Nonlinear Neumann Boundary Conditions Research Articles

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

Doubly reflected generalized BSDEs with jumps and an obstacle problem of parabolic IPDEs with nonlinear Neumann boundary conditions

Abstract A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions.

Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${\mathcal {C}}^1$$ variation of the domain

Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${\mathcal {C}}^1$$ variation of the domain

Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation on with initial conditions and a nonlinear Neumann boundary condition at x = 0. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at x = 0 have enough regularity to solve the initial value problem with their own initial data.

$$\pmb N$$-Laplacian Equation with a Nonlinear Neumann Boundary Condition and a Singular Nonlinearity

$$\pmb N$$-Laplacian Equation with a Nonlinear Neumann Boundary Condition and a Singular Nonlinearity

Global existence of solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary conditions

We consider classical solutions to the chemotaxis system with logistic source, au−μu2, under nonlinear Neumann boundary conditions ∂u∂ν=|u|p with p>1 in a smooth convex bounded domain Ω⊂Rn, where n≥2. This paper aims to show that if p<32, and μ>0, n=2, or μ is sufficiently large when n≥3, then the parabolic-elliptic chemotaxis system admits a unique nonnegative global-in-time classical solution that is bounded in Ω×(0,∞). The similar result is also true if p<32, n=2, and μ>0 or p<75, n=3, and μ is sufficiently large for the parabolic-parabolic chemotaxis system.

Blow-up phenomena and global existence for nonlinear parabolic problems under nonlinear boundary conditions

&lt;abstract&gt;&lt;p&gt;In this paper, we consider an initial-boundary value parabolic problem under nonlinear Neumann boundary conditions. By virtue of the modified differential inequality, lower bounds for the blow-up time of the solution are derived in higher dimensional spaces. An upper bound for the blow-up time are specified under appropriate assumptions on the functions $ a, b, f, g, h $ and $ u_0 $.&lt;/p&gt;&lt;/abstract&gt;

Reflected generalized discontinuous BSDEs with rcll barrier and an obstacle problem of IPDE with nonlinear Neumann boundary conditions

Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.

Viscosity solutions of system of PDEs with interconnected obstacles and nonlinear Neumann boundary conditions

This paper investigates the Hamilton-Jacobi-Bellman system of equations associated with the m-states optimal switching problem in finite horizon when the state process is constrained to live in a connected bounded closed domain. We show existence and uniqueness of the solution in viscosity sense of the system. The main tool is the notion of systems of generalized reflected backward stochastic differential equations with oblique reflection and the Feynman-Kac representation of their solutions in the Markovian framework.

Nonlinear Neumann Problems for Fully Nonlinear Elliptic PDEs on a Quadrant

We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub- and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a $C^{1,1}$ test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general $n$-dimensional orthant by Biswas et al. [SIAM J. Control Optim., 55 (2017), pp. 365--396], where the functions ($H_i$ in the main text) describing the boundary condition are required to be positively one-homogeneous, and the result in this paper removes the positive homogeneity in two dimensions. An existence result for solutions is also presented.

Averaging principle for stochastic variational inequalities with application to PDEs with nonlinear Neumann conditions

Stochastic variational inequalities have been widely used in various areas. In this paper we establish averaging principles for a separated time-scale system of fully coupled stochastic system characterized by stochastic variational inequalities. Under non-Lipschitz continuous conditions, we show that the classical weak convergence result holds for this type of stochastic systems. Strong convergence is also studied for the cases when the diffusion coefficients of the slow motions do not depend on the fast motion components. As an application, we study the homogenization of generalized backward SDEs and semilinear parabolic variational inequalities with nonlinear Neumann boundary conditions.

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required for the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required for the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is monotone in the solution variable. Note that the existence of a viscosity solution to this problem has been established in [13] then completed in [15]. In the present paper, we construct a sequence of penalized systems of decoupled forward backward stochastic differential equations (FBSDEs) then we directly show its strong convergence. This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1, 3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem.

Parametric superlinear double phase problems with singular term and critical growth on the boundary

In this paper, we study quasilinear elliptic equations driven by the double phase operator along with a reaction that has a singular and a parametric superlinear term and with a nonlinear Neumann boundary condition of critical growth. Based on a new equivalent norm for Musielak–Orlicz Sobolev spaces and the Nehari manifold along with the fibering method, we prove the existence of at least two weak solutions, provided the parameter is sufficiently small.

Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-viscosity sense. The method we use is based on backward stochastic differential equations and their S-tightness. This work is motivated by the fact that many PDEs in physics have discontinuous coefficients. As a consequence, it follows that if the uniqueness holds, then the solution can be constructed by a penalization.

Homogenization of periodic elliptic degenerate PDEs with non-linear Neumann boundary condition

In this paper, a semi-linear elliptic partial differential equation (PDE) with non linear Neumann boundary condition and rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of R d . Our fully probabilistic method is based on the connection between PDEs and backward stochastic differential equations (BSDEs) with random terminal time and the weak convergence of a class of diffusion processes.

Stochastic viscosity solutions for stochastic integral-partial differential equations

In order to study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs), a new class of generalized backward doubly stochastic differential equations with general jumps is investigated. The definition of stochastic viscosity solutions of SIPDEs is introduced. A probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions is given.

Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation(−Δ)σu=|x|αupinB1\\{0} with an isolated singularity at the origin, where σ∈(0,1) and the punctured unit ball B1\\{0}⊂Rn with n≥2. When −2σ<α<2σ and n+αn−2σ<p<n+2σn−2σ, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (1981) [21], but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on S+n. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

AbstractA parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.