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In this paper we study an initial and boundary value problem for damped wave equations with nonlinear singular terms concentrating away from the boundary of the domain, on an interior neighbourhood of a hyper-surface M that collapses to M as ɛ goes to zero. We describe the conditions for well posedness of both the approximating and limit problems, as well as the convergence, at the singular limit, of the solutions of the former to solutions of the latter, when the parameter ɛ goes to zero. [ABSTRACT FROM AUTHOR]

This paper discusses a rearrangement minimization problem related to a boundary value problem where the differential operator is the p -Laplacian and the external force is indefinite . By applying the rearrangement theory established by G. R. Burton, we show that the minimization problem has a unique solution in the weak closure of the admissible set in an appropriate function space. What makes this paper different from many others in the literature is that the admissible set here is allowed to be indefinite (takes on both positive and negative values). We show that in this case the optimal solution will be a weak limit point of the admissible set and cannot be a rearrangement element itself. Optimality conditions are derived based on the mean value of the generator. In case the generator is a three valued function, we shall show that the optimality conditions lead to free boundary value problems that are of interest independently. We also discuss a minimization problem where the admissible set is not seemingly related to rearrangement classes but we shall prove otherwise, and apply the earlier results of the current paper to draw interesting conclusions regarding the optimal solution. We have also discussed the case where the domain of interest is radial, and shown that the optimal solutions are radial as well. [ABSTRACT FROM AUTHOR]

This paper is concerned with a new integrable multi-component Camassa–Holm system, which has been proven to be integrable and has peakon solutions. This system includes many one-component and two-component Camassa–Holm type systems as special cases. In this paper, we first establish the local well-posedness and a continuation criterion for the system, then we present several global existence and blow-up results for two important integrable two-component subsystems. Our obtained results cover and improve considerably recent results in Gui et al. (2013), Yan et al. (2015). [ABSTRACT FROM AUTHOR]

We consider the nonlinear boundary value problem consisting of the equation − u ″ = f ( u ) , on ( − 1 , 1 ) , where f : R → R is continuous, together with general Sturm–Liouville type, multi-point boundary conditions at ± 1 . We obtain Rabinowitz-type global bifurcation results, and then use these to obtain ‘nodal’ solutions of the problem. We conclude with a nonresonance result for an inhomogeneous form of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation − u ″ = λ u , on ( − 1 , 1 ) , together with the multi-point boundary conditions. In a previous paper it was shown that, under certain ‘optimal’ conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm–Liouville problem with single-point boundary conditions. In particular, for each integer k ⩾ 0 there exists a unique, simple eigenvalue λ k , whose eigenfunctions have ‘oscillation count’ equal to k , where the ‘oscillation count’ was defined in terms of a complicated Prüfer angle construction. Unfortunately, it seems to be difficult to apply the Prüfer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above. [ABSTRACT FROM AUTHOR]

This paper deals with boundary value problems of the form − ∑ i = 1 n D i (a i (x , u (x) , D u (x))) = f (x) , x ∈ Ω , u (x) = 0 , x ∈ ∂ Ω. Assume that there exist c 1 , ν , θ > 0 such that for almost all x ∈ Ω and all (s , z) ∈ R × R n , | a i (x , s , z) | ≤ c 1 1 + | z i | p i − 1 , i = 1 , ... , n , and ν ∑ i = 1 n | z i | p i (1 + | s |) θ ≤ ∑ i = 1 n a i (x , s , z) z i. We let f ∈ L m (Ω) and we derive regularity results for weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem Δ u = b (x) f (u) , x ∈ Ω , u > 0 in Ω , u | ∂ Ω = + ∞ , where Ω is a C 2 -bounded domain in R N , (N ≥ 3), f ∈ C 1 ((0 , ∞) , (0 , ∞)) is nondecreasing on (0 , ∞) and b is a function in C l o c γ (Ω) , (0 < γ < 1) such that there exist b 1 , b 2 > 0 satisfying for each x ∈ Ω , 0 < b 1 = lim d (x) ⟶ 0 inf b (x) h (d (x)) ≤ lim d (x) ⟶ 0 sup b (x) h (d (x)) = b 2 < ∞ , where d (x) = d i s t (x , ∂ Ω) and h (t) ≔ c t − λ exp ∫ t η z (s) s d s , η > d i a m (Ω) , λ ≤ 2 such that z is a continuous function on [ 0 , η ] with z (0) = 0. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up phenomenon of the initial boundary value problem for the periodic rotation-two-component Camassa–Holm (R2CH) system. The results include a precise blow-up criterion, when σ > 0 , and two new blow-up statements and the exact blow-up rate for strong solutions of the system. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik's Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretical results, we present formulations and numerical simulations for the controllability problem, the results were obtained through the finite element method and a quasi-Newton method, simulated in the software Free fem++. Finally, large time null controllability, stabilization and others possible extensions and open problems concerning other systems are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will give conditions which will guarantee the existence of positive solutions for a variety of second order boundary value problems, using the well-known Krasnoselskii fixed point theorem. We will deal with a specific differential equation meeting a specific initial condition and use a general boundary condition, involving a not necessarily linear functional. Our purpose is to pose conditions on that functional, which will guarantee that the Krasnoselskii fixed point theorem can be applied. It is important to notice that only the values of this functional on two specific functions are involved in the conditions we pose. This paper unifies the way we deal with a wide variety of boundary value problems and provides results which, to the best of our knowledge, are new. [Copyright &y& Elsevier]

