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This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary-value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn- and Friedrichs-type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L2 norm of a vector-valued function from H1(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L2 norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.

This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the inhomogenenous quasi-linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:C1[0, T], Ψ[·]:C1[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x, t)) in the quasi-linear parabolic equation u t (x, t) = (k(u(x, t))u x (x, t)) x , with Dirichlet boundary conditions u(0, t) = ψ 0 , u(1,t) = ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input-output mappings Φ[·]: K →C 1 [0, T], Ψ[·]: Κ → C 1 [0, T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[·] and Ψ[·] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0, t))u x (0, t) or/and h(t):=k (u(1,t))u x (1, t), the values k(ψ 0 ) and k(ψ 1 ) of the unknown diffusion coefficient k(u(x,t)) at (x, t) = (0,0) and (x,t) = (1,0), respectively, can be determined explicitly. In addition to these, the values k u (ψ 0 ) and k u (ψ 1 ) of the unknown coefficient k(u(x, t)) at (x,t)=(0,0) and (x, t) = (1, 0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input-output mappings Φ[·]:Κ→ C 1 [0, T], Ψ[·]:K→ C 1 [0, T] are given explicitly in terms of the semigroup.

Consider the two adjacent rectangular wedges K1, K2 with common edge in the upper halfspace of ℝ3 and the operator A (=−Laplacian multiplied by different constant coefficients a1, a2 in K1, K2, respectively) acting on a subspace of ∏2j=1L2(Kj). This subspace should consist of those sufficiently regular functions u=(u1,u2) satisfying the homogeneous Dirichlet boundary condition on the bottom of the upper halfspace. Moreover, the coincidence of u1 and u2 along the interface of the two wedges is prescribed as well as a transmission condition relating their first one-sided derivatives. We interpret the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. In this paper it is shown (by Friedrichs' extension) that A is selfadjoint in a suitable Hilbert space. Applying the Fourier (-sine) transformations we reduce our problem with singularities along the z-axis to a non-singular Klein–Gordon equation in one space dimension with potential step. The resolvent, the limiting absorption principle and expansion in generalized eigenfunctions of A are derived (by Plancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theorem for unbounded selfadjoint operators on Hilbert spaces yields the solution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physical geometry of the problem and its spectral properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

In this paper, we establish existence and finite time blow up of weak solutions of a parabolic equation of p(x)-Laplacian type with the Dirichlet boundary condition. Moreover, we obtain upper and lower bounds for the blow up time of solutions, by employing concavity method and differential inequality technique respectively.

In this paper, we study the stability of a one-dimensional Bresse system with infinite memory type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. Unlike [6], [10], and [22], we consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions and generalizes that of [6], [10], and [22].

The main purpose of this paper is to analyze the convergence and regularity of our proposed algorithm of the finite element methods coupled with a theta time discretization scheme for evolutionary Hamilton-Jacobi-Bellman equations with linear source terms with respect to the Dirichlet boundary conditions (Appl. Math. Comput., 262 (2015), 42.55 ). Also, an optimal error estimate with an asymptotic behavior in uniform norm is given. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we study the global existence and asymptotic behavior of the Boussinesq-Burgers system subject to the Dirichlet boundary conditions. Based on the Lp(p > 2) estimates of the solution, which are different from the standard L2-based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider semilinear elliptic equations of the form Δu + f(u) = 0 over a quarter space with Dirichlet boundary conditions. Given a suitable positive root z of f, we show how to construct a non-negative bounded solution u converging to a one-dimensional limiting profile V with V . This is established using Perron's method by constructing sub-solutions and super-solutions and employing a sliding argument. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the Neimark–Sacker bifurcation of a diffusive food-limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank-Nicolson difference scheme, and nonstandard finite-difference scheme. The existence of Neimark-Sacker bifurcation at the equilibrium is obtained. Our results show that Crank-Nicolson and nonstandard finite-difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove the Saint-Venant compatibility conditions in Lp for p∈(1,+∞), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in Lp to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p∈(1,+∞). Copyright © 2015 John Wiley & Sons, Ltd.

