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Predators impact prey populations directly through consumption and indirectly via trait-mediated effects like predator-induced emigration (PIE), where prey alter movement due to predation risk. While PIE can significantly influence prey dynamics, its combined effect with direct predation in fragmented habitats is underexplored. Habitat fragmentation reduces viable habitats and isolates populations, necessitating an understanding of these interactions for conservation. In this paper, we present a reaction–diffusion model to investigate prey persistence under both direct predation and PIE in fragmented landscapes. The model considers prey growing logistically within a bounded habitat patch surrounded by a hostile matrix. Prey move via unbiased random walks internally but exhibit biased movement at habitat boundaries influenced by predation risk. Predators are assumed constant, operating on a different timescale. We examine three predation functional responses—constant yield, Holling Type I, and Holling Type III—and three emigration patterns: density-independent, positive density-dependent, and negative density-dependent emigration. Using the method of sub- and supersolutions, we establish conditions for the existence and multiplicity of positive steady-state solutions. Numerical simulations in one-dimensional habitats further elucidate the structure of these solutions. Our findings demonstrate that the interplay between direct predation and PIE crucially affects prey persistence in fragmented habitats. Depending on the functional response and emigration pattern, PIE can either mitigate or amplify the impact of direct predation. This underscores the importance of incorporating both direct and indirect predation effects in ecological models to better predict species dynamics and inform conservation strategies in fragmented landscapes. [ABSTRACT FROM AUTHOR]

We establish an explicit L^\infty (\Omega) a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their H^1(\Omega) norms. To prove our result, we combine in a novel way Moser type estimates together with elliptic regularity and Gagliardo–Nirenberg interpolation inequality. We illustrate our result with an application to subcritical problems satisfying Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

This paper presents a methodology for accurately incorporating the nonlinearity of boundary conditions (BCs) into the mode shapes, natural frequencies, and dynamic behaviour of analytical beam models. Such models have received renewed interest in recent years as a result of their successful implementation in state-of-the-art multiphysics problems. To address the need for this boundary nonlinearity to be more completely captured in the equations of motion, a nonlinear algebra expansion of the classical linear approach for developing solvability conditions for natural frequencies and mode shapes is presented. The method is applicable to any BC that can be accurately represented in polynomial form, either explicitly or through the application of a Taylor expansion; this is the only assumption made in removing the need for the use of analytical approximations of the dynamics themselves. By reducing the BCs of the beam to a system of polynomials, it is possible to utilise the tensor resultant to develop these solvability conditions analogous to the conditions placed on the matrix determinant in linear, classical cases. The approach is first derived for a general set of nonlinear BCs before being applied to two example systems to investigate the importance of including nonlinear tip behaviour in the BCs to accurately predict the system response. In the first, a theoretical, symmetric system, in which a beam is supported by nonlinear springs, is used to explore both the applicability of the methodology and the improvements it can make to the accuracy of the model. Then, the more practical example of a cantilever beam with repulsive magnetic interaction at the tip is used to more explicitly assess the importance of properly incorporating boundary nonlinearity into multiphysics problems.

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

We are concerned with Liouville-type theorems for the nonlinear Schrödinger equation - Δ u + λ | x | α u = | x | β | u | p - 1 u in R + N , with ∂ u ∂ ν = | x | γ | u | q - 1 u on Σ 1 , where Σ 1 : = { x = (x 1 , ... x N) ∈ R N ; x N = 0 , x 1 > 0 } . Here, N ≥ 2 , p , q > 1 , α , β , γ > - 2 and λ is a positive real parameter. We prove the nonexistence of weak sign-changing solutions which are stable or with finite Morse index, possibly unbounded. The main ideas we use here are integral estimates, a Pohozaev type identity, and a monotonicity formula. [ABSTRACT FROM AUTHOR]

We prove the existence of sign-changing solution to the problem - Δ u - 1 2 x · ∇ u = λ u , in R + N , ∂ u ∂ ν = | u | 2 ∗ - 2 u , on ∂ R + N , where R + N = { (x ′ , x N) : x ′ ∈ R N - 1 , x N > 0 } is the upper half-space, 2 ∗ : = 2 (N - 1) / (N - 2) , N ≥ 7 , ∂ u ∂ ν is the partial outward normal derivative and the parameter λ > 0 interacts with the spectrum of the linearized problem. In the proof, we apply variational methods. [ABSTRACT FROM AUTHOR]

We establish some results for positive solutions to class of Laplacian systems with positive parameter, sign change weight functions and nonlinear boundary conditions, in particular, we discuss the existence of positive solutions for a certain range of our parameter using the method of sub-super solutions. Additionally, we introduce novel conditions to ensure the existence of positive solutions for the given system.