Abstract: There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems. [Copyright &y& Elsevier]

This paper concerns the existence of global smooth solutions to the initial boundary value problem for a three-dimensional compressible quantum hydrodynamic model with damping and heat diffusion in a bounded domain in R 3. Based on the continuation argument and the uniform a priori estimates with respect to the time, we obtain the existence of global solutions in a bounded smooth domain provided that the initial perturbation around a constant state is small enough. The key difficulty is to deal with the higher order quantum terms, which do play an essential role in establishing the a priori estimates. The boundary conditions finally adopted are the insulating boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1) τ ∂ t 2 u (x , t) − Δ u (x , t) + a (x) ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ C Σ × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ C Σ × (0 , T) , u (x , 0) = ε f (x) , x ∈ C Σ , τ ∂ t u (x , 0) = τ ε g (x) , x ∈ C Σ , where C Σ is a cone-like domain in R N (N ≥ 2) defined as C Σ = int r ω ∈ R N ; r ≥ 0 , ω ∈ Σ with a connected open set Σ in S N − 1 with smooth boundary ∂ Σ. If N = 1 , then we only consider two cases C Σ = (0 , ∞) and C Σ = R. Here a (x) is a non-zero coefficient of ∂ t u which could be complex-valued and space-dependent, λ ∈ ℂ is a fixed constant, and ε > 0 is a small parameter. The constants τ = 0 , 1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study existence and uniqueness of a solution for a nonlinear elliptic problem subject to nonlocal boundary condition. Moreover, we prove the equivalence between this kind of problem and nonlinear problem with very large diffusion around the boundary. [ABSTRACT FROM AUTHOR]

Abstract Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. The corresponding study of higher critical points was recently initiated in Jerison and Perera (2018, 0000). In particular, the existence and nondegeneracy of a mountain pass point in a superlinear and subcritical free boundary problem related to plasma confinement was proved in Jerison and Perera (2018). In this paper we study ground states of a critical free boundary problem related to the Brézis–Nirenberg problem (1983). We extend the results of Jerison and Perera (2018) to this problem by combining the method introduced there with the concentration compactness principle to overcome the difficulties arising from lack of compactness. [ABSTRACT FROM AUTHOR]

Abstract: The present paper shows Hölder continuity of discrete Morse flows to a regularized Alt–Caffarelli variational functional generating free boundaries; the continuity is uniform with respect to the discrete Morse flows and the regularizations. The uniformity enables to construct Morse flows to the Alt–Caffarelli functional, which shall be dealt with in another paper. [Copyright &y& Elsevier]

Abstract: The paper first study the steady-state thin film type equation with Navier boundary conditions in multidimensional space. By the truncation method, a fixed point argument and some energy estimates, the existence and asymptotic limit for the positive weak solutions are given. Second, the parabolic equation with a Navier boundary in one-dimensional space is researched. The existence is obtained by applying a semi-discrete method for the time variable and solving the corresponding elliptic problem. The uniqueness is shown for depending on an energy estimate. In addition, the iteration relation of the semi-discrete problem gives an exponential decay result for the time . The thin film equation, which is usually used to describe the motion of a very thin layer of viscous in compressible fluids along an inclined plane, is a class of nonlinear fourth-order parabolic equations and the maximum principle does not hold directly. For applying the classic theory of partial differential equation, the paper transforms the fourth-order problem into a second-order elliptic–elliptic system or a second-order parabolic–elliptic system. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial–boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial–boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time. [Copyright &y& Elsevier]

