On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

The Bitsadze–Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ? H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λ_j are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity

We consider the chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ? ?ⁿ. The chemotactic sensitivity function is assumed to generalize the prototype It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data (with some r > n), the corresponding initial‐boundary value problem possesses a unique global solution that is uniformly bounded (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of bounded trajectories for some non‐autonomous systems

We prove, by variational arguments, the existence of a solution to the boundary value problem in the half line ((0.1)) where c ≥ 0 and a belongs to a certain class of positive functions. The existence of such a solution in the case c = 0 means that the system (0.1) behaves in significantly different way from its autonomous counterpart. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solutions in a fully parabolic chemotaxis system with singular sensitivity

The Neumann boundary value problem for the chemotaxis system is considered in a smooth bounded domain Ω??ⁿ, n?2, with initial data and v₀∈W^{1, ∞}(Ω) satisfying u₀?0 and v₀>0 in . It is shown that if then for any such data there exists a global‐in‐time classical solution, generalizing a previous result which asserts the same for n=2 only. Furthermore, it is seen that the range of admissible χ can be enlarged upon relaxing the solution concept. More precisely, global existence of weak solutions is established whenever . Copyright © 2010 John Wiley & Sons, Ltd.     

Higher order multi‐point boundary value problems

We consider the boundary value problem where n ? 2 and m ? 1 are integers, t_j ∈ [0, 1] for j = 1, …, m, and f and g_i, i = 0, …, n ? 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi‐point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Local existence for solutions of fully non-linear wave equations

Let Ω be a domain in ?ⁿ and let m? ?; be given. We study the initial-boundary value problem for the equation with a homogeneous Dirichlet boundary condition; here u is a scalar function, $ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $ and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T>0 such that a unique classical solution u exists on [0, T]×Ω. Furthermore, we show that T → ∞ if the data tend to zero.     

On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation

We consider an initial boundary‐value problem for the generalized Benjamin–Bona–Mahony equation. A three‐level conservative difference schemes are studied. The obtained algebraic equations are linear with respect to the values of unknown function for each new level. It is proved that the scheme is convergent with the convergence rate of order k – 1, when the exact solution belongs to the Sobolev space of order . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 301–320, 2014     

On the Eigenvalues of Half‐Linear Boundary Value Problems

We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .     

Single cell discretization of O(kh2 + h4) for the estimates of ${\partial u}\over{\partial n}$ for the two‐space dimensional quasi‐linear parabolic equation

In this article, new stable two‐level explicit difference methods of O(kh² + h⁴) for the estimates of for the two‐space dimensional quasi‐linear parabolic equation are derived, where k > 0 and h > 0 are grid sizes in time and space directions, respectively. We use a single computational cell for the methods, which are applicable to the problems both in cartesian and polar coordinates. The proposed methods have the simplicity in nature and employ the same marching type technique of solution. Numerical results obtained by the proposed methods for several different problems were compared with the exact solutions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 250–261, 2001     

Periodic solutions of the 1D Vlasov–Maxwell system with boundary conditions

We study the 1D Vlasov–Maxwell system with time‐periodic boundary conditions in its classical and relativistic form. We are mainly concerned with existence of periodic weak solutions. We shall begin with the definitions of weak and mild solutions in the periodic case. The main mathematical difficulty in dealing with the Vlasov–Maxwell system consist of establishing L^∞ estimates for the charge and current densities. In order to obtain this kind of estimates, we impose non‐vanishing conditions for the incoming velocities, which assure a finite lifetime of all particles in the computational domain ]0,L[. The definition of the mild solution requires Lipschitz regularity for the electro‐magnetic field. It would be enough to have a generalized flow but the result of DiPerna Lions (Invent. Math. 1989; 98 : 511–547) does not hold for our problems because of boundary conditions. Thus, in the first time, the Vlasov equation has to be regularized. This procedure leads to the study of a sequence of approximate solutions. In the same time, an absorption term is introduced in the Vlasov equation, which guarantees the uniqueness of the mild solution of the regularized problem. In order to preserve the periodicity of the solution, a time‐averaging vanishing condition of the incoming current is imposed: \def\d{{\rm d}}\def\incdist#1#2{\int_{0}^{T}\d t\int_{v_{x}#10}\int_{v_{y}}v_xg_{#2}(t,v_x,v_y)\,\d v}$$\incdist{>}{0}+\incdist{<}{L}=0$$\nopagenumbers\end (1) where g₀, g_L are incoming distributors (2) (3) The existence proof uses the Schauder fixed point theorem and also the velocity averaging lemma of DiPerna and Lions (Comm. Pure Appl. Math. 1989; XVII : 729–757). In the last section we treat the relativistic case. Copyright © 2000 John Wiley & Sons, Ltd.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time periodic solutions of porous medium equation

