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In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value μ ⁎ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for 0 μ μ ⁎ and linearly unstable for μ > μ ⁎ . In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for 0 μ μ ⁎ when neglecting translations.

In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n=3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. The angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When c>0 is sufficiently small, the stability of the steady state solution is studied, where c is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given.

This chapter is devoted to general boundary value problems for second-order elliptic differential operators. We begin in Sect. 6.1 with a summary of the basic facts about existence, uniqueness and regularity of solutions of the Dirichlet problem in the framework of Holder spaces. In Sect 6.2, using the calculus of pseudo-differential operators, we prove existence, uniqueness and regularity theorems for the Dirichlet problem in the framework of Sobolev spaces. In Sect. 6.3 we formulate general boundary value problems, and show that these problems can be reduced to the study of pseudo-differential operators on the boundary. The virtue of this reduction is that there is no difficulty in taking adjoints after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints. This allows us to discuss the existence theory more easily.

This paper is concerned with a nonlinear free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, set with a Robin boundary condition, in which, both nonnecrotic tumors and necrotic tumors are taken into consideration. The well-posedness and asymptotic behavior of solutions are studied. It is shown that there exist two thresholds, denoted by σ and σ ∗ , on the surrounding nutrient concentration σ . If σ ≤ σ , then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if σ > σ , then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if σ σ ≤ σ ∗ and necrotic if σ > σ ∗ . The connection and mutual transition between the nonnecrotic and necrotic phases are also given.

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ = σ ( x ) and the inner pressure p = p ( x ) . Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20] . In this paper we study existence of nonradial solutions by using the bifurcation method. Let { γ k } k = 2 ∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k ⁎ ⩾ 2 such that in the two dimension case for any k ⩾ k ⁎ , γ k is a bifurcation point, and in the three dimension case for any even k ⩾ k ⁎ , γ k is also a bifurcation point.

We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy fo...

This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration $\sigma =\sigma (r,t)$ occupies the unknown tumor region $r< R(t)$ and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius $R=R(t)$ satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for $\sigma $ is imposed, in this paper the model uses the Robin boundary condition for $\sigma $ . We prove existence and uniqueness of a global in-time classical solution ( $\sigma (r,t),R(t)$ ) for arbitrary $c>0$ and establish asymptotic stability of the unique stationary solution ( $\sigma _{s}(r),R_{s}$ ) for sufficiently small $c$ , where $c$ is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set boundary conditions which result in a stochastic differential equation for the trace of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

In this paper, we try to construct the seamless immersed boundary phase-field method for simulating the two-phase flows. The seamless immersed boundary method is one of the Cartesian grid approaches, in which the forcing term is added to the incompressible Navier–Stokes equations in order to satisfy the velocity condition on the boundary. In the seamless immersed boundary method, the forcing term is added not only on the grid points near the boundary but also on the grid points inside the boundary. This method is applied to the phase-field equation, i.e., the Cahn–Hilliard equation for the two-phase flow analysis. By using the Taylor series expansion in multi-variable, the correction term for satisfying the Neumann boundary condition is estimated. The phase separation in a rotated square cavity is considered, in order to validate the present approach. It is found that the rotated solutions obtained on the Cartesian coordinates are the same as the original solution at any time. Then, it is concluded that the present seamless immersed boundary phase-field method is very fruitful for simulating the complicated two-phase flows.

A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn-Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn-Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and rigorously discussed., Key words: Cahn-Hilliard system, convection, dynamic boundary condition, initial-boundary value problem, well-posedness, regularity of solutions

We study the qualitative behavior of the Boussinesq–Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H 1 × H 2 initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data.

A SPH-FDM boundary method is proposed for the analysis of thermal process in homogeneous media with a discontinuous interface in this study, in which the smoothed particle hydrodynamics (SPH) method is used in the inner computational domain; and the finite difference method (FDM) is used as the function approximation near the boundary. This mixed method not only can improve the calculation accuracy under the first-type boundary conditions (i.e., Dirichlet), but also can convert the second- and third-type boundary conditions (i.e., Neumann and Robin) into the first-type boundary conditions in solving heat conduction problems of homogeneous media. As a result, a second-order accuracy can be achieved in the entire solution domain. The proposed SPH-FDM boundary method is applicable to the analysis of heat conduction in various media, including the problems with discontinuous interface in the computational domain and the solidification of materials with a moving phase transition boundary. Numerical results show that the proposed SPH-FDM boundary method overcomes the difficulties of the conventional SPH method in dealing with the second- and third-type boundary conditions and has a very high calculation accuracy.

