Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.     

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

Critical exponents and lower bounds of blow-up rate for a reaction–diffusion system

This paper is concerned with a reaction–diffusion system with absorption terms under Dirichlet boundary conditions, modelling the cooperative interaction of two diffusion biological species. By constructing blow-up sub-solutions and bounded super-solutions, we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions. Moreover, for a special case, we derive the lower bound estimates of blow-up rate.     

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.     

Blow-up time and boundary layer for solutions in parabolic equations with different diffusion

This paper deals with parabolic equations with different diffusion coefficients and coupled nonlinear sources, subject to homogeneous Dirichlet boundary conditions. We give many results about blow-up solutions, including blow-up time estimates for all of the spatial dimensions, the critical non-simultaneous blow-up exponents, uniform blow-up profiles, blow-up sets, and boundary layer with or without standard conditions on nonlocal sources. The conditions are much weaker than the ones for the corresponding results in the previous papers.     

A Quasilinear Parabolic System with Nonlocal Boundary Conditions and Localized Sources

This paper investigates the properties of solutions to a quasilinear parabolic system with nonlocal boundary conditions and localized sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and blow-up rate estimates are also derived.     

一类具强阻尼Kirchhoff方程的能量衰减和解的爆破

The initial boundary value problem for a Kirchhoff equation with Lipschitz type continuous coefficient is studied on bounded domain. Under some conditions, the energy decaying and blow-up of solution are discussed. By refining method, the exponent decay estimates of the energy function and the estimates of the life span of blow-up solutions are given.     

Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for semilinear parabolic systems with nonlinear boundary conditions. The upper and lower bounds of blow-up rates are obtained.     

The blow-up rates for systems of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.     

含梯度项的椭圆方程组的边界爆破解

研究了含梯度项的椭圆方程组的边界爆破解的性质,其中权函数a(x),b(x)为正并且满足一定的条件.利用上下解的方法及比较原则证明了正解的存在性与唯一性,并得到了边界爆破速率的估计.     

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.     

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.     

Blow-Up Properties of Solutions to Parabolic Systems Coupled in Equations and Boundary Conditions

The paper deals with the estimates of the blow-up rate and theblow-up set of positive solutions to semilinear parabolic systemscoupled in the equations and boundary conditions. The explicitupper and lower bounds of the blow-up rate are obtained. Moreover,it is proved that the blow-up set contains a single point x= 1 under the proper conditions on the initial data.     

Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources

This paper deals with blow-up properties of positive solutions to semi-linear parabolic system with nonlocal sources, subject to null Dirichlet boundary conditions. We first obtain some sufficient conditions for blow-up in finite time and a necessary condition for the two components of the solution blow up simultaneously, and then establish uniform blow-up profiles in the interior and estimates of the boundary layer.     

Blow-up and blow-up rate for a reaction–diffusion model with multiple nonlinearities

This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear diffusion, nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities. The necessary and sufficient blow-up conditions are established together with blow-up rate estimates for the positive solutions of the problem.     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition

This paper deals with the blow-up rate estimates of solutions for semilinear parabolic systems coupled in an equation and a boundary condition. The upper and lower bounds of blow-up rates have been obtained.     

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0.