In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition: The existence of nontrivial nonnegative solution is established provided that 0≤α<m. The existence is also proved in the case α=m but with an additional assumption $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition: The existence of nontrivial nonnegative solution is established provided that 0≤α<m. The existence is also proved in the case α=m but with an additional assumption $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1$, where λ₁ is the first eigenvalue of the operator ?Δ under the homogeneous Dirichlet boundary condition. We also show that the support of these solutions is independent of time by providing a priori estimates for their upper bounds using Moser iteration. Further, we establish the attractivity of maximal periodic solution using the monotonicity method. Copyright © 2010 John Wiley & Sons, Ltd.     

Ein schiefes Randwertproblem zu einer elliptischen Differentialgleichung zweiter Ordnung mit einer expliziten L2-Abschätzung für die zweiten Ableitungen der Lösung

In connection with the free boundary value problem of determining the earth's surface from measurements of gravitational potential and force-field (“the geodetic boundary problem”), an oblique derivative problem arises, where D₀ is some bounded domain, star shaped with respect to the origin. In order to prove a uniquencess theorem for the geodetic boundary problem, it is essential to give estimates for (weighted) L₂-norms of the second derivatives of the solutions so that their bounds can be estimated numerically if bounds for the function describing the boundary are known. In this paper a Fredholm inverse for the above problem is constructed and the second derivatives of the solutions are estimated in the desired form.     

Power concavity on nonlinear parabolic flows

Our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p‐Laplace equation, called power concavity, and that the power is determined by the homogeneity of the parabolic operator. In the parabolic p‐Laplace equation for the density u, the concavity of u^(p?2)/p is considered, which indicates why the log‐concavity has been considered in heat flow, p = 2. In addition, the long time existence of the classical solution of the parabolic p‐Laplacian equation can be obtained if the initial smooth data has ‐concavity and a nondegenerate gradient along the initial boundary. © 2004 Wiley Periodicals, Inc.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Phase transition for potentials of high‐dimensional wells

For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N^± in , we discuss the asymptotics, as ? → 0, of minimizers u_? of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of E _? (u_?) with respect to , we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N⁺ and N^?, and the zeroth‐order term by the energy of minimizing harmonic maps into N^± both under the Dirichlet boundary condition on ?Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

On solution of Lamé equations in axisymmetric domains with conical points

Partial Fourier series expansion is applied to the Dirichlet problem for the Lamé equations in axisymmetric domains ??³ with conical points on the rotation axis. This leads to dimension reduction of the three‐dimensional boundary value problem resulting to an infinite sequence of two‐dimensional boundary value problems on the plane meridian domain Ω_a?? of with solutions u _n(n=0,1,2,…) being the Fourier coefficients of the solution û of the 3D BVP. The asymptotic behaviour of the Fourier coefficients u _n (n=0,1,2,…) near the angular points of the meridian domain Ω_a is fully described by singular vector‐functions which are related to the zeros α_n of some transcendental equations involving Legendre functions of the first kind. Equations which determine the values of α_n are given and a numerical algorithm for the computation of α_n is proposed with some plots of values obtained presented. The singular vector functions for the solution of the 3D BVP is obtained by Fourier synthesis. Copyright © 2004 John Wiley & Sons, Ltd     

Hyperbolic limit of the Jin‐Xin relaxation model

We consider the special Jin‐Xin relaxation model We assume that the initial data ( ) are sufficiently smooth and close to ( ) in L^∞ and have small total variation. Then we prove that there exists a solution ( ) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz‐continuously in the L¹ norm with respect to time and the initial data. Letting , the solution converges to a unique limit, providing a relaxation limit solution to the quasi‐linear, nonconservative system These limit solutions generate a Lipschitz semigroup on a domain containing the functions with small total variation and close to . This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1). © 2005 Wiley Periodicals, Inc.