Abstract We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s ∈ (0 , 1) and summability growth p ∈ (1 , ∞) , whose model is the fractional p -Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Hölder continuity up to the boundary, and many others. We then discuss the good definition of (s , p) -superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned (s , p) -superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma. We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper. [ABSTRACT FROM AUTHOR]

Abstract The objects of our studies are vector valued parabolic minimizers u associated to a convex Carathéodory integrand f obeying a p -growth assumption from below and a certain monotonicity condition in the gradient variable. Here, the functions being considered are defined on a metric measure space (X , d , μ). For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u 0 on the parabolic boundary of a space–time-cylinder Ω × (0 , T) with an open subset Ω ⊂ X and T > 0 , we prove existence in the parabolic Newtonian space L p (0 , T ; N 1 , p (Ω ; R N)). In this paper we generalize results from Bögelein et al. (2014,2015) to the metric setting and argue completely on a variational level. [ABSTRACT FROM AUTHOR]

Abstract The asymptotic wave behavior of solution for the Euler equations with damping in R + n is investigated around a given constant equilibrium in this paper. Three classical boundary condition cases are considered here. New approaches and techniques are introduced to deal with the multi-dimensional case for the system. For the linearized problem, by comparing symbols in the transformed tangential–spatial and time space, we show that its Green's function can be described in terms of the fundamental solution for the Cauchy problem and reflected fundamental solution coupled with a boundary operator. Our approaches help us to simplify the Green's function to the essential part and benefit the follow-up nonlinear analysis. For the nonlinear problem, we prove the pointwise decaying rate with the help of the Duhamel's principle. [ABSTRACT FROM AUTHOR]

Abstract We study the boundary differentiability for inhomogeneous infinity Laplace equations on convex domains Ω with the inhomogeneous term f ∈ C (Ω) ∩ L ∞ (Ω) and differentiable boundary data g. At a flat point (the boundary point where the blow up of the domain is the half-space), u is differentiable due to a previous result of the second author in Hong (2014). At a corner point (the boundary point where the blow up of the domain is not the half-space), we establish a slope estimate for u , and provide a necessary and sufficient condition for the boundary differentiability of u in this paper. [ABSTRACT FROM AUTHOR]

We consider the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space. The goal of this paper is to study the long-time behavior of the solution of this initial–boundary value problem based on the asymptotic analysis of an associated matrix Riemann–Hilbert problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, combining the method of lower and upper solutions with the localization method, we establish the boundary blow-up rate of the large positive solutions to the singular boundary value problem where is a smooth bounded domain in . The weight function is a non-negative continuous function in the domain, which vanishes on the boundary of the underlying domain at different rates according to the point of the boundary. is locally Lipschitz continuous satisfying the Keller–Osserman condition and is increasing on . It is worth emphasizing that we obtain the main results for a large class of nonlinear terms , which is regularly varying at infinity with index (that is for all , ), instead of the restriction: for sufficiently large and some positive constants , as in the series of papers of J. L. Gómez. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and multiplicity of classical solutions for a second-order impulsive differential equation with periodic boundary conditions. By using a variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions and infinitely many solutions when the parameter pair lies in different intervals, respectively. Some examples are given in this paper to illustrate the main results. [Copyright &y& Elsevier]

Abstract: This paper studies a first-order periodic boundary value problem for impulsive integro-differential equations and establishes some new maximum principles. Some points of a recent paper are also corrected and extended. [Copyright &y& Elsevier]

Abstract: This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation. [Copyright &y& Elsevier]

Abstract: This paper investigates the boundary control problem of the viscosity Degasperis–Procesi equation in [0, 1]. Results of the existence of a solution to the viscosity Degasperis–Procesi equation in a short time interval under the boundary condition, are given in the paper. It proves the global exponential stability under the given boundary condition in , , , space. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial boundary value problem for the linear wave equation where , , , , , , , are given functions and is a given constant. The paper consists of four main parts. In Part 1, under conditions (, , , , , ) , , , , a.e. ; the function continuous, , for all , with , are given constants and some other conditions, we prove that, the problem (1) has a unique weak solution . The proof is based on the Faedo–Galerkin method associated with the weak compact method. In Part 2 we prove that the unique solution belongs to , with , , if we assume , and some other conditions. In Part 3, with , , we obtain an asymptotic expansion of the solution of the problem (1) up to order in a small parameter . Finally, in Part 4, we prove that the solution of this problem is stable with respect to the data (, , , , , ). [Copyright &y& Elsevier]

