driven by Levy noise on a bounded domain of R 2 . Global existence and uniqueness of strong probabilistic solutions are established, which improves the existing corresponding results for martingale solutions in literature.