Nonlinear Dynamic Boundary Conditions Research Articles

Strong Stability for a Viscoelastic Transmission Problem under a Nonlocal Boundary Control

The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Benaissa. Namely, they focused on a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with an emphasis on rough interfaces and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface, where a transmission problem in the presence of a boundary control condition of a nonlocal type was investigated in these papers. Owing to the semigroup theory, we prove the question of well-posedness. For the very rare cases, we combined between the frequency domain approach and the Borichev–Tomilov theorem to establish strong stability results.

Local existence and blow up for the wave equation with nonlinear logarithmic source term and nonlinear dynamical boundary conditions combined with distributed delay

Local existence and blow up for the wave equation with nonlinear logarithmic source term and nonlinear dynamical boundary conditions combined with distributed delay

General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

In this work we investigate a viscoelastic wave equation involving a logarithmic nonlinear source and dynamic Wentzell boundary condition. Making some assumptions on the memory kernel function and using convex function theory and Lyapunov method, we establish the general decay estimate of the solutions. Finally we give two examples to illustrate our results.

On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

Phragmén–Lindelöf theorems for a weakly elliptic equation with a nonlinear dynamical boundary condition

We establish two Phragmén–Lindelöf theorems for a fully nonlinear elliptic equation. We consider a dynamical boundary condition that includes both spatial variable and time derivative terms. As a spatial term, we consider a non-linear Neumann-type operator with a strict monotonicity in the normal direction of the boundary on the spatial derivative term. Our first result is for an elliptic equation on an epigraph in $$\mathbb {R}^n$$ . Because we assume a good structural condition, which includes wide classes of elliptic equations as well as uniformly elliptic equations, we can benefit from the strong maximum principle. The second result is for an equation that is strictly elliptic in one direction. Because the strong maximum principle need not necessarily hold for such equations, we adopt the strategy often used to prove the weak maximum principle. Considering such equations on a slab we can approximate the viscosity subsolutions by functions that strictly satisfy the viscosity inequality, and then obtain a contradiction.

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.

Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of p-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and mathbb {R}^N with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allows us to cover the nonlocal counterparts of a large scope of local diffusion problems like, for example, Stefan problems, Hele–Shaw problems, diffusion in porous media problems and obstacle problems. Nonlinear semigroup theory is the basis for this study.

A qualitative analysis of a second-order anisotropic phase-field transition system endowed with a general class of nonlinear dynamic boundary conditions

The paper is concerned with the study of a nonlinear second-order anisotropic phase-field transition system of Caginalp type, subject to nonlinear and in-homogeneous dynamic boundary conditions (in both unknown functions). Under certain hypothesis on the input data: $ f_{_1}(t,x) $, $ f_{_2}(t,x) $, $ w_{_1}(t,x) $, $ w_{_2}(t,x) $, $ u_0(x) $, $ \alpha_0(x) $, $ \varphi_0(x) $ and $ \xi_0(x) $, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $ W^{1,2}_p(Q)\times W^{1,2}_p(\Sigma) $, $ W^{1,2}_\nu(Q)\times W^{1,2}_p(\Sigma) $. Here we extend the previous results concerned with nonlinearity of cubic type, allowing to the present mathematical model to be more capable for describing the complexity of a wide class of real physical phenomena (moving interface problems, image processing, the phase changes at the boundary $ \partial\Omega $, etc.).

Velocity Stabilization of a Wave Equation With a Nonlinear Dynamic Boundary Condition

This paper deals with a one-dimensional wave equation with a nonlinear dynamic boundary condition and a Neumann-type boundary control acting on the other extremity. We consider a class of nonlinear stabilizing feedbacks that only depend on the velocity at the controlled extremity. The uncontrolled boundary is subject to a nonlinear first-order term, which may represent nonlinear boundary anti-damping. Initial data is taken in the optimal energy space associated with the problem. Exponential decay of the mechanical energy is investigated in different cases. Stability and attractivity of suitable invariant sets are established.

Strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape

We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Holder nonlinear \(\sigma(u)\). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on \(L^2(0,T)\), independently of the regularity of \(\sigma\).

Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Appearance of a Nonlocal Monotone Operator in Transmission Conditions when Homogenizing the Poisson Equation in a Domain Perforated Along an (n − 1)-Dimensional Manifold by Sets of Arbitrary Shape and Critical Size with Nonlinear Dynamic Boundary Conditions on the Boundary of Perforations

We construct and justify an efficient model arising when homogenizing the Poisson equation in an e-periodically perforated domain along an (n − 1)-dimensional manifold by sets of an arbitrary shape and critical size on the boundary of which we impose a nonlinear dynamic condition containing an absorption coefficient of the form e−k, where k takes the critical value (n − 1)/(n − 2), n ≥ 3. We show that the transmission conditions on the manifold contain a nonlocal monotone operator.

A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications

This paper is devoted to a game-theoretic approach to the level-set curvature flow equation with nonlinear dynamic boundary conditions. Under the comparison principle for the dynamic boundary problem, we construct a family of deterministic discrete games, whose value functions approximate the unique viscosity solution. We also apply the game approximation to study the convexity preserving properties and the fattening phenomenon for this geometric dynamic boundary problem.

Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Numerical solution of nonlinear fractional optimal control problems using generalized Bernoulli polynomials

AbstractThis article introduces a new class of basis functions, namely, the generalized Bernoulli polynomials (GBP). The GBP are adopted for solving nonlinear fractional optimal control problems (NFOCP) generated by nonlinear fractional dynamical systems (NFDS) and boundary conditions (BC). The corresponding operational matrices (OM) of fractional derivatives (FD) expand the solution of the problem in terms of the GBP. The method transforms the NFOCP into systems of nonlinear algebraic equations. First, the state and control variables are approximated by the GBP with unknown coefficients and parameters and substituted in the objective function, NFDS and BC. Then, the Gaussian quadrature rule and the OM of FD allow the formulation of a constrained problem, which is solved using Lagrange multipliers. The accuracy of the method is tested by means of several examples and the results confirm its good performance.

Local flow at plate edge during water entry

The local flow near the edge of a horizontal plate impacting a flat liquid surface is investigated through velocity potential flow theory. The inner solution is matched with the outer solution. The far field of the inner solution is assumed to be far away from the other edge of the plate, and thus, its effect can be neglected. The effects of surface tension, viscous friction, and gravity are accounted for in the fully nonlinear dynamic boundary condition on the free surface. When one of these effects is dominant and the other two can be ignored, it is then possible to use self-similar variables to describe the local flow if the entry speed varies with time in a corresponding manner. Detailed results for various self-similar solutions are provided, and the relative importance of the Weber number, Reynolds number, and Froude number is investigated. Simulations are also undertaken for general non-similar flow, and the comparison with the experimental data is also made.

Viscoelastic string-beam coupled vibro-impact system: modeling and dynamic analysis

This paper presents the vibro-impact responses of a viscoelastic string-beam (SB) coupled system in detail for the first time. The Kelvin-Voigt viscoelasticity and the inelastic instantaneous impact are considered in this model. The generalized Hamilton's principle is used to derive the nonlinear dynamic equations and boundary conditions of the viscoelastic SB coupled system. The Galerkin technique truncates the complicated governing partial differential equations. The relationship between the momentum and kinetic energy before and after impact is used to obtain the impact conditions. The frequency-response curves, bifurcation diagrams, time histories, and phase trajectories are utilized to investigate the viscoelastic effect on the complex dynamics and vibro-impact responses of the SB coupled system and determine the underlying mechanism. Finally, the existence of grazing bifurcation is numerically analyzed to ascertain the critical value for the occurrence of impact.

Higher-order exponential integrators for constrained semi-linear parabolic problems

This paper provides an introduction to exponential integrators for constrained parabolic systems. In addition, building on existing results, schemes with an expected order of convergence of three and four are established and numerically tested on parabolic problems with nonlinear dynamic boundary conditions. The simulations reinforce the subjected error behaviour.

Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition

<p style='text-indent:20px;'>For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space <inline-formula><tex-math id="M1">$ H_0^1(\Omega) $</tex-math></inline-formula> at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.</p>

A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions

This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.