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]

We study the existence of solution to the system of differential equations (ϕ (u ′)) ′ = f (t , u , u ′) with nonlinear boundary conditions g (u (0) , u , u ′) = 0 , h (u ′ (1) , u , u ′) = 0 , where f : [ 0 , 1 ] × R n × R n → R n , g , h : R n × C ([ 0 , 1 ] , R n) × C ([ 0 , 1 ] , R n) → R n are continuous, ϕ : ∏ i = 1 n (- a i , a i) → R n , 0 < a i ≤ + ∞ , ϕ (s) = ϕ 1 (s 1) , ⋯ , ϕ n (s n) and ϕ i : (- a i , a i) → R is a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions. [ABSTRACT FROM AUTHOR]

We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and L¹ data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models. [ABSTRACT FROM AUTHOR]

This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global dynamics of a Lotka-Volterra competition-diffusion-advection system with nonlinear boundary conditions, including the existence, nonexistence and global stability of coexistence steady states. We start with the investigation of the principal eigenvalue of linearized system to get the local stability of steady states and then discuss the global dynamics in terms of competition coefficients. [ABSTRACT FROM AUTHOR]

Considering the solutions of a class of noncooperative Kirchhoff-type p (x) -Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (P S) c * condition using another method, diverging from the approaches cited in the previous literature. [ABSTRACT FROM AUTHOR]

In this paper, we study positive solutions of a Steklov problem driven by the (p, 2)- Laplacian operator by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems. [ABSTRACT FROM AUTHOR]

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we present the numerical solution of the Stefan problem to calculate the temperature of the tungsten sample heated by the laser pulse. Mathematical modeling is carried out to analyze field experiments, where an instantaneous heating of the plate to 9000K is observed due to the effect of a heat flow on its surface and subsequent cooling. The problem is characterized by nonlinear coefficients and boundary conditions. An important role is played by the evaporation of the metal from the heated surface. Basing on Samarskii's approach, we choose to implement the method of continuous counting considering the heat conductivity equation in a uniform form in the entire domain using the Dirac delta function. The numerical method has the second-order of approximation with respect to space, the interval of smoothing of the coefficients is 5K. As a result, we obtain the temperature distributions on the surface and in the cross section of the sample during cooling. [ABSTRACT FROM AUTHOR]

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti–Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of eigenvalues for the problem (ϕp(u′(t)))′+λh(t)ϕp(u(t))=0,t∈(0,1),Au(0)−A′u′(0)=0,Bu(1)+B′u′(1)=0\left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t){\phi }_{p}\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ Au\left(0)-A^{\prime} u^{\prime} \left(0)=0,\hspace{1em}Bu\left(1)+B^{\prime} u^{\prime} \left(1)=0\end{array}\right. under hypotheses that ϕp(s)=∣s∣p−2s,p>1{\phi }_{p}\left(s)={| s| }^{p-2}s,p\gt 1, and hh is a nonnegative measurable function on (0,1)\left(0,1), which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem: (ϕp(u′(t)))′+λh(t)f(u(t))=0,t∈(0,1),u(0)=0,au′(1)+c(λ,u(1))=0,\left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t)f\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ u\left(0)=0,\hspace{1em}au^{\prime} \left(1)+c\left(\lambda ,u\left(1))=0,\end{array}\right. where λ\lambda is a real parameter, a≥0a\ge 0, f∈C((0,∞),(0,∞))f\in C\left(\left(0,\infty ),\left(0,\infty )) satisfies infs∈(0,∞)f(s)>0{\inf }_{s\in \left(0,\infty )}f\left(s)\gt 0 and limsups→0sαf(s)0\alpha \gt 0.

