Search

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.

The purpose of this work is to study the existence and multiplicity of positive solutions for a class of singular elliptic systems involving the p(x)- Laplace operator and nonlinear boundary conditions.

We study positive solutions to the two-point boundary value problem: $$\begin{aligned} \begin{matrix} -u''=\lambda h(t) f(u)~;~(0,1) \\ u(0)=0\\ u'(1)+c(u(1))u(1)=0,\end{matrix} \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$h \in C^1((0,1],(0,\infty ))$$ is a decreasing function, $$f \in C^1((0,\infty ),\mathbb {R}) $$ is an increasing concave function such that $$\lim \limits _{s \rightarrow \infty }f(s)=\infty $$ , $$\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0$$ , $$\lim \limits _{s \rightarrow 0^+}f(s)=-\infty $$ (infinite semipositone) and $$c \in C([0,\infty ),(0,\infty ))$$ is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for $$\lambda \gg 1$$ .

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.

In this work, we peruse a class of Kirchhoff type problems with nonlinear boundary conditions on a bounded domain, and earn the uniqueness and existence of positive solutions for a class of those problems by the variational methods. Also, we earn the multiplicity of positive solutions of those problems by using the Nehari manifold.

For a one-dimensional semilinear wave equation, a mixed problem with nonlinear boundary condition is considered. The uniqueness and the local and global solvability of the problem under consideration are studied depending on the type of nonlinearities in the equation and in the boundary conditions. The cases of nonexistence of a solution not only globally but even locally are considered, as well as the case where this problem has a blow-up solution.

Under study are the boundary value problems that describe the equilibria of two-dimensional elastic bodies with thin weakly curved inclusions in the presence of delamination, which means that there is a crack between the inclusions and an elastic body. Some inequality-type nonlinear boundary conditions are imposed on the crack faces that exclude mutual penetration. This puts the problems into the class of those with unknown contact area. We assume that the inclusions have a contact point, find boundary conditions at the junction point, and justify passage to infinity with respect to the rigidity parameter of the thin inclusion. In particular, we obtain and analyze limit models.

We study a two-point boundary-value problem describing steady states of a population dynamics model with diffusion, logistic growth, and nonlinear density-dependent dispersal on the boundary. In particular, we focus on a model in which the population exhibits hump-shaped density-dependent dispersal on the boundary, and explore its effects on existence, uniqueness and multiplicity of steady states.

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.

In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problem with resonance part and nonlinear boundary conditions. Our approach is variational and is based on the well-known Landesman-Laser-type conditions.

In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)&gt, 0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)&lt, 0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing.

A homogenized model is constructed (with rigorous justification) for a boundary-value problem for the Poisson equation in a periodically perforated domain with a nonlinear Robin condition on the boundary of the cavities. This condition contains a parameter depending on the period of the structure and a function σ(x, u) responsible for the nonlinearity. The cavities can have an arbitrary shape and the parameters of the problem have “critical values,” which results in a homogenized problem with a different type of nonlinearity.

We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small inclusions in the form of balls with absorbing surfaces. The process of absorption is described by the Robin nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We deduce a homogenized equation for the main term of the asymptotics and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.

We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter.

The paper concerns an equilibrium problem for an elastic plate with a thin rigid inclusion. Both vertical and horizontal displacements of the plate are considered in the frame of the model. It is assumed that the inclusion crosses the external boundary of the plate. Moreover, the inclusion is delaminated from the plate which leads to appearance of a crack between the inclusion and the plate. To provide a mutual nonpenetration between crack faces, we consider nonlinear boundary conditions with unknown set of a contact. We prove a solution existence of the equilibrium problem and analyze an inverse problem with unknown elasticity tensor. To formulate the inverse problem, additional data are imposed to be determined from a boundary measurement. An existence of a solution to the inverse problem is proved, and stability properties are established.

We studied the existence of a positive solution to Schrodinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with $C^{1,\alpha}(\overline{\Omega})$-regularity for the superlinear case.

