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In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ-Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

The main goal of this work is to study an initial boundary value problem for a Kirchhoff-type equation with nonlinear boundary delay and source terms. This paper is devoted to prove the global existence, decay, and the blow up of solutions. To the best of our knowledge, there are not results on the Kirchhoff type-equation with nonlinear boundary delay and source terms. [ABSTRACT FROM AUTHOR]

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow‐up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow‐up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality. [ABSTRACT FROM AUTHOR]

We show that a specific skew-symmetric formulation of the nonlinear terms in the compressible Navier-Stokes equations leads to an energy rate in terms of surface integrals only. No dissipative volume integrals contribute to the energy rate. We also discuss boundary conditions that bounds the surface integrals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term. [ABSTRACT FROM AUTHOR]

The main crux of this work is to study the existence of extremal solutions for a new class of nonlinear sequential fractional differential equations (NSFDEs) with nonlinear boundary conditions (NBCs) under the ψ -Caputo operator. The obtained outcomes of the proposed problem are derived by means of the monotone iterative technique (MIT) associated with the method of upper and lower solutions. Lastly, the desired findings are well illustrated by an example. [ABSTRACT FROM AUTHOR]

We study positive solutions to the two-point boundary value problem: - u ′ ′ = λ h (t) f (u) ; (0 , 1) u (0) = 0 u ′ (1) + c (u (1)) u (1) = 0 , where λ > 0 is a parameter, h ∈ C 1 ((0 , 1 ] , (0 , ∞)) is a decreasing function, f ∈ C 1 ((0 , ∞) , R) is an increasing concave function such that lim s → ∞ f (s) = ∞ , lim s → ∞ f (s) s = 0 , lim s → 0 + f (s) = - ∞ (infinite semipositone) and c ∈ C ([ 0 , ∞) , (0 , ∞)) is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for λ ≫ 1 . [ABSTRACT FROM AUTHOR]

A multiple variational iteration method (VIM) is proposed to effectively solve the second-order nonlinear two-point boundary value problems. For problems where convergence speed of the original VIM is slow or the original method is divergent, the multiple VIM method (MVIM) presented in this paper can readily improve the rate of convergence. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach: The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula. Findings: When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions. Research limitations/implications: Basically, the authors' strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides. Practical implications: Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied. Originality/value: Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast. [ABSTRACT FROM AUTHOR]

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. Firstly, the energy rate in terms of a surface integral with boundary terms is derived. Secondly, we bound the surface integral by deriving new nonlinear boundary procedures for boundary conditions with non-zero data. The new nonlinear boundary procedure generalises the well known characteristic boundary procedure for linear problems to the nonlinear setting. To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations. [ABSTRACT FROM AUTHOR]

We study positive solutions to (singular) boundary value problems of the form:$\left\{ \begin{align} & -\left({{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.$where \begin{document}$\varphi_p(u): = |u|^{p-2}u$\end{document} with \begin{document}$p>1$\end{document} is the \begin{document}$p$\end{document}-Laplacian operator of \begin{document}$u$\end{document}, \begin{document}$λ>0$\end{document}, \begin{document}$0≤α<1$\end{document}, \begin{document}$c:[0,∞)\rightarrow (0,∞)$\end{document} is continuous and \begin{document}$h:(0,1)\rightarrow (0,∞)$\end{document} is continuous and integrable. We assume that \begin{document}$f∈ C[0,∞)$\end{document} is such that \begin{document}$f(0)<0$\end{document}, \begin{document}$\lim_{s\rightarrow ∞}f(s) = ∞$\end{document} and \begin{document}$\frac{f(s)}{s^{α}}$\end{document} has a \begin{document}$p$\end{document}-sublinear growth at infinity, namely, \begin{document}$\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$\end{document}. We will discuss nonexistence results for \begin{document}$λ≈ 0$\end{document}, and existence and uniqueness results for \begin{document}$λ \gg 1$\end{document}. We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient explicit method based on the spectral parametric iteration method and quasilinearization scheme, which can be used for the efficient numerical solution of nonlinear stiff/nonstiff two-point boundary value problems. The method derived here has the advantage that it does not require the solution of nonlinear systems of equations. We derive the method, which requires one evaluation of the Jacobian and one LU decomposition per step. Some numerical experiments on nonlinear stiff/nonstiff problems show the efficiency and accuracy of the method. Moreover, the method provides us a simple way to control and modify the convergence rate of the solution. [ABSTRACT FROM AUTHOR]