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a p (x) -biharmonic operator { Δ p (x) 2 u = λ f (x) | u | q (x) − 2 u , x ∈ Ω , ∂ (| Δ u | p (x) − 2 Δ u) ∂ n = g (x) | u | r (x) − 2 u , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain, Δ p (x) 2 u = Δ (| Δ u | p (x) − 2 Δ u) is the operator of fourth order called the p (x) -biharmonic operator, p (x) , q (x) , r (x) ∈ C (Ω ¯) , and f ∈ C (Ω ¯) , g ∈ C (∂ Ω) are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces L p (x) (Ω) and W m , p (x) (Ω). [ABSTRACT FROM AUTHOR]

The non-planar vibration response of a beam with initial geometric imperfections is investigated through the development of a geometrically nonlinear beam model. The equations of motion are derived through the application of Lagrange's equation and accounting for the interaction of the initial deviations from straightness and the transverse bending deformations. The experimental frequency response of a nuclear fuel rod supported by two prototypical spacer grids is described by the response in the driven direction and the response in the non-driven direction. The beam model also includes a hysteretic restoring moment at the ends to represent the constraint of the spacer grid and the associated reduction in resonant frequency as the applied force is increased. A pseudo-arc-length continuation algorithm is used to solve the equations of motions and calculate the periodic solution. Comparison between the nonlinear beam model response and the experimental data shows good agreement around the resonant frequency of the first bending mode for both the driven and companion modes. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new generalized result to study the existence of solutions of nonlinear fractional boundary value problems (FBVPs). The proposed results unify the existence criteria of certain FBVPs including periodic and antiperiodic as special cases that have been previously studied separately in the literature. The method we employ is topological in its nature and manifests themselves in the forms of differential inequalities (lower and upper solutions, and coupled lower and upper solutions (CLUSs)). Two examples are given to demonstrate the applicability of the developed theoretical results. [ABSTRACT FROM AUTHOR]

We study the solvability of a class of parabolic equations having p(x)-growth structure and involving nonlinear boundary conditions with irregular data. We tackle our problem in a suitable functional setting by considering the so-called Lebesgue and Sobolev spaces with variable exponents. We establish two existence and uniqueness results of solutions to the studied problem. In the first one, we use a subdifferential approach to investigate the existence and uniqueness of a weak solution when the data are regular enough. Second, we assume that the data belong only to the space L 1 and we show the existence and uniqueness of an integral solution to the considered model. Our main idea relies essentially on the use of accretive operator theory and involves some new techniques. [ABSTRACT FROM AUTHOR]

We consider a multiphysics problem of equilibrium for a two-dimensional elastic body containing a thin rigid and thin elastic inclusions. The inclusions delaminate from the elastic matrix, forming a crack; therefore, the problem is posed in a nonsmooth domain with a cut. The mathematical model of the delaminated thin rigid inclusion was developed on the assumption that the function of rigid inclusion displacement has a prescribed structure. The elastic inclusion is modeled in the framework of theory of Timoshenko beam. The problem statement is presented both in a form of a variational inequality and in the form of a boundary value problem including the junction conditions in a common point of inclusions. The boundary condition on the crack faces has a form of inequality and as a result, the problem is non-linear. As a consequence, the construction of an algorithm for the numerical solution of the problem requires the use of additional analytical methods. The methods of domain decomposition, the method of Lagrange multipliers, and the finite element method are used. An algorithm for the numerical solution of the problem is constructed, and a computational example is provided. [ABSTRACT FROM AUTHOR]

We establish existence results for differential equations of the form: (φ(u′))′=f(t,u,u′),$$ {\left(\varphi \left({u}^{\prime}\right)\right)}^{\prime }=f\left(t,u,{u}^{\prime}\right), $$where f$$ f $$ is continuous and φ$$ \varphi $$ is a homeomorphism, with nonlinear boundary conditions: g(u(0),G(u′))=h(u′(1),H(u))=0;g(u′(0),G(u))=h(u(1),H(u′))=0;g(u(0),G(u))=h(u′(1),H(u′))=0org(u′(0),G(u′))=h(u(1),H(u))=0,$$ {\displaystyle \begin{array}{cc}\hfill g\left(u(0),G\left({u}^{\prime}\right)\right)=h\left({u}^{\prime }(1),H(u)\right)=& 0;\kern6.05pt g\left({u}^{\prime }(0),G(u)\right)=h\left(u(1),H\left({u}^{\prime}\right)\right)=0;\hfill \\ {}\hfill g\left(u(0),G(u)\right)=h\left({u}^{\prime }(1),H\left({u}^{\prime}\right)\right)=& 0\kern6.05pt \mathrm{or}\kern6.05pt g\left({u}^{\prime }(0),G\left({u}^{\prime}\right)\right)=h\left(u(1),H(u)\right)=0,\hfill \end{array}} $$where g,h$$ g,h $$ are continuous and G,H$$ G,H $$ are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators. [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]

The article studies a 3D contact problem with Coulomb friction for an elastic body resting on a rigid support. The solution of the quasi-variational formulation of the problem is defined as a fixed point of some mapping that associates the given force of the normal reaction of the support with the value of the normal stress in the contact zone. The fixed point is sought by the method of successive approximations, the convergence of which is proved using modified Lagrange functionals. The results of the numerical solution using finite element modeling and the proximal gradient method are presented. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem a(u, v) = λb(u, v) + μm(u, v) + εF(u, v), for a triple (a, b, m) of continuous, symmetric bilinear forms on a real separable Hilbert space V and nonlinear form F. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results. [ABSTRACT FROM AUTHOR]

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results.