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L2-based Sobolev spaces. A useful intermediary result is the well-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L2-based Sobolev spaces. We obtain this well-posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well-posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the global existence of solution for the quasilinear chemotaxis system with Dirichlet boundary conditions, and further we show that the blow up properties of the solution depend only on the first eigenvalue. Copyright © 2014 John Wiley & Sons, Ltd.

This paper describes well-posedness, spectral representations, and approximations of solutions of uniformly elliptic, second-order, divergence form elliptic boundary value problems on exterior regions U in when N ≥ 3. Inhomogeneous Dirichlet, Neumann, and Robin boundary conditions are treated. These problems are first shown to be well-posed in the space E1(U) of finite-energy functions on U using variational methods. Spectral representations of these solutions involving Steklov eigenfunctions and solutions subject to zero Dirichlet boundary conditions are described. Some approximation results for the A-harmonic components are obtained. Positivity and comparison results for these solutions are given. Copyright © 2014 John Wiley & Sons, Ltd.

We study in this paper the Dirichlet boundary value problem of second-order differential system in the form where u ∈ Rn,A,B ∈ Rn,M ∈ Rn × n is a symmetric matrix, F : [0,1] × Rn Rn such a system comes from a model describing the vibration of a multi-storey building. By using the saddle point theorem, we prove an existence theorem for the solutions to the given system. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with a class of fourth-order nonlinear difference equations. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Dirichlet boundary value problems and give some new results. Our results successfully complement the existing ones. Copyright (c) 2014 John Wiley & Sons, Ltd.

In this paper, we consider the Petrov–Galerkin spectral method for fourth-order elliptic problems on rectangular domains subject to non-homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher-order problems. By applying these results to a fourth-order problem, we establish the H2-error and L2-error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we consider the existence of solutions for second-order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.

This paper deals with the following doubly degenerate parabolic system with null Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ RN, where m, n ≥ 1, p, q ≥ 2, r1, r2, s1, s2 ≥ 1, α, β

In this paper, we study the existence of solutions for damped nonlinear impulsive differential equations with Dirichlet boundary conditions. By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution. Some recent results are extended and significantly improved. Finally, some examples are presented to illustrate our main results. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we consider the existence and multiplicity for second-order nonlinear impulsive differential equations with Dirichlet boundary condition and a parameter. By using critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution or infinitely many solutions, assuming that the impulsive functions satisfy the superlinear growth condition and the parameter inequality is reverse. Our results extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem CDαu(t)+f(t,u(t))=0,0

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.

Communicated by W. Sprosig This paper deals with radial solutions to localized reaction-diffusion equations with variable exponents, subject to homogeneous Dirichlet boundary conditions. The global existence versus blow-up criteria are studied in terms of the variable exponents. We proposed that the maximums of variable exponents are the key clue to determine blow-up classifications and describe blow-up rates for positive solutions. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper we study well-posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ℝ2; ω⩾0, ωλ1+µ>0 with λ1 being the first eigenvalue of −Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.

In the paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel oscillating boundary. This domain consists of the body, a large number of thin periodically situated cylinders joining to the body through thin random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems. Moreover we derive estimates of deviation of the solution to initial problem from the solution to the homogenized problem in different cases. It appears that depending on small parameters in Fourier boundary conditions of initial problem one can obtain Dirichlet, Neumann or Fourier boundary conditions in the homogenized problem. We estimate the convergence of solutions in these three cases.

In this paper, we consider the following Kirchhoff type equation: with initial condition and zero Dirichlet boundary condition. We established sufficient conditions on the initial data with arbitrarily high energy such that the solution blows up in finite time. This result generalizes and improves the earlier results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.

This paper describes existence, uniqueness and special eigenfunction representations of H1-solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H1(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H1(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, we study the asymptotic relation between the solutions to the one-dimensional viscous conservation laws with the Dirichlet boundary condition and the associated inviscid solution. We assume that the viscosity matrix is positive definite, then we prove the existence and the stability of the weak boundary layers by discussing nonlinear well-posedness of the inviscid flow with certain boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.