We consider the problem ..., where a > 0, c ≥ 0, α ∈ (0, 1), h:(0, 1] → (0, ∞) is a continuous function which may be singular at t = 0, but belongs to L¹(0, 1) ∩ C¹(0, 1), and g:[0, ∞) → [0, ∞) is a continuous function. We discuss existence, uniqueness, and non existence results for positive solutions for certain values of a, b and c. [ABSTRACT FROM AUTHOR]

In this paper, we consider the strongly nonlinear Nernst-Planck equations coupled with the quasi-linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi-component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well-posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a-priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the existence of positive solutions of the nonlinear first order problem with a nonlinear nonlocal boundary condition given by x'(t) = r(t)x(t) + Σi=1mfi(t,x(t)), t ∈ [0,1] λx(0) = x(1) + Σj=1nΛj(τj,x(τj)), τj ∈ [0,1], where r : [0, 1] → [0,∞) is continuous, the nonlocal points satisfy 0 ≤ τ1 < ⋯< τn ≤ 1, the nonlinear functions fi and Λj are continuous mappings from [0, 1] × [0,∞) → [0,∞) for i = 1, 2,⋯,m and j = 1, 2, ⋯, n respectively, and λ > 1 is a positive parameter. The Leray—Schauder theorem and Leggett—Williams fixed point theorem were used to prove our results. [ABSTRACT FROM AUTHOR]

By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical results, we study some problems that occur in applied mathematics, namely models of chemical reactors, beams and thermostats. We also apply our theory in order to prove the existence of nontrivial radial solutions of systems of elliptic boundary value problems subject to nonlocal, nonlinear boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of nonlinear boundary value problems (BVPs) consisting of more general class of sequential hybrid fractional equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We study necessary conditions required forexistence ofsolutions tothe suggested system ofBVPsunder the Caratheodory conditions using the technique of measure of noncompactness and degree theory. We also develop conditions for uniqueness results and also on stability analysis. [ABSTRACT FROM AUTHOR]

In this article, we study the initial boundary value problem for nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions ux(0, t) + λ|u(0, t)|r u(0, t) = 0, λ ∈ R - {0}, r > 0. We discuss the local well-posedness when the initial data u0 = u(x, 0) belongs to an L²-based inhomogeneous Sobolev space Hs(R+) with s ∈ (1/2, 7/2) - {3/2}. We deal with the nonlinear boundary condition by first studying the linear Schrödinger equation with a time-dependent inhomogeneous Neumann boundary condition ux(0, t) = h(t) where h ∈ H ...(0, T). [ABSTRACT FROM AUTHOR]

The paper is concerned with equilibrium problems for 2D elastic bodies having thin inclusions with different properties. In particular, rigid and semirigid inclusions are considered. It is assumed that inclusions have a joint point, and we analyze a junction problem for these inclusions. Existence of solutions is proved for each equilibrium problem, and different equivalent formulations of these problems are discussed. In particular, junction conditions at the joint point are found. A delamination of the inclusions is also assumed. This means that we have a crack, and inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. We discuss a convergence to zero and infinity of a rigidity parameter of the semirigid inclusion and analyze limit problems. [ABSTRACT FROM AUTHOR]

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the existence of positive solutions for a certain class of semilinear elliptic equation with nonlinear boundary. Our approach is based on the variational methods. [ABSTRACT FROM AUTHOR]

We study positive radial solutions to singular boundary value problems of the form: { − Δ u = λ K ( | x | ) f ( u ) u α , in Ω , ∂ u ∂ η + c ˜ ( u ) u = 0 , | x | = r 0 , u ( x ) → 0 , | x | → ∞ , where Δ u : = div ( ∇ u ) is the Laplacian operator of u , Ω = { x ∈ R N ‖ x | > r 0 > 0 , N > 2 } , λ > 0 , K ∈ C ( [ r 0 , ∞ ) , ( 0 , ∞ ) ) is such that K ( s ) ≤ 1 s N + β ˆ for s ≫ 1 for some β ˆ > 1 , α < min ⁡ { 1 , β ˆ N − 2 } and ∂ u ∂ η is the outward normal derivative of u on | x | = r 0 . Here, f ∈ C 1 ( [ 0 , ∞ ) , R ) is such that f ( s ) s 1 + α → 0 as s → ∞ , and c ˜ ∈ C ( [ 0 , ∞ ) , ( 0 , ∞ ) ) . We analyse the cases when ( a ) f ( 0 ) > 0 and ( b ) f ( 0 ) < 0 . We discuss existence, non-existence, multiplicity and uniqueness results. We prove our existence results by the method of sub and supersolutions. [ABSTRACT FROM AUTHOR]