The equilibrium problem for a two-dimensional elastic body containing a thin semirigid inclusion is considered. The inclusion delaminates 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 semirigid inclusion was developed on the assumption that the rigidity of the material differs in different directions. The problem statement is presented both in the form of a variational inequality and in the form of a boundary value problem. The boundary condition on the crack faces has a form of inequality and, as a result, the problem is non-linear. Consequently, 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., Журнал «Математические заметки СВФУ», Выпуск 1 (109) 2021, Pages 51-66

In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

In this paper, we shall classify all positive solutions of [Formula: see text] on the upper half space [Formula: see text] with nonlinear boundary condition [Formula: see text] on [Formula: see text] for parameters [Formula: see text] and [Formula: see text]. We will prove that for [Formula: see text] or [Formula: see text], [Formula: see text] (and [Formula: see text]) all positive solutions are functions of last variable; for [Formula: see text] (and [Formula: see text]) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.

Using variational methods , we prove in a different cases the existence and multiplicity of $a$ - harmonic solutions for the following elleptic problem: \begin {equation*} \begin{gathered} div(a(x, \nabla u))=0, \quad \text{in }\Omega, \\ a(x, \nabla u).\nu=f(x,u), \quad \text{on } \partial\Omega, \end{gathered} \end {equation*} where $\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain of smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector on $\partial\Omega$ . $f: \partial\Omega\times \mathbb{R} \rightarrow \mathbb{R},$ $a: \overline{\Omega}\times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N},$ are fulfilling appropriate conditions.

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

We consider a class of p(x)-Laplacian equations involving nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method proposed by Drabek and Pohozaev, together with the recent idea from Brown and Wu.

In this paper, the one-sided exact boundary null controllability of entropy solutions is studied for a class of general strictly hyperbolic systems of conservation laws, whose negative (or positive) characteristic families are all linearly degenerate. The authors first prove the well-posedness of semi-global solutions constructed as the limit of e-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions and they establish various properties of both the e-approximate front tracking solutions and such solutions. By means of essential modifications of the strategy suggested by the first author in [17] originally for the local exact boundary controllability in the framework of classical solutions, the one-sided local exact boundary null controllability of entropy solutions can then be realized via boundary controls acting on one side of the boundary, where the incoming characteristics are all linearly degenerate.

Рассматриваются вариационные задачи теории упругости в области с разрезом, описывающие равновесие пластины с трещиной. Предполагается, что под действием определенных заданных нагрузок пластина имеет деформации с определенной заранее известной конфигурацией кромок вблизи трещины. На кривой в срединной плоскости, соответствующей трещине, заданы краевые условия в виде двух неравенств. В предположении, что параметр описывает возмущение трещины, находится производная функционала энергии по отношению к . Результаты получены для новых математических моделей с новыми краевыми условиями, описывающими специальный характер контактного взаимодействия кромок пластины., Equilibrium problems for elastic plates with a rectilinear crack are studied. It is assumed that under the action of certain given loads, plates have deformations with a certain predetermined configuration of edges near the crack. On the crack curve, we impose a nonlinear boundary condition as an inequality describing the nonpenetration of the opposite crack faces. Assuming that the parameter describes the crack perturbation, the derivative of the energy functional with respect to is found. The results are obtained for new mathematical models with new nonlinear boundary conditions describing special character of the mechanical contact interaction of the plate edges., №4(104) (2020)

Bifurcation theory is used to prove the existence of positive solutions of some classes of semi-positone problems with nonlinear boundary conditions $$\begin{aligned} {\left\{ \begin{array}{ll} -u''=\lambda f(t, u), \qquad t\in (0,1),\\ u(0)=0, \quad u'(1)+c(u(1))u(1)=0,\\ \end{array}\right. } \end{aligned}$$where $$c:[0, \infty )\rightarrow [0, \infty )$$ is continuous, $$f:[0, \infty )\rightarrow \mathbb {R}$$ is continuous and $$f(t,0)

In this paper we study a model of population which is described by positive solutions to the nonlinear boundary value problem $$ \left\{\begin{array}{ll} -\Delta u = au-bu^{2}-c\frac{u^{2}}{1+u^{2}}-\epsilon, & x\in \Omega,\\ \mathbf{n}.\nabla u+ g(u)=0 , & x\in\partial \Omega.\\ \end{array}\right. $$ Here $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N},$ $\Delta$ is the Laplacian operator, $a,$ $b,$ $c,$ $\epsilon$ are positive parameters and $g \in C^{1}\Big([0,\infty),$ $[\theta,\infty)\Big)$ is decreasing for some $\theta > 0.$ This model describes the dynamics of the fish populations. Our existence results are established via the well-known sub-super solution method.