In Liu and Lou (2015), the authors considered the reaction–diffusion equation u t = u x x + f ( u ) with Robin and free boundary conditions. For the initial data σ ϕ , there exists σ ∗ > 0 such that spreading happens when σ > σ ∗ and vanishing happens when σ σ ∗ . In the transition case that σ = σ ∗ , the solution u ( t , x ) converges to the ground state with a suitable shift: V ( ⋅ − ξ ( t ) ) , and ξ ( t ) tends to a finite number (Case 1) or to ∞ (Case 2) as t → ∞ . For both cases, the right free boundary h ( t ) always propagate to infinity. In this paper, we will discuss the expanding speed of h ( t ) of these two cases. Actually, h ( t ) = 1 − f ′ ( 0 ) ln t + O ( 1 ) in Case 1 and h ( t ) = 3 2 − f ′ ( 0 ) ln t + O ( 1 ) in Case 2.

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical...

Fourth-order boundary value problems with discontinuous boundary conditions are studied. We prove that the eigenvalues depend not only continuously but smoothly on the coefficients and on the boundary conditions and find formulas for the derivatives with respect to each of these parameters.

In this paper, using the Nehari manifold and fibering maps, we study the existence of at least two positive solutions for the following p-Kirchhoff equation where is a bounded domain, with a, b, , is the p-Laplacian operator, is a parameter.

We investigate the Markov process on the boundary of a bounded Lipschitz domain associated to the Neumann and Robin boundary value problems. We first construct L p -semigroups of sub-Markovian contractions on the boundary, generated by the boundary conditions, and we show that they are induced by the transition functions of the forthcoming processes. As in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected Brownian motion. The Robin problem is treated with a Kato type L p -perturbation method, using the Revuz correspondence. An exceptional (polar) set occurs on the boundary. We make the link with the Dirichlet forms approach.

In this paper, we study the non-reflecting boundary condition for the time-harmonic Maxwell's equations in homogeneous waveguides with an inhomogeneous inclusion. We analyze a series representation of solutions to the Maxwell's equations satisfying the radiating condition at infinity, from which we develop the so-called electric-to-magnetic operator for the non-reflecting boundary condition. Infinite waveguides are truncated to a finite domain with a fictitious boundary on which the non-reflecting boundary condition based on the electric-to-magnetic operator is imposed. As the main goal, the well-posedness of the reduced problem will be proved. This study is important to develop numerical techniques of accurate absorbing boundary conditions for electromagnetic wave propagation in waveguides.

In this paper, we study a coupled system of fractional boundary value problems subject to integral boundary conditions. By applying a recent fixed point theorem in ordered Banach spaces, we investigate the local existence and uniqueness of positive solutions for the coupled system. We show that the unique positive solution can be found in a product set, and that it can be approximated by constructing iterative sequences for any given initial point of the product set. As an application, an interesting example is presented to illustrate our main result.

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth.

We consider the stability of a system of two strongly coupled wave equations by means of only one boundary feedback. We show that the stability of the system depends in a very complex way on all of the involved factors such as the type of coupling, the hinged regularity and the accordance of boundary conditions. We first show that the system is uniformly exponentially stable if the undamped equation has Dirichlet boundary condition, while it is only polynomially stable if the undamped equation is subject to Neumann boundary condition.Next, by a spectral approach, we show that this sensibility of stability with respect to the boundary conditions on the undamped equation is intrinsically linked with the transmission of the vibration as well as the dissipation between the equations.