Abstract: In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique -periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of -periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of positive solutions for a class of nonlinear second-order four-point boundary value problems with alternating coefficient. Our approach relies on the Krasnosel’skii fixed point theorem. The result of this paper is new and extent the previously known result. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and uniqueness of a nontrivial solution for the following third-order -Laplacian -point eigenvalue problems on time scales: where is -Laplacian operator, i.e., is a parameter, . We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. The conditions we used in the paper are different from those in [Z.C. Hao, L. Debnath, On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. Math. Lett. 18 (2005) 543–553; D. Jiang, R.P. Agarwal, A uniqueness and existence theorem for a singular third-order boundary value problem on , Appl. Math. Lett. 15 (2002) 445–451; S.H. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl. 323 (2006) 413–425; Z. Liu, J.S. Ume, D.R. Anderson, S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, J. Math. Anal. Appl. 334 (2007) 299–313; Q. Yao, Positive solutions to a special singular second-order boundary value problem, Math. Comput. Model. (in press); Q. Yao, Successive iteration of positive solution for a discontinuous third-order boundary value problem, Comput. Math. Appl. 53 (2007) 741–749]. [Copyright &y& Elsevier]

Abstract: The main goal of this paper is to prove existence and comparison results for elliptic differential inclusions governed by a quasilinear elliptic operator and a multivalued function given by Clarke’s generalized gradient of some locally Lipschitz function. These kinds of problems have been treated in the past by various authors including the authors of this paper. However, in all the works we are aware of, additional assumptions on the structure of the elliptic operator and/or the generalized Clarke’s gradient are needed to get comparison results in terms of sub-supersolutions. Comparison principles were obtained recently, e.g., in the case where the elliptic operator is of potential type, or Clarke’s gradient is required to satisfy some one-sided growth condition, or the sub-supersolutions are supposed to satisfy additional properties. The novelty of this paper is that we are able to obtain a comparison principle without assuming any of the above restrictions. To the best of our knowledge this is the first mathematical treatment of the considered elliptic inclusion in its full generality. The obtained results of this paper complement the development of the sub-supersolution method for nonsmooth problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu. [Copyright &y& Elsevier]

Abstract: This paper deals with finite-time quenching for the nonlinear parabolic system with coupled singular absorptions: , in subject to positive Dirichlet boundary conditions, where , is a bounded domain in with smooth boundary. We obtain the sufficient conditions for global existence and finite-time quenching of solutions, and then determine the blow-up of time-derivatives and the quenching set for the quenching solutions. As the main results of the paper, a very clear picture is obtained for radial solutions with : the quenching is simultaneous if , and non-simultaneous if or ; if with , then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. In determining the non-simultaneous quenching criteria of the paper, some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous Dirichlet boundary value conditions, in addition to techniques frequently used in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial–boundary value problem for the linear wave equation where , are given constants and , , are given functions, and the unknown function and the unknown boundary value satisfy the following nonlinear integral equation: where , , , are given constants and , are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1.1)–(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of , Part 2 is devoted the study of the asymptotic behavior of the solution as . Finally, in Part 3 we obtain an asymptotic expansion of the solution of the problem (1.1)–(1.5) up to order in four small parameters , , , . [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation. [Copyright &y& Elsevier]

Abstract: This paper establishes exponential decay bounds for steady magnetohydrodynamic geophysical finite pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of C.O. Horgan, L.T. Wheeler [Spatial decay estimates for the Navier–Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978) 97–116] and K.A. Ames, L.E. Payne [Decay estimates in steady pipe flow, SIAM J. Math. Anal. 20 (1989) 789–815] the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is symmetric like in the work of J.C. Song, [Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Anal. 54 (2003) 1029–1044] and it is fully developed at the exit section. Using a coupling parameter, we derive an integro-differential inequality that leads to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial-boundary value problem for the linear wave equationwhere K, are given constants and , , , are given functions, the unknown function and the unknown boundary value satisfy the following linear integral equationwhere g, k, , are given functions. The paper consists of four parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1)–(2). The proof is based on a Galerkin type approximation associated to various energy estimates-type bounds, weak-convergence and compactness arguments. In Part 2, it is devoted to the study of the regularity of the solution , with respect to the functions , , f, g, k, . In Part 3 we prove that the solution , of this problem is stable with respect to the data , , , , f, g, k, . Finally, in Part 4 we obtain an asymptotic expansion of the solution , of this problem up to order in , , for , sufficiently small. [Copyright &y& Elsevier]

