Search

In this paper we study the initial boundary value problem for the nonlinear wave equation with combined power-type nonlinearities with variable coefficients. Existence and uniqueness of local weak solutions are proved. The global behavior of the solutions with non-positive and sub-critical energy is completely investigated. The threshold between global existence and finite time blow up is found. For super-critical energy, two new sufficient conditions guaranteeing blow up of the solutions for a finite time, are given. One of them is proved for an arbitrary sign of the scalar product of the initial data, while the other one is derived only for a positive sign. [ABSTRACT FROM AUTHOR]

Abstract We consider exterior boundary value problems with nontrivial boundary conditions, including three types of semilinear equations: heat equation, wave equation and damped wave equation. Using a test function approach with a choice of a suitable family of test functions, we obtain the critical exponents in the sense of Fujita for the considered problems. Our obtained results close some open questions from the literature. [ABSTRACT FROM AUTHOR]

We are concerned with the problem of boundary stabilization for wave equations with damping only on the nonlinear Wentzell boundary, and the damping is also nonlinear. We obtain boundary stabilization results (explicit energy decay rates), as well as the wellposedness, for the systems. Our method consists in establishing a new type of non-uniform integral inequality of energy, and exploiting it to derive the decay rates. The results are set up in all space dimensions. Even for the one dimensional linear system, the result is new, and we also indicate that the obtained decay rate is optimal in this case, that is, the obtained decay rate for linear strings is optimal. [ABSTRACT FROM AUTHOR]

We consider the nonlinear initial boundary value problem of one dimensional Schrödinger equation with the power nonlinearity of order q in boundary conditions. Our purpose in this paper is to prove that if q > 2 and the initial data are small in suitable norms, then we have a global in time of solutions in a scale invariant space. Furthermore, we present a blow up result of small amplitude solutions when 1 < q ≤ 2 and derive an upper-bound of existence time Our results indicate that the quadratic nonlinearity is a critical exponent. [ABSTRACT FROM AUTHOR]

This paper deals with existence results for a nonlinear boundary value problem derived from a model introduced by Lebowitz and Rubinow (1974) describing a proliferating cell population. Cells are distinguished by age a and cycle length l . The cycle length is viewed as an inherited property determined at birth. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. In our framework, daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. In this work, the cycle length l is allowed to be infinite, that is, l ∈ [ l 1 , + ∞ ) . This hypothesis introduces some mathematical difficulties which are overcome by using domination arguments (in the lattice sense) and recent fixed point theorems involving continuous weakly compact operators on non reflexive Banach spaces. [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]

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give rather complete descriptions of the global bifurcation diagrams of positive solutions for the logistic elliptic equation with nonlinear boundary conditions in three cases of an included weight function. Besides the abstract theory of local and global bifurcation, careful observation of the reduced bifurcation equation in finite dimensional space and blow up arguments for the corresponding initial boundary value problem play an important role in characterizing the global behavior of the bifurcating positive solutions for values of a parameter. The critical case is dealt with by using a topological technique proposed by Whyburn. [Copyright &y& Elsevier]

Abstract: The paper establishes the existence, estimate, uniqueness and regularity for the solution of a nonlinear parabolic system (a two-phase Caginalp type system) with non-homogeneous Cauchy–Stefan–Boltzmann and homogeneous Neumann boundary conditions and non-constant thermal conductivity. It extends the already studied types of boundary conditions which makes the mathematical model to be richer and more flexible to describe the real physical phenomena, including phase separation. [Copyright &y& Elsevier]

Abstract: In this paper we consider a second-order boundary value problem of the form , , . We demonstrate that if the nonlinear functional satisfies certain asymptotic behavior, then the problem will possess at least one positive solution. These results generalize and improve on some recent results in the literature on boundary value problems with nonlocal, nonlinear boundary conditions, and we provide some examples, which illustrate these generalizations and improvements. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods. [Copyright &y& Elsevier]

