A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

A porous medium equation with nonlocal boundary condition and a localized source

This article studies the blow-up properties of solutions to a porous medium equation with nonlocal boundary condition and a general localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, it is proved that the unique solution has global blow-up property whenever blow-up occurs. Blow-up rate estimates are also obtained for some special cases.     

Blow-up for degenerate parabolic equations with nonlocal source

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source in subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain.

     

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow-up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.     

一类非局部非线性扩散方程解的全局爆破

主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破.     

A degenerate parabolic system with nonlocal boundary condition

In this article, we investigate the blow-up properties of the positive solutions to a degenerate parabolic system with nonlocal boundary condition. We give the criteria for finite time blow-up or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate for small weighted nonlocal boundary.     

Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries

This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries.By using super-and sub-solution techniques,we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively,and then give the necessary and sufficient conditions that two components u and v blow up simultaneously.Finally,the uniform blow-up profiles in the interior are presented.     

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

Global blow-up for a localized quasilinear parabolic system with nonlocal boundary conditions

In this article, we investigate the positive solution of a localized quasilinear parabolic system with nonlocal boundary conditions. Under certain conditions, the global existence and finite time blow-up criteria are established, and the global blow-up behaviour is also obtained.     

Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition

In this article, a porous medium equation with nonlocal boundary condition and a localized source is studied. The results of the existence of global solutions or blow-up of solutions are given. The blow-up rate estimates are also obtained under some conditions.     

A nonlinear parabolic equation with a nonlocal boundary term

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe’s method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.     

Propagations of singularities in a parabolic system with coupling nonlocal sources

This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles are established. The critical exponent of the system is determined also. This work was supported by the National Natural Science Foundation of China (Grant No. 10771024)     

Attractors for a plate equation with nonlocal nonlinearities

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of R ⁿ with smooth boundary, γ ,ρ ?0 and are nonlocal functions. The main result states that the dynamical system {S (t )}_{t ?0} associated with this problem has a compact global attractor. In addition, in the limit case γ  = 0, it is also shown that {S (t )}_{t ?0} has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.     

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source.Conditions for the existence of global or bl...     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.