Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

On the sufficient condition for regularity of a boundary point for singular parabolic equations with non‐standard growth

We investigate the continuity of solutions for general nonlinear parabolic equations with non‐standard growth near a nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for regularity of a boundary point.     

Blow‐up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions

This paper deals with the blow‐up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow‐up occurs at some finite time. We also obtain upper and lower bounds for the blow‐up time when blow‐up occurs. Copyright © 2014 John Wiley & Sons, Ltd.     

Time delayed parabolic systems with coupled nonlinear boundary conditions

The aim of this paper is to show the existence and uniqueness of a solution for a system of time-delayed parabolic equations with coupled nonlinear boundary conditions. The time delays are of discrete type which may appear in the reaction function as well as in the boundary function. The approach to the problem is by the method of upper and lower solutions for nonquasimonotone functions.

     

Existence of solution for boundary value problem of nonlinear fractional differential equation

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation.By means of Schauder fixed-point theorem,an existence result of solution is obtained.     

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Ω₀ which is interior to the physical domain Ω⊂Rn. We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Ω₀ and converges uniformly to a continuous and positive function in .     

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

非线性三阶常微分方程的非线性三点边值问题解的存在性

基于上下解方法,在一定条件下,得到了一类带有非线性混合边界条件的三阶常微分方程的非线性三点边值问题解的存在性,作为上述存在性结果的应用,在推论中给出了一类三阶非线性微分方程三点边值问题解的存在性.     

一类二阶微分议程边值问题三解的存在性

§ 1　 IntroductionWe are interested in the existence ofthree-solutions ofthe following second-order dif-ferential equations with nonlinear boundary value conditionsx″=f( t,x,x′) ,　　 t∈ [a,b] ,( 1 .1 )g1 ( x( a) ,x′( a) ) =0 ,　　 g2 ( x( b) ,x′( b) ) =0 ,( 1 .2 )where f:[a,b]×R1 ×R1 →R1 ,gi:R1 ×R1 →R1 ( i=1 ,2 ) are continuous functions.The study ofthe existence of three-solutions ofboundary value prolems forsecond or-der differential equations was initiated by Amann[1 ] .In[1 …     

Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions

We prove the existence of nonnegative solutions of the problem , , for a physically motivated class of nonlinearity . The results, which are established using a ``forbidden value' argument, are new even in the case of linear .

     

Existence and regularity of solution of mixed boundary value problem of a Busemann-equation with nonlinear absorb term

The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R². With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.     

A PDAE formulation of parabolic problems with dynamic boundary conditions

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential–algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Ω⊂RN be a bounded domain with Lipschitz boundary, with a>0 on . Let σ be the restriction to ∂Ω of the (N−1)-dimensional Hausdorff measure and let be σ-measurable in the first variable and assume that for σ-a.e. x∈∂Ω, B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞), the operator Ap:=div(a|∇u|^p−2∇u) with nonlinear Wentzell-Robin type boundary conditions

     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Nonlinear partial differential equations of fourth order under mixed boundary conditions

Solutions to nonlinear partial differential equations of fourth order are studied. Boundary regularity is proved for solutions that satisfy mixed boundary conditions. Various geometric situations including so called triple points are considered. Regularity is measured in Sobolev-Slobodeckii spaces and the results are sharp in this scale. The approach is based on the use of a first order difference quotient method.