Abstract: We use a fixed point theorem due to Avery and Peterson to establish the existence of at least three non-negative solutions of some nonlocal boundary value problems to third order differential equations with advanced arguments. An example is given to illustrate the main results. [Copyright &y& Elsevier]

Abstract: We consider a population model with diffusion, a strong Allee effect per capita growth function, and constant yield harvesting. In particular, we focus our study on a population living in a patch, with , that satisfies a certain nonlinear boundary condition. We establish our existence results by the method of sub-supersolutions. [Copyright &y& Elsevier]

Abstract: We consider a semilinear Neumann problem with an asymptotically linear reaction term. We assume that resonance occurs at infinity. Using variational methods based on the critical point theory, together with the reduction technique and Morse theory, we show that the problem has at least four nontrivial smooth solutions. [Copyright &y& Elsevier]

Abstract: We study the nonlinear boundary value problem consisting of the equation and a boundary condition involving a Riemann–Stieltjes integral. By relating it to the eigenvalues of the corresponding linear Sturm–Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. The shooting method and an energy function are used to prove the main results. [Copyright &y& Elsevier]

Abstract: The existence and multiplicity of unbounded solutions is investigated for second-order three-point differential equations with nonlinear boundary conditions on , where the growth of the nonlinear term with respect to the third variable is allowed to be quadratic. By applying the techniques of unbounded upper and lower solutions and the topological degree theory, the existence of at least one solution and three solutions, respectively, is shown for the above problem. One example is worked out to indicate the applications of our main results. [ABSTRACT FROM AUTHOR]

Abstract: We study the nonexistence, existence and multiplicity of positive solutions for the nonlinear Neumann boundary value problem involving the -Laplacian of the form where is a bounded smooth domain in , and for . Using the sub–supersolution method and the variational principles, under appropriate assumptions on and , we prove that there exists such that the problem has at least two positive solutions if , has at least one positive solution if and has no positive solution if . [Copyright &y& Elsevier]

Abstract: We discuss solvability for the semilinear equation of the vibrating string in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions. [Copyright &y& Elsevier]

Abstract: This paper deals with the blow-up of the solution to a semilinear second-order parabolic equation with nonlinear boundary conditions. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time and when blow-up does occur upper and lower bounds for the blow-up time are obtained. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented. [Copyright &y& Elsevier]

Abstract: We prove the existence of -solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions, of the form: where is an -accretive operator acting on the infinite-dimensional Banach space , is an almost strongly weakly u.s.c. multi-function which satisfies an appropriate “sign” condition, while is a continuous function. [Copyright &y& Elsevier]

Abstract: We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987)  and Del Pino, Manasevich and Murua (1992) . The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré–Birkhoff Fixed Point Theorem. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of maximal and minimal periodic solutions of a class of quasilinear parabolic systems with nonlinear boundary conditions. Our approach to the problem is based on the method of upper and lower solutions and its associated monotone iterations. An application is also made to the reaction–diffusion system of Lotka–Volterra competition model. [Copyright &y& Elsevier]

Abstract: We study nonlinear elliptic Dirichlet problems driven by the -Laplacian and with a nonlinearity which consists of a -superlinear term perturbed by a -linear one. Using critical point theory, coupled with suitable truncation techniques, we show that the problem has two positive smooth solutions. When the conditions are symmetric in the two semiaxes, we have four nontrivial smooth solutions of constant sign. Finally for the semilinear problem, using Morse theory, we produce a fifth nontrivial smooth solution. [Copyright &y& Elsevier]

Abstract: We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Łojasiewicz–Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms are real analytic. Moreover, we provide an estimate for the convergence rate. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of non-zero solutions to a Neumann boundary value problem is established. Multiplicity results are also pointed out. The approach is based on a novel critical point theorem. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type. The existence of a weak solution is proved by using the Faedo–Galerkin method. The uniqueness, regularity and asymptotic expansion of the solution are also discussed. [Copyright &y& Elsevier]