In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre-malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, a constant heat transfer coefficient present in a nonlinear Robin-type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one-dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.

This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or boundary blowup problems without gradient term, for which the corresponding existence results are also derived, which is a partial extension and supplement to the previous works. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper the nonlinear viscoelastic wave equation in canonical form |u t | ρ u tt - Δu - Δu tt + ∫ t 0 g(t - τ)Δu(τ) dτ = b|u| P-2 u with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set.

The paper deals with theoretical analysis of non-stationary incompressible flow through a cascade of profiles. The initial-boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.

SUMMARY In this paper, we consider the non-linear wave equation utt − � ut − div(|∇u| m ∇u )+ a|ut| � ut = b|u| p u a; b?0, associated with initial and Dirichlet boundary conditions. Under suitable conditions on � , m, and p, we give precise decay rates for the solution. In particular, we show that for m = 0, the decay is exponential. This work improves the result by Yang (Math. Meth. Appl. Sci. 2002; 25:795 – 814). Copyright ? 2005 John Wiley & Sons, Ltd.

This article deals with a boundary value problem for Laplace equation with a non-linear and non-local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non-linear and non-local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.

In this paper we construct the scattering operator for the forced non-linear Schrodinger equation with a potential on the half-line. Moreover, in the case where the force is zero, and the solutions satisfy the homogeneous Dirichlet boundary condition at zero, we prove that the scattering operator determines uniquely the potential and the non-linearity and we give a method for the reconstruction of both. Copyright © 2005 John Wiley & Sons, Ltd.

In this paper we consider the non-linear wave equation u tt - Δu t - div(|⊇u| α-2 ⊇u)-div(|⊇u t | β-2 ⊇u t )+a|u t | m-2 u t =b|u| p-2 u a,b>0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α, β, m, p and for negative initial energy, a global non-existence theorem. This improves a result by Yang (Math. Meth. Appl. Sci. 2002; 25:825-833), who requires that the initial energy be sufficiently negative and relates the global non-existence of solutions to the size of Ω.

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of time-harmonic acoustic scattering of an incident field by a sound-soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double- and single-layer potential and a Dirichlet half-plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α1=α2) of solution are analysed. For , the blow-up time, blow-up rate and blow-up set of blow-up solution are estimated and the asymptotic behaviour of solution near the blow-up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper positive solutions of the heat equation with a nonlinear Neumann boundary conditions in an upper halfspace are studied. The optimal result on blow-up rate, valid for all solutions which blow up in finite time, is established under the assumption that the exponent of a nonlinear boundary condition is subcritical in the Sobolev sense. This complements results derived for the bounded domain case in [10, 13] either for monotonous solutions or under a stronger restriction on the exponent of a boundary condition.

The paper deals with the Dirichlet problem for the Stokes linear equation in a domain exterior to an open surface. With the help of the theory of boundary integral (pseudo-differential) equations uniqueness and existence theorems are proved in the Bessel-potential and Besov spaces and C a -smoothness (with α < 1/2) of solution is established in the neighbourhood of the boundary of the open surface.

Large class of non-Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three-dimensional domain is established for subcritical ps, i.e. for p>11/5. Our method works for ‘all’ physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper a non-linear system of equations in a one-dimensional domain is considered, where both the viscosity, v, and R are positive. The existence of solutions in the case of Neumann boundary conditions for θ and a polynomial form of p(θ, ϵ) is proved. When θ satisfies the Dirichlet boundary condition, the existence of solutions is obtained under additional assumptions limiting the growth of p(θ, ϵ) with respect to θ. In both cases the uniqueness of the solutions is also established.

The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (ut – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for um – um-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of UΔt and ∇UΔt, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Holder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates xΔt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.

The present paper employs potential well arguments of a previous work by the authors [6] to obtain a sharp existence-nonexistence alternative for solutions to the linear heat equation subject to a non-linear boundary condition.

In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of Rn(n ⩾ 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R3 with vanishing Dirichlet or Neumann boundary condition.

A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.