We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by... where r : [0, 1] → [0,∞) is continuous; the nonlocal points satisfy 0 ≤T1 0 is a positive parameter. [ABSTRACT FROM AUTHOR]

In this paper, the fractional comparison result of order α ∈ ( 1 , 2 ) is established. By using the monotone iterative technique and the method of upper and lower solutions, we investigate the existence of extremal solutions for a nonlinear fractional differential equation with three-point nonlinear boundary conditions. As an application, an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

Vibrations in an elastic beam supported by nonlinear supports at both ends under the influence of harmonic forces are analyzed in this study. It is hypothesized that the elastic Bernoulli-Euler beam is supported by cubic springs to simulate nonlinear boundary conditions. The dynamic behavior of the beam is described by using the Fourier expansion and the Bessel functions. The Hankel transform is then applied to obtain particular (nonhomogeneous) solutions. This study succeeds in describing the 'jump' phenomenon (instantaneous transition of the system from one position to another) of the vibrating system at certain frequencies. Models based on linear boundary conditions are unable to capture this phenomenon. A larger modulus of elasticity in nonlinear supports increases the frequency of unstable vibrations in the first mode and also widens the frequency region of system instability. This influence is less prominent in the second mode, in which the largest amplitude is smaller than those observed in the first mode. [ABSTRACT FROM AUTHOR]

We study positive radial solutions to: − Δ u = λ K ( | x | ) f ( u ) ; x ∈ Ω e , where λ > 0 is a parameter, Ω e = { x ∈ R N : | x | > r 0 , r 0 > 0 , N > 2 } , Δ is the Laplacian operator, K ∈ C ( [ r 0 , ∞ ) , ( 0 , ∞ ) ) satisfies K ( r ) ≤ 1 r N + μ ; μ > 0 for r ≫ 1 and f ∈ C 1 ( [ 0 , ∞ ) , R ) is a concave increasing function satisfying lim s → ∞ ⁡ f ( s ) s = 0 and f ( 0 ) < 0 (semipositone). We are interested in solutions u such that u → 0 as | x | → ∞ and satisfy the nonlinear boundary condition ∂ u ∂ η + c ˜ ( u ) u = 0 if | x | = r 0 where ∂ ∂ η is the outward normal derivative and c ˜ ∈ C ( [ 0 , ∞ ) , ( 0 , ∞ ) ) is an increasing function. We will establish the uniqueness of positive radial solutions for large values of the parameter λ . [ABSTRACT FROM AUTHOR]

We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions where we impose asymptotic conditions on both non-linearities in the differential equation and on the boundary in such a way that resonance occurs at a generalized eigenvalue; which is an eigenvalue of the linear problem in which the spectral parameter is both in the differential equation and on the boundary. The proofs are based on some variational techniques and topological degree arguments. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the bifurcation of semi-trivial solutions for the Lotka–Volterra competition model with nonlinear boundary conditions over a smooth bounded domain. Applying the Crandall–Rabinowitz local bifurcation theorem Crandall and Rabinowitz (1971) we prove the existence of a smooth curve bifurcating from the appropriate semi-trivial branch. [ABSTRACT FROM AUTHOR]

We provide a characterization for maximal monotone realizations for a certain class of (nonlinear) operators in terms of their corresponding boundary data spaces. The operators under consideration naturally arise in the study of evolutionary problems in mathematical physics. We apply our abstract characterization result to Port–Hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco-elasticity. [ABSTRACT FROM AUTHOR]

Abstract: We consider one dimensional parabolic equations with nonlinear boundary conditions: in , on , in . This equation admits a family of positive stationary solutions ( ) such that if . The main purpose of this paper is to study the stability of these stationary solutions. Furthermore we discuss the large time behavior of global solutions. In particular, we prove that every global solution is uniformly bounded and converges to one of the stationary solutions. [Copyright &y& Elsevier]

