On the Laplace equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Diffusive logistic equation with non-linear boundary conditions

We analyze the solutions of a population model with diffusion and logistic growth. In particular, we focus our study on a population living in a patch, Ω⊆Rn with n?1, that satisfies a certain non-linear boundary condition and on its survival when constant yield harvesting is introduced. We establish our existence results by the method of sub-super solutions.     

The heat equation with stochastic non-linear boundary conditions

In this paper we examine the problem of the heat equation with non-linear boundary conditions of stochastic type.     

Heat equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Asymptotic analysis for the eikonal equation with the dynamical boundary conditions

We study the dynamical boundary value problem for Hamilton‐Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton‐Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large‐time asymptotics of solutions of the limit equation.     

A degenerate diffusion problem with dynamical boundary conditions Diffusion problem with dynamical boundary conditions

The purpose of this paper is to study the existence, the uniqueness and the limit in , as of solutions of general initial-boundary-value problems of the form and in a bounded domain with dynamical boundary conditions of the form Received: 5 December 2000 / Revised version: 20 November 2001 / Published online: 4 April 2002     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system

We consider a structural acoustic wave equation with nonlinear acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with boundary conditions on the interface. We prove wellposedness in the Hadamard sense for strong and weak solutions. The main tool used in the proof is the theory of nonlinear semigroups. We present the system of partial differential equations as a suitable Cauchy problem . Though the operator A is not maximally dissipative we are able to show that it is a translate of a maximally dissipative operator. The obtained semigroup solution is shown to satisfy a suitable variational equality, thus giving weak solutions to the system of PDEs. The results obtained (i) dispel the notion that the model does not generate semigroup solutions, (ii) provide treatment of nonlinear models, and (iii) provide existence of a correct state space which is invariant under the flow-thus showing that physical model under consideration is a dynamical system. The latter is obtained by eliminating compatibility conditions which have been assumed in previous work (on the linear case).     

On the wave equation with semilinear porous acoustic boundary conditions

The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function.     

On a stochastic wave equation with unilateral boundary conditions

We prove the existence and uniqueness of solutions to the initial boundary value problem for a one-dimensional wave equation with unilateral boundary conditions and random noise. We also establish the existence of an invariant measure.

     

Hele-Shaw type problems with dynamical boundary conditions

In this paper, we study the nonlinear evolution equation of Hele-Shaw type with dynamical boundary conditions. That is, the equation ut=Δw+f where u∈H(w) and H is the Heaviside function, with boundary condition μ(x,w)t_∂w+k∇w⋅ν=g, where ν denotes the outward normal vector of the fixed boundary of the domain. We prove existence, uniqueness and some qualitative properties of the solution.     

On the solution of boundary value problems for the laplace equation on the plane with a three-layer film inclusion

We derive analytic formulas that, on the basis of given harmonic functions, permit one to construct solutions of boundary value problems on a plane containing an infinitely thin three-layer film by simple quadratures with the preservation of singular points of the harmonic functions. The three-layer film consists of alternating strongly and weakly permeable layers and has the form of a ray, a segment, or a circle arc.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

Stefan problem for the laplace equation with regard for the curvature of the free boundary

We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

A Nystrm method for the numerical solution of Laplace's equation with non-linear boundary conditions on a polygon

A non-linear boundary value problem for Laplace's equation ona polygonal domain is solved with a Nystrm method applied toa boundary integral equation reformulation. The method usesmeshes which are graded toward the corners. Numerical examplesare included.     

On non-linear elliptic equation with weight

In this paper, we prove the existence and uniqueness theorem of the Dirichlet problem for certain non-linear elliptic operators with Muckenhoupt weight.