An equilibrium problem for a 2D elastic body with thin inclusions and defects is analyzed. The presence of defects means that the problem is formulated in a non-smooth domain. The defects are characterized by a positive damage parameter. Nonlinear boundary conditions at the defect faces are imposed to prevent a mutual penetration between the faces. An existence of solutions is proved, and different formulations of the problem are proposed. We study an asymptotics of solutions with respect to the damage parameter and analyze the limit models. Moreover, we study the dependence of the solution on the rigidity parameter of the inclusions. In particular, passages to infinity and to zero of the rigidity parameter are investigated.

In this paper, we study the existence of multiple positive solutions to problem \[\left \{\begin{aligned} &\bigg (a+b \int _\Omega (|\nabla u|^2+|u|^2)\,dx\bigg )(-\Delta u+u)=|u|^{4}u &&\mbox {in } \Omega, \\ &\frac {\partial u}{\partial \nu }=\lambda |u|^{q-2}u &&\mbox {on } \partial \Omega,\end{aligned} \right . \] where $\Omega \subset \mathbb {R}^{3}$ is a smooth bounded domain, $a, b \gt 0$, $\lambda \gt 0$ and $1\lt q\lt 2$. Based on the Nehari manifold and variational methods, we prove that the problem has at least two positive solutions, and one of the solutions is a positive ground state solution.

We study the quenching behavior of the solution of a semi- linear reaction-diffusion system with nonlinear boundary conditions. We prove that the solution quenches in finite time and its quenching time goes to the one of the solution of the differential system. We also obtain lower and upper bounds for quenching time of the solution.

In this paper we study the existence of at least two nontrivial solutions for the nonlinear problem p-Laplacian, with nonlinear boundary conditions. We establish that there exist at least two solutions, which are opposite signs. For this reason, we characterize the first eigenvalue of an intermediary eigenvalue problem by the minimization method. In fact, in some sense, we establish the non-resonance below the first eigenvalues of nonlinear Steklov-Robin.

We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by \[\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}\] where \(r:[0,1] \rightarrow [0,\infty)\) is continuous; the nonlocal points satisfy \(0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1\), the nonlinear function \(f_i\) and \(\tau_j\) are continuous mappings from \([0,1] \times [0,\infty) \rightarrow [0,\infty)\) for \(i = 1,2,\ldots ,m\) and \(j = 1,2,\ldots ,n\) respectively, and \(\lambda \gt 0\) is a positive parameter.

In this paper, a semilinear elliptic equation with a nonlinear boundary condition and a perturbation in the reaction term is studied. The existence of a positive solution and another non-zero solution to the problem is proved when $$|\lambda |$$ is small enough without any specific assumptions on the perturbation term. Moreover, it is shown that the non-zero solution becomes a positive one for small $$\lambda >0$$ under suitable assumptions on the perturbation term.

In this paper, we consider the blow-up problem of the following porous medium equations with nonlinear boundary conditions $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t} =\Delta u^{m}+k(t)f(u) &{}\hbox { in } \Omega \times (0,t^{*}), \\ {}\displaystyle \frac{\partial u}{\partial \nu }=g(u) &{}\hbox { on } \partial \Omega \times (0,t^{*}), \\ {}\displaystyle u(x,0)=u_{0}(x) &{} \hbox { in } \overline{\Omega }, \end{array} \right. \end{aligned}$$ where $$m>1$$ , $$\Omega \subset \mathbb {R}^{n} \ (n\ge 2)$$ is a bounded convex domain with smooth boundary. Under appropriate assumptions on the data, a criterion is given to guarantee that solution u blows up at finite time, and an upper bound for blow-up time is derived. Moreover, a lower bound for blow-up time is also obtained.

