Global and Blow-up Solutions to a p-Laplace Equation with Nonlocal Source and Nonlocal Boundary Condition

This paper deals with an evolution p-Laplace equation with nonlocal source subject to weighted nonlocal Dirichlet boundary conditions. We give sufficient conditions for the existence of global and non-global solutions.     

Initial Motion of the Free Boundary for a Non-linear Diffusion Equation

The free boundary between v > 0 and v = 0 for the porousmedium equation v₁= |v|² +nv²v can remain stationary for somepositive waiting-time and then start moving. It is of interestto know the way in which any part of the boundary first moves.It is already known that if the waiting time is given by purelylocal considerations then the boundary speed is continuous,i.e. the initial speed is zero. For cases where the boundary moves before the time found bysimply considering v(x, 0) close to the boundary, a perturbationanalysis indicates that it starts moving with positive speed.Two-dimensional problems show the possible formation of verticesin the boundary. At these points the normal velocity jumps fromzero to some positive value.     

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

The solvability of the nonlocal boundary value problem

Blow-Up of Solutions for a Singular Nonlocal Viscoelastic Equation

We study the nonlinear one-dimensional viscoelastic nonlocal problem： uff-1/x（xux）x＋∫t0g（t-s）1/x）xux（x,s））xds=｜u｜p-2u,
with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].     

Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

一类具有非局部反应扩散方程的奇摄动问题

本文讨论一类具非局部反应扩散方程的奇摄动初始边值问题，利用迭代法及微分不等式，研究了初始边值问题的存在唯一及其渐近性态。     

拟线性方程非局部边值问题解的衰减估计

本文讨论一类源于拟静态热弹性力学和控制理论等领域的边值问题,通过构造适当的上下解得到拟线性方程解的衰减估计。     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

An Explicit Solution of a Parabolic Equation with Nonlocal Boundary Conditions

We consider a parabolic differential equation with nonlocal boundary conditions. We find an explicit solution of the problem and prove the uniqueness of the solution. Analysis of the explicit solution reveals that it has three physically different linear components. The first component is of standing wave type, and the other two are of right- and left-going wave types, respectively. The speed of propagation of the heat waves depends on constants present in nonlocal boundary conditions. We give examples of right-going heat waves that have constant energy. This work was partially supported by Lithuanian State Science and Studies Foundation grant No. C-07/2003. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 315–332, July–September, 2005.     

Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach

We give a unified approach for studying the existence of multiplepositive solutions of nonlinear differential equations of theform where g, f are non-negativefunctions, subject to various nonlocal boundary conditions.We study these problems via new results for a perturbed integralequation, in the space C[0, 1], of the form where [u], ß[u] are linear functionalsgiven by Stieltjes integrals but are not assumed to be positivefor all positive u. This means we actually cover many more differentialequations than the simple equation written above. Previous resultshave studied positive functionals only, but even for positivefunctionals our methods give improvements on previous work.The well-known m-point boundary value problems are special casesand we obtain sharp conditions on the coefficients, which allowssome of them to have opposite signs. We also use some optimalassumptions on the nonlinear term.     

Construction and Investigation of Exact Solutions with Free Boundary to a Nonlinear Heat Equation with Source

Siberian Advances in Mathematics - The article is devoted to the construction and investigation of exact solutions with free boundary to a second-order nonlinear parabolic equation. The solutions...     

A Class of Nonlocal Boundary Value Problemsfor Semilinear Elliptic Equation of Fourth Order

in this paper a class of singularly perturbed nonlocal problem for semilinear ellip tic equation of fourth order is considered.     

A Diffusion Equation Illustrating Spectral Theory for Boundary Layer Stability

The stability of a special solution of a nonlinear diffusion equation is examined. It is shown that the spectral resolution of the linearized stability equation is best accomplished by requiring sufficiently strong decay of the eigenfunctions at infinity. The techniques employed illustrate what is involved in determining the spectral resolution for the equations of hydrodynamic stability when the boundary layer has a transverse component at infinity.