We consider an initial-boundary value problem for a fully nonlinear coupled parabolic system with nonlinear boundary conditions modeling hygro-thermal behavior of concrete at high temperatures. We prove a global existence of a weak solution to this system on any physically relevant time interval. The main result is proved by an approximation procedure. This consists in proving the existence of solutions to mollified problems using the Leray-Schauder theorem, for which a priori estimates are obtained. The limit then provides a weak solution for the original problem. A practical example illustrates a performance of the model for a problem of a concrete segment exposed to transient heating according to three different fire scenarios. Here, the focus is on the short-term pore pressure build up, which can lead to explosive spalling of concrete and catastrophic failures of concrete structures. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of positive solutions of a nonlinear second order ordinary differential equation with symmetric nonlinear boundary conditions where both of the nonlinearities are of power type. [ABSTRACT FROM AUTHOR]

A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the blow-up of the solution to a non-local reaction diffusion problem under nonlinear boundary conditions. Lower bounds for blow-up time are derived when the blow-up does occur. [Copyright &y& Elsevier]

Abstract: For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal -regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level. [Copyright &y& Elsevier]

We consider the elliptic problem with nonlinear boundary conditions: -Δu + bu = f(x, u) in Ω, -∂vu = |u|q-1u - g(u) on ∂Ω, where Ω is a bounded domain in Rn. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since Lq+1(∂Ω) ⊂ H¹(Ω) does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations. [ABSTRACT FROM AUTHOR]

Abstract: We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) -a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times. [Copyright &y& Elsevier]

Abstract: The method of upper–lower solutions for nonlinear parabolic systems without the assumption of quasi-monotonicity is obtained. An application is provided, by using the method developed in this paper, involving the existence of positive solutions to certain time-dependent interaction systems arising in biological and medical sciences. Furthermore, the existence of the ω-limit of given system is studied. [Copyright &y& Elsevier]

Abstract: This paper investigates extremal solutions of the boundary value problem for impulsive functional integro-differential equations with nonlinear boundary conditions and deviating arguments. In the presence of a lower solution and an upper solution with , existence of extremal solutions is proved by establishing a new comparison principle and using the monotone iterative technique. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the existence of solutions of first order impulsive functional differential equations with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence of solutions by establishing a new comparison principle and using the monotone iterative technique. A concrete example is presented and solved to illustrate the obtained results. [Copyright &y& Elsevier]

Abstract: In this article, an approach of finite element method has been introduced to predict the distribution of potential and current on electrode surfaces. In this method, the bulk of electrolyte solution and the main body of solid electrodes are modeled to be normal conductive media; while the surface of the electrode, whose current–potential (I–V) curve satisfies experimental data, is modeled to be special nonlinear boundary conditions. This modeling approach for electrochemistry system is successfully verified by experimental data, and would contribute to a better understanding of the distribution of electrical potential and current at electrode surfaces in practical systems with high voltage coupling effect. [Copyright &y& Elsevier]

Abstract: In this paper, we study certain fractional differential equations with nonlinear boundary conditions. By means of the Amann theorem and the method of upper and lower solutions, some new results on the multiple solutions are obtained. [Copyright &y& Elsevier]

Abstract: This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion which is interior to the physical domain . We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in and converges uniformly to a continuous and positive function in . [Copyright &y& Elsevier]

In this work, we consider a class of semilinear elliptic problems with nonlinear boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies on the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x,u) on Γ and u = 0 on Γ', where ν is the outward unit normal to Γ while Γ,Γ' are open, Γ ∩ Γ' = ∅, $\partial\Omega = \overline{\Gamma} \cup \overline{\Gamma'}$, but $\overline{\Gamma}, \overline{\Gamma'}$ need not be disjoint. [ABSTRACT FROM AUTHOR]

A two-dimensional mathematical model for corrosion of metallic bipolar plates in a PEM fuel cell (FC) is presented. Numerical approximations by a novel finite difference technique to deal with nonlinear and mixed boundary conditions are illustrated. Numerical computations and comparisons with experimental data obtained with a laboratory PEMFC are reported. Good qualitative agreement of the simulations has been found with the time dependence of the average current density and with the space distribution of the corrosion rate. [ABSTRACT FROM AUTHOR]

Abstract: We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional -manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the Łojasiewicz–Simon approach, but are of local nature. [Copyright &y& Elsevier]

We investigate the existence of weak solutions to a class of quasilinear elliptic equations with nonlinear Neumann boundary conditions in exterior domains. Problems of this kind arise in various areas of science and technology. An important model case related to the initial data problem in general relativity is presented. As an application of our main result, we deduce the existence of the conformal factor for the Hamiltonian constraint in general relativity in the presence of multiple black holes. We also give a proof for uniqueness in this case. [ABSTRACT FROM AUTHOR]