Article Highlights Fully coupled-thermoelastic analysis of a thick-wall cylinder consider temperature-dependent material properties. Investigation on the effect of thermal accumulation due to cyclic thermo-mechanical loads. A comparative study between the impact of considering TIMP and TDMP on the accuracy of the results.

In this work, we investigate the continuum of one‐sign solutions of the nonlinear one‐dimensional Minkowski‐curvature equation with nonlinear boundary conditions by using unilateral global bifurcation techniques. We obtain the existence and multiplicity of one‐sign solutions for this problem and give the global structure of one‐sign solutions set according to different asymptotic behaviors of nonlinearity near zero. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which arise in the thermal explosion model. Firstly, we prove a comparison principle for some kinds of parabolic systems under nonlinear boundary conditions. Using this, we improve a new theorem of the sub-and-super solution. Secondly, based on the new sub-and-super solution theorem, the sufficient conditions that the solution exists and blows up uniformly in finite time are presented. Then, we generalize some of the lemmas related to uniform blow-up solutions, which are used to introduce the uniform blow-up profiles of solutions. Finally, we give several numerical simulations to illustrate the existence and uniform blow-up of solutions. [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.

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]

In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. [ABSTRACT FROM AUTHOR]

In this work, we study positive solutions of a Steklov problem driven by the (p, 2)-Laplacian operator by using the variational method. A sufficient condition for the existence of positive solutions is characterized by the eigenvalues of a linear eigenvalue problem and another nonlinear eigenvalue problem. [ABSTRACT FROM AUTHOR]

In this article, we study the second-order difference equations boundary value problem - Δ [ p (t - 1) Δ u (t - 1) ] = λ g (t) f (u) , t ∈ [ 1 , T ] Z , a 1 u (0) - H 1 (Δ u (0)) = 0 , a 2 u (T + 1) + H 2 (Δ u (T)) = 0 , where T > 1 is an integer, [ 1 , T ] Z = 1 , 2 , ... , T , Δ is the forward difference operator defined by Δ u (t) = u (t + 1) - u (t) , Δ 2 u (t) = Δ (Δ u (t)) , H i : R → R can be nonlinear, i = 1 , 2 , f : [ 1 , T ] Z × R + → R is superlinear or sublinear at ∞ and is allowed be singular (± ∞) at 0, and λ > 0 is a parameter. By constructing a new comparison theorem and using a fixed-point theorem of Krasnosel'skii's type, we obtain not only the existence of positive solutions of this kind of problems, but also the asymptotic behavior of the positive solution as λ → 0 + or λ → ∞ . [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]

Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized- α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems. [ABSTRACT FROM AUTHOR]

Abstract We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner. It is assumed that both inclusions are delaminated from an elastic matrix, thus forming an interfacial crack between the inclusions and the surrounding elastic media. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual nonpenetration of opposite crack faces. The problems of the family depend on a parameter specifying the coordinate of a connection point of the inclusions. For the considered family of problems, we formulate a new inverse problem of finding unknown coordinates of a hinge joint point. The continuity of solutions of the problems on this parameter is proved. The solvability of this inverse problem has been established. Using a passage to the limit, a qualitative connection between the problems for plates with flat and bulk hinged inclusions is shown.

This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, the partial differential equations which govern the steady-state heat transfer process in bodies with temperature-dependent thermal conductivity and temperature-dependent internal heat source. In addition, it will be considered a large class of nonlinear boundary conditions, especially those arising from the description of the heat exchange process from/to bodies at high temperature levels. Proofs of the solution's existence and uniqueness are presented. [ABSTRACT FROM AUTHOR]

In this work, the localized method of fundamental solution (LMFS) is extended to Signorini problem. Unlike the traditional fundamental solution (MFS), the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes. The idea of the LMFS is similar to the localized domain type method. The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix. The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function (NCP-function). Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems. [ABSTRACT FROM AUTHOR]

Abstract In this article we extend the known theory of solution regions to encompass nonlinear boundary conditions. We both provide results for new boundary conditions and recover some known results for the linear case.