This paper addresses the problem of heat transport in an elliptical channel in the presence of a temperature gradient parallel to its axis. The Williams equation is used as the basic equation describing the kinetics of the process and a model of diffusive reflection is used as the boundary conditions on the channel wall. The deviation of the gas condition from the equilibrium is assumed to be small. In order to find a linear correction to the local equilibrium function of distribution, a boundary problem consisting of a linear homogeneous partial differential equation of the first order with a homogeneous boundary condition has been built. The solution of the built boundary value problem has been found by the method of characteristics. The value of the heat flow through the cross section of the channel is found by using numerical procedures implemented by the computer algebra Maple 17 system. The results were compared with the analogous results found in the open press.

A nonlinear second-order differential equation with a small parameter at the derivatives is considered in the case where the limit algebraic equation has a multiple root. The matching method is applied to construct an asymptotic expansion of the solution of the boundary value problem. Two boundary layer variables with different scales are used to describe the asymptotic solution near the boundary.

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).

A polynomial solution of the classical one-phase Stefan problem with a Neumann boundary condition is presented. As a result of the multiple integration of the heat-conduction equation, a sequence of identical equalities has been obtained. On the basis of these equalities, solutions were constructed in the form of the second-, third-, fourth-, and fifth-degree polynomials. It is shown by test examples that the approach proposed is highly efficient and that the approximation errors of the solutions in the form of the fourth- and fifth-degree polynomials are negligible small, which allows them to be considered in fact as exact. The polynomial solutions obtained substantially surpass the analogous numerical solutions in the accuracy of determining the position of the moving interphase boundary in a body and are in approximate parity with them in the accuracy of determining the temperature profile in it.

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right)\] with coefficient $a(x)$ that is thegeneralized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by thereduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.

This article studies the solutions in H 1 of a steady transport equation with a divergence-free driving velocity that is W 1 , ∞ , in a two-dimensional bounded polygon. Since the velocity is assumed fully non-homogeneous on the boundary, existence and uniqueness of the solution require a boundary condition on the open part Γ − , where the normal component of u is strictly negative. In a previous article, we studied the solutions in L 2 of this steady transport equation. The methods, developed in this article, can be extended to prove existence and uniqueness of a solution in H 1 with Dirichlet boundary condition on Γ − only in the case where the normal component of u does not vanish at the boundary of Γ − . In the case where the normal component of u vanishes at the boundary of Γ − , under appropriate assumptions, we construct local H 1 solutions in the neighborhood of the end-points of Γ − , which allow us to establish existence and uniqueness of the solution in H 1 for the transport equation with a Dirichlet boundary condition on Γ − .

We prove that we can uniquely recover the coefficient of a one-dimensional heat equation from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well-known Gelfand–Levitan–Gasymov inverse spectral theory of Sturm–Liouville operators.

It is considered the (slow) flow of the viscous incompressible fluid in the Hele-Shaw cell at presence of a rigid obstacle in the flow. A novel model for such a flow is proposed in terms of the parametrization of the boundary of the fluid front and Green’s function (or the Robin–Neumann function) for the Laplace equation in a doubly connected domain subject of the mixed boundary value problem (the Neumann problem on the boundary of the obstacle and the Robin problem on the fluid front). Preliminary results for asymptotic study of such Green’s function are presented too.

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.

A hybrid method involving boundary analysis and boundary collocation is used to obtain an approximate solution for a plane problem of uncoupled thermoelasticity with mixed thermal and mechanical boundary conditions in a square domain with one curved side. The unknown functions in the cross-section are obtained in the form of series expansions in Cartesian harmonics. A boundary analysis reveals the singular behavior of the solution at the transition points. In order to simulate the weak discontinuities of the temperature function and the discontinuities of stress, these expansions are enriched with proper harmonic functions with a singular behavior at the transition points. The results are discussed, and the functions of practical interest are represented on the boundary and also inside the domain. The locations where possible debonding of the fixed part of the boundary may take place are noted.

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next, in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.

We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.

In this paper we consider the existence and general energy decay rate of global solution to the mixed problem for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. In order to prove the existence of solutions, we employ the Galerkin method and compactness arguments. Besides, we establish an explicit and general decay rate result using the perturbed modified energy method and some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve some earlier results, i.e., exponential or polynomial decay rates.