Abstract: In this paper we study Sturm-Liouville problems with periodic nonlinearities . We are interested in the resonant case. We generalize results known by Schaaf and Schmitt and Cañada and Roca to the situations in which the corresponding nontrivial solution to the linear part change sign. Our main tools are Lyapunov-Schmidt reduction and asymptotic methods based on the stationary phase argument. These methods have been used previously by Dancer to treat the particular case . In this paper we essentially modify the last method to get results also in the case when cannot be expressed as a finite sum of sines and cosines. [Copyright &y& Elsevier]

In this paper we will use the degree theory to give an existence result to the boundary value problem of nonlinear perturbations of the scalar p-Laplacian with the multi-point boundary condition involving the Lebesgue-Stieltjes integral. Among the boundary conditions considered in this paper, we will find also the optimal bounds on the growth of the nonlinear perturbations in our existence result. [Copyright &y& Elsevier]

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of RN. We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F(y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L2-norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments. [Copyright &y& Elsevier]

In this paper we use the linear theory developed in Cano-Casanova and Lo´pez-Go´mez (J. Differential Equations 178 (2002) 123) to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the boundary operator. Besides working with mixed non-classical boundary conditions, this paper considers problems where bifurcation from infinity occurs. Treating this kind of problems is far from easy, since a complete knowledge of the set of values of the parameters where they exhibit a positive solution is imperative. Our results are far from immediate, as a consequence of the vanishing weight functions involved in our general setting. Now a days, the analysis of these general problems, where the coefficients are allowed to vary and eventually vanishing or changing sign, is focusing a great deal of attention. [Copyright &y& Elsevier]

In this paper, we investigate the spatial decay of some diffusion-reaction equations. In the stationary case we obtain results of exponential decay. In the transient case, we find a decay which is as fast as the exponential of a quadratic polynomial. The main tool of the paper is the maximum principle together with the usual methods of decay estimates for linear problems. We also obtain an estimate for the amplitude term by means of the boundary conditions. [Copyright &y& Elsevier]

In this paper, first we obtain a necessary and sufficient condition for the existence of positive solution to a singular impulsive Dirichlet boundary value problem, then we discuss the existence of multiple solutions to a singular impulsive BVP. In the last section of this paper, to illustrate the application of our main results, we discuss the existence of positive radial solutions to a semilinear elliptic radial solutions problem. [Copyright &y& Elsevier]

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, ( − Δ + m 2 − m ) u = | u | p − 1 u in Ω , with the Dirichlet boundary condition u = 0 on ∂ Ω . Here, p ∈ ( 1 , ∞ ) and the operator ( − Δ + m 2 − m ) is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of p , m and Ω . Precisely, we show that ( i ) if p is not H 1 subcritical ( p ≥ n + 2 n − 2 ) and Ω is star-shaped, the equation has no nontrivial solution for all m > 0 ; ( i i ) if p is not H 1 ∕ 2 supercritical ( 1 < p ≤ n + 1 n − 1 ), then there exists a least energy solution for all m > 0 and any bounded domain Ω ; ( i i i ) finally, in the intermediate range ( n + 1 n − 1 < p < n + 2 n − 2 ), the problem has a nontrivial solution, provided that m is sufficiently large and the problem (0.1) − Δ u = | u | p − 1 u in Ω , u = 0 on ∂ Ω admits a non-degenerate nontrivial solution, for example, when Ω is a ball or an annulus. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fréchet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the partial viscosity Boussinesq equations on a 3D thin periodic domain. The global well-posedness of strong solution with the initial data ( u 0 , θ 0 ) ∈ H p e r 2 ( Ω ϵ ) × H p e r 1 ( Ω ϵ ) is established, where Ω ϵ = [ 0 , L 1 ] × [ 0 , L 2 ] × [ 0 , ϵ ] is with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we study Dirichlet problems for noncoercive linear elliptic equations with discontinuous coefficients in unbounded domains. Exploiting a nonlinear approach, we achieve existence, uniqueness and regularity results. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous Dirichlet boundary condition under the condition p ( x ) > 2 n n + 2 , n = 2 , 3 . In particular, we improve a priori estimate for weak solutions to the problem with min p ( x ) < 2 , which is an improvement even for constant p . [ABSTRACT FROM AUTHOR]