Abstract: We study the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition Sufficient conditions are obtained for the problem to have at least one positive solution and to have no solution, respectively. Moreover, under certain conditions, we prove that there exists a bounded and continuous surface separating into two disjoint subsets and with such that the problem has at least two positive solutions for each , one positive solution for each , and no positive solution for any . [Copyright &y& Elsevier]

Abstract: In this paper we consider a parabolic problem as well as its stationary counterpart of a model arising in angiogenesis. The problem includes a chemotaxis type term and a nonlinear boundary condition at the tumor boundary. We show that the parabolic problem admits a unique positive global in time solution. Moreover, by bifurcation methods, we show the existence of coexistence states and also we study the local stability of the semi-trivial states. [Copyright &y& Elsevier]

Abstract: In this paper, we show the existence of infinite solutions to the Kirchhoff type quasilinear elliptic equation in a smooth bounded domain with nonlinear boundary condition on . The method we used here is based on “Fountain Theorem” and “Dual Fountain Theorem”. [Copyright &y& Elsevier]

Abstract: In this paper, we study a linear Sturm–Liouville impulsive problem and a nonlinear Sturm–Liouville impulsive problem. By applying a new approach via critical point theory and variational methods, the existence results of positive solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions. By using the methods of upper–lower solutions, truncations and penalization, we give some existence results of the extremal (greatest and smallest) solutions. Our approach is based on the theories of fixed points, maximal monotone operators and ordered Banach spaces. [Copyright &y& Elsevier]

Abstract: The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52] where the system is claimed to be nondissipative. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence and multiplicity of solutions for the elliptic system with nonlinear coupling at the smooth boundary given by where is a bounded domain of with smooth boundary, is the outer normal derivative. The proofs are done under suitable assumptions on the Hamiltonian, and based on the local linking theorem and a multiple critical points theorem of the critical point theory. [Copyright &y& Elsevier]

Abstract: This paper deals with the non-simultaneous and simultaneous blow-up for some parabolic systems coupled via nonlinear boundary flux . For radially symmetric solutions, we obtain that one component can blow up by itself and may provide sufficient help to the blow-up of the other ones under suitable initial data. In particular, such phenomena happen for every initial data in some exponent regions. It is interesting that there exist initial data such that any two components blow up simultaneously, either of which blows up depending on itself and also can give sufficient help to the other components blowing up simultaneously. A necessary and sufficient condition is obtained on the simultaneous blow-up of at least two components for all initial data. Moreover, the non-simultaneous and simultaneous blow-up rates and sets are determined. [Copyright &y& Elsevier]

Abstract: In this work, we consider a nonlinear elliptic problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a subsolution and a supersolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. [Copyright &y& Elsevier]

Abstract: Using the technique of Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336], we present a note on the paper [T.F. Wu, A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential, Electron. J. Differential Equations 131 (2006) 1–15] by Wu. Indeed, we extend the multiplicity results for a class of semilinear problems to the quasilinear elliptic problems of the form: Here denotes the -Laplacian operator defined by , ( if , if ), is a bounded domain with smooth boundary, is the outer normal derivative, , the weight is a bounded function with and are continuous functions which change sign in . [Copyright &y& Elsevier]

Abstract: In this paper we use Krasnoselskii’s fixed point theorem on cone expansions to prove a new fixed point theorem for nondecreasing operators on ordered Banach spaces. Moreover we apply this abstract result to prove the existence of a positive periodic solution for a nonlinear boundary value problem. [Copyright &y& Elsevier]

Abstract: A class of reaction-diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions. The diffusion term in the equation includes the so-called slow diffusion , fast diffusion and super diffusion , and the reaction term may involve a singular function. The aim of this paper is to show the existence of a global time-dependent solution, the existence of positive minimal and maximal steady-state solutions, including the uniqueness of the positive solution, and the asymptotic behavior of the time-dependent solution in relation to the positive steady-state solutions. Of particular concern is the convergence of the time-dependent solution to a unique positive steady-state solution and the global attraction property of the reaction-diffusion problem. Applications are given to several specific problems which are of considerable interest in the current literature. [Copyright &y& Elsevier]