Исследуется краевая задача, описывающая равновесие двумерного упругого тела с двумя тонкими сопрягающимися включениями при наличии отслоения. При этом одно из включений является упругим, а другое – полужестким. Наличие отслоения означает существование трещины между включениями и упругим телом. На берегах трещины задаются нелинейные краевые условия вида неравенств, которые не позволяют противоположным берегам трещин проникать друг в друга. Указанные краевые условия приводят к формулировке проблемы в виде задачи с неизвестным множеством контакта. Приведена как дифференциальная постановка в виде краевой задачи, так и вариационная постановка в виде задачи минимизации функционала энергии на выпуклом множестве допустимых перемещений. Обоснована однозначная разрешимость поставленной задачи. Показана эквивалентность дифференциальной и вариационной постановок. Исследован предельный переход по параметру жесткости тонкого упругого включения. Найдены условия сопряжения как для исходной задачи, так и для предельной., An equilibrium problem for elastic bodies with a thin elastic inclusion and a thin semi-rigid inclusion is investigated. The inclusions are assumed to be delaminated from the elastic bodies, forming therefore a crack between the inclusions and the elastic matrix. Nonlinear boundary conditions are considered at the crack faces to prevent mutual penetration between the crack faces. The inclusions have a joint point. We present both differential formulation in the form of a boundary value problem and a variational formulation in the form of a minimization problem for an energy functional on a convex set of admissible displacements. The unique solvability of the problem is substantiated. Equivalence of differential and variational statements is shown. Passage to the limit is investigated as the rigidity parameter of the elastic inclusion goes to infinity. The limit model is analyzed. Junction boundary conditions are found at the joint point for the considered problem as well as for the limit problem., №1 (2018)

We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem $$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\lambda ^{2} u&=Q(x)|u|^{p-2}u \qquad&\text {in} ~~~~\mathbb {R}^{N}_{+}, \\ \frac{\partial u }{\partial n}&=f(x,u) \qquad&\text {on}~~\partial \mathbb {R}^{N}_{+}, \end{aligned}\right. \end{aligned}$$ where $$\lambda $$ is a large number, $$p\in (2,\frac{2N}{N-2})$$ for $$N\ge 3$$ , f(x, u) is subcritical about u and Q is positive and has some non-degenerate critical points in $$\mathbb {R}^{N}_{+}$$ . For $$\lambda $$ large, we can get solutions which have peaks near the non-degenerate critical points of Q.

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H1an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted Ls space, the existence of nontrivial solutions is obtained via variational methods.

By applaying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:\begin{equation*}\begin{gathered}-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad \text{in }\Omega, \\a(x, \nabla u).\nu=\mu g(x,u), \quad \text{on } \partial\Omega,\end{gathered}\end{equation*}where $\lambda$, $\mu \in \mathbb{R}^{+},$$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain ofsmooth boundary $\partial\Omega$ and $\nu$ is the outward normalvector on $\partial\Omega$. $p: \overline{\Omega} \mapsto\mathbb{R}$, $a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto\mathbb{R}^{N},$ $f: \Omega\times\mathbb{R} \mapsto \mathbb{R}$and $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ arefulfilling appropriate conditions.

We investigate the asymptotic behavior, as e → 0, of the solution ue to the boundary value problem for the equation −Δpue = f in a domain Ωe ⊂ ℝn perforated by very thin arbitrarily shaped cavities separated by an O(e) distance in the case of p = n ≥ 3 with a nonlinear third boundary condition of the form \(\partial _{v_p } u_\varepsilon \equiv \left| {\nabla u_\varepsilon } \right|^{n - 2} \left( {\nabla u_\varepsilon ,v} \right) = - \beta ^{n - 1} \left( \varepsilon \right)\sigma \left( {x,u_\varepsilon } \right)\) specified on their boundary, where ν is the outward unit normal vector on the boundary of the cavities. The adsorption coefficient β(e) and the perforation radius ae satisfy conditions that are critical to the given problem. A homogenized model is constructed, and the solutions ue are proved to converge weakly, as e → 0, to the solution of the homogenized problem.

We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-\Delta u=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\partial\nu=u^q$. Here $\nu$ denotes the outer unit normal on the boundary of the halfspace ${\mathbb R}^n_+$ and the exponents $p,q>1$ satisfy $p2$ (or $p2$ and some symmetry of the solutions is assumed). As a typical application of our nonexistence results we provide optimal universal estimates for positive solutions of related problems in bounded and unbounded domains.