Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.     

A generalization of the Archimedean double sequence

A reaction-diffusion system involving an integrodifferential linear parabolic equation and a nonlinear ordinary differential equation is studied, under third type boundary conditions. Such a system may be used to describe the evolution of a class of man-environment epidemics. Conditions are given for the existence and the asymptotic stability of a unique nontrivial equilibrium solution, by using monotone iteration techniques.     

Automatic control problems for reaction-diffusion systems

A class of nonlinear reaction-diffusion systems is considered. We formulate some automatic control problems based on feedback devices located on the boundary. Two different types of devices are analyzed: relay switch and Preisach hysteresis operator. The resulting models lead to a nonlinear integrodifferential parabolic system with nonlinear and nonlocal boundary conditions. We prove global existence and uniqueness of solutions in both the cases considered.     

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the \(W^{1,\infty }\) norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain \(W^{1,\infty }\) estimates, which are the key to the numerical analysis of these problems.     

Renormalized solutions of nonlinear parabolic equations with diffuse measure data

Given a parabolic cylinder Ω × (0, T), where Ω is a bounded domain of ${\mathbb{R}^N}$ , we consider IBV problems involving equations of the type $$b(u)_{t} - \Delta_{p} u = \mu$$ where b is a increasing C ¹-function and μ is a diffuse measure. We prove the existence and uniqueness of a renormalized solution for this class of nonlinear parabolic equations.     

The Temporal Evolution of a System in Combustion Theory. II

A model consisting of two nonlinear coupled parabolic equations governing the combustion of a material is considered. Under certain assumptions, the concentration of the combustible material is shown to be largely spatially homogeneous. Construction of upper and lower solutions for the equation governing the temperature is then given in terms of the solution of an associated ordinary integrodifferential equation. Estimates for the critical parameters for the three type-A geometries are obtained.     

Difference Schemes of Fully Nonlinear Parabolic Systems of Second Order

The general difference schemes for the first boundary problem of the fully nonlinear parabolic systems of second order f(x, t, u, u_x, u_{xx}, u_t) = 0 are considered in the rectangular domain Q_T = {0 ≤ x ≤ l, 0 ≤ t ≤ T}, where u(x, t) and f(x, t, u, p, r, q) are two m-dimensional vector functions with m ≥ 1 for (x, t) ∈ Q_T and u, p, r, q ∈ R^m. The existence and the estimates of solutions for the finite difference system are established by the fixed point technique. The absolute and relative stability and convergence of difference schemes are justified by means of a series of a priori estimates. In the present study, the existence of unique smooth solution of the original problem is assumed. The similar results for nonlinear and quasilinear parabolic systems are also obtained.     

Nonlinear Integroparabolic Equations on Unbounded Domains: Existence of Classical Solutions with Special Properties

Classical solvability is established for a certain nonlinear integrodifferential parabolic equation, on unbounded domains in several dimensions. The model equation of the Fokker-Planck type represents a regularized version of an equation recently derived by J. A. Acebrón and R. Spigler for the physical problem of describing the time evolution of large populations of nonlinearly globally coupled random oscillators. Precise estimates are obtained for the decay of convolutions with fundamental solutions of linear parabolic equations on unbounded domains in R ⁿ . Existence of a classical solution with special properties is established.     

Local Existence of Strong Solutions to the 3D Zakharov-Kuznetsov Equation in a Bounded Domain

We consider here the local existence of strong solutions for the Zakharov-Kuznetsov (ZK) equation posed in a limited domain $\mathcal{M}=(0,1)_{x}\times(-\pi/2, \pi/2)^{d}$ , d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution on a short time interval, whose length only depends on the given data. We use the parabolic regularization of the ZK equation as in Saut et al. (J. Math. Phys. 53(11):115612, 2012) to derive the global and local bounds independent of ? for various norms of the solution. In particular, we derive the local bound of the nonlinear term by a singular perturbation argument. Then we can pass to the limit and hence deduce the local existence of strong solutions.     

Stability for abstract linear functional differential equations

A class of parabolic partial integrodifferential equations with discrete and distributed delays in the spatial derivatives of maximum order is considered. After the study of well posedness of the initial value problem the asymptotic behaviour of the solutions is investigated through the spectral properties of the infinitesimal generator of the solution semigroup.     