Abstract: The optimal tracking control (OTC) problem for a class of affine nonlinear composite systems with similar structure is considered. By using a modeling technique, the nonlinear similar composite system is first transformed into some quasi-decoupled subsystems. Then the high-order, strongly coupled, nonlinear two-point boundary value (TPBV) problem is transformed into a sequence of linear decoupled TPBV problems through a successive approximation procedure. The obtained OTC law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence. A suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term. [Copyright &y& Elsevier]

Abstract: By using the fixed-point principle in cone and the fixed-point index theory for strict-set-contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear three-point boundary value problems of nth-order differential equations in ordered Banach spaces. In addition, an example is included to demonstrate the main results. [Copyright &y& Elsevier]

Abstract: This paper deals with the nonlinear elliptic equation in a bounded smooth domain with a nonlinear boundary value condition. The existence results are obtained by the sub–supersolution method and the Mountain Pass Lemma. And nonexistence is also considered. [Copyright &y& Elsevier]

Abstract: This paper investigates the multiplicity of positive solutions for a class of nonlinear boundary-value problems of fourth-order differential equations with integral boundary conditions in ordered Banach spaces. The arguments are based upon a specially constructed cone and the fixed point index theory in cone for strict set contraction operator. The nonexistence of positive solution is also studied. Meanwhile, an example is worked out to demonstrate the main results. [Copyright &y& Elsevier]

Abstract: This paper deals with boundary value problems in a bounded domain for the stationary one-dimensional Kawahara equation. For reasonable boundary conditions we prove existence and uniqueness of a strong solution and its asymptotics when the coefficient of a higher order derivative approaches zero. Formulations of well-posed and ill-posed problems for odd-order equations are discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence and non-existence of global solutions of the non-Newtonian polytropic filtration equations with nonlinear boundary conditions. We first obtain the critical global existence curve by constructing various self-similar supersolutions and subsolutions. And then the critical Fujita curve is conjectured with the aid of some new results. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the critical exponents and non-extinction property for a nonlinear boundary value problem of a fast diffusive polytropic filtration equation. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of triple positive solutions for Sturm–Liouville boundary value problems of second-order nonlinear differential equation on a half line. By using a fixed point theorem in a cone due to Avery–Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with a multidimensional Timoshenko system subjected to boundary conditions of memory type. We establish general rate decay results. The usual exponential and polynomial decay rates are only special cases. [Copyright &y& Elsevier]

Abstract: Consider the initial–boundary value problem for the nonlinear 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, under the conditions , , , , , a.e. ; , ; , , , , , the function supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others, we prove that the problem (1) and (2) has a weak solution . If, in addition, , , and some other conditions, then the solution is unique. The proof is based on the Faedo–Galerkin method and the weak compact method associated with a monotone operator. For the case of , in Part 2 we prove that the unique solution belongs to , with , , , , if we assume , and some other conditions. Finally, in Part 3, with ; , , , , we obtain an asymptotic expansion of the solution of the problem (1) and (2) up to order in two small parameters , . [Copyright &y& Elsevier]

Abstract: We study the nonlinear boundary value problem consisting of the equation and a general separated homogeneous linear boundary condition. By comparing this problem with a corresponding linear Sturm–Liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem. More specifically, let , be the -th eigenvalues of the corresponding linear Sturm–Liouville problem. Then under certain assumptions, the boundary value problem has a solution with exactly zeros in if is in the interior of the range of ; and does not have any solution with exactly zeros in if is outside of the range of . These conditions become necessary and sufficient when is monotone. The existences of multiple and even an infinite number of solutions are derived as consequences. We also discuss the changes of the number and the types of nontrivial solutions as the interval shrinks, as increases in a given direction, and as the boundary condition changes. [Copyright &y& Elsevier]