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps, whose data (the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.     

Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

In this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.     

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form

$$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$

where \(T>0\) and \(\Omega \) is a Reifenberg domain. We suppose that the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) and to more general setting of Lorentz spaces.

     

Approximate Controllability of Nonlinear Delay Integro Differential Evolution Equations with Random Impulses

The basic concept of this research is to analyse the approximate controllability (AC) of a nonlinear delay integrodifferential evolution system (NDIDES) with random impulse of the type \begin{align*}&z''(\zeta)=\mathfrak{A}(\zeta)z(\zeta)+(\mathfrak{B}x)(\zeta)+\int_{0}^{\zeta}\mathcal{H}(\zeta, s,z(\beta(s))), \ \sigma_{q} <\zeta < \sigma_{q+1}, \ \zeta\in [\zeta_{0}, \mathcal{T}], \\ &z(\sigma_{q})=a_{q}(\tau_{q})z(\sigma^{-}_{q}), ~~q = 1,2,\ldots,\\ &z_{\zeta_{0}}=\upsilon,\end{align*} by assuming that the linear system is approximately controllable. The existence and uniqueness of the mild solution to above system have been determined by using the Banach contraction principle and trajectory accessible sets. We generalize the results for NDIDES with and without fixed-type impulsive moments.     

A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of nonlinear integrodifferential impulsive differential equations with nonlocal conditions $$\left\{\begin{array}{l} x'(t)=Ax(t)+f\left(t,x(t), \int_{0}^{t}k(t,s,x(s))ds\right), \quad t\in J=[0,b], \,\, t\neq t_{i},\\ x(0)=g(x)+x_{0},\\ \Delta x(t_{i})=I_{i}(x(t_{i})), \quad i=1,2,\ldots,p, \,\, 0=t_{0} < t_{1} < \cdots < t_{p} < t_{p+1}=b.\end{array} \right.$$ Using the generalized Ascoli-Arzela theorem given by us, some fixed point technique including Schaefer fixed point theorem and Krasnoselskii fixed point theorem, and theory of operators semigroup, some new results are obtained. At last, some examples are given to illustrate the theory.     

数值积分对完全非线性抛物型积分微分方程有限元方法的影响*

抛物型积分微分方程在带有记忆性的热传导 ,扩散 ,生物力学等实际问题中有广泛的应用 本文考虑如下模型 :ｃ(ｘ ,ｕ) ｔ = · {ａ(ｘ ,ｕ) ｕ + ∫ｔ0 ｂ(ｘ ,ｔ,τ ,ｕ(ｘ ,τ) ) ｕ(ｘ ,τ)ｄτ}+ ｆ(ｘ ,ｔ ,ｕ)　　　　　　　　　　　　　　　　 (ｘ ,ｔ) ∈Ω× [0 ,Ｔ]ｕ(ｘ ,ｔ) =0 ,　　 (ｘ ,ｔ)∈ Ω× [0 ,Ｔ]ｕ(ｘ ,0 ) =ｕ0 (ｘ) ,　　ｘ ∈Ω ,其中Ω Ｒｎ 为多角型区域 , Ω为边界 不用数值积分 ,对这类问题有限元方法研究已有很多工作[5 -8,11] ,导出了最优Ｌ2 及Ｈ1模估计 众所周知 ,解偏微分方程的有限元方法最终…     

Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

Here we give an existence and uniqueness result of a renormalized solution for a class of nonlinear parabolic equations \(\displaystyle {\partial b(u) \over \partial t} - \mathrm{div}(a(x,t,\nabla u))+\mathrm{div}(\Phi (x,t, u))=\mu \), where the right side is a measure data, b is a strictly increasing \(C^1\)-function, \(- \mathrm{div}(a(x,t,\nabla u))\) is a Leray–Lions type operator with growth \(|\nabla u|^{p-1}\) in \(\nabla u\) and \(\Phi (x,t, u)\) is a nonlinear lower order term.     

A model of liquid level measurements

A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level.     

Superconvergence phenomena in nonlinear two-point boundary-value problems

Superconvergence estimates are derived for a class of strongly nonlinear two-point boundary-value problems. The analysis considers solution and gradient superconvergence points as well as flux postprocessing formulas. Finally, the extension to parabolic problems is considered. © 1993 John Wiley & Sons, Inc.