Blow-up solutions and global solutions for nonlinear parabolic equations with mixed boundary conditions

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to mixed boundary condition. 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 “blow-up time”, blow-up rate and global solutions are obtained also. The results improve and extend importantly the findings obtained by A. Friedman and R. Sperb.     

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.     

Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition

In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate.     

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.     

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.     

一类具非局部源抛物型方程组解的一致爆破性质

考虑带有齐次Dirichlet边界条件,具非局部源项的半线性抛物型方程组正解的爆破性质,首先给出了该问题的解在有限时刻爆破的充分条件,以及解的两个分量同时爆破的必要条件,并建立了解的一致爆破模式.     

Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources

This paper investigates the global existence of the nonnegative solution and the finite time blow-up of solutions of nonlinear parabolic equation with a more complicated source term, which is a product of localized source, local source, and weight function; we also study the blow-up rate of solution to this 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.     

A degenerate parabolic system with localized sources and nonlocal boundary condition

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

On the Cauchy problem for the periodic generalized Degasperis-Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis-Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.     

Blow-Up of the solution of the initial boundary-value problem for the generalized Boussinesq equation with nonlinear boundary condition

The initial boundary-value problem for a nonlinear equation of pseudoparabolic type with nonlinear Neumann boundary condition is considered. We prove a local theorem on the existence of solutions. Using the method of energy inequalities, we obtain sufficient conditions for the blow-up of solutions in a finite time interval and establish upper and lower bounds for the blow-up time.     

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 phenomena for a class of quasilinear parabolic problems under Robin boundary condition

This note deals with a class of heat emission processes in a medium with a non-negative source, a nonlinear decreasing thermal conductivity and a linear radiation (Robin) boundary condition. For such heat emission problems, we make use of a first-order differential inequality technique to establish conditions on the data sufficient to guarantee that the blow-up of the solutions does occur or does not occur. In addition, the same technique is used to determine a lower bound for the blow-up time when blow-up occurs.     

Blow-up rate for a nonlinear diffusion equation

In this work we study the blow-up rate for a nonlinear diffusion equation with an inner source and a nonlinear boundary flux, which is equivalent to a porous medium equation with convection. Depending upon the sign of a parameter included, the source can be positive or negative (absorption). By the scaling method, we obtain that the blow-up rate is independent of a negative source, while for the situation with a positive source, the blow-up rate is determined by the interaction between the inner source and the boundary flux. Comparing with the previous results for the porous medium model without convection, we observe that the gradient term included here does not affect the blow-up rates of solutions.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

In this paper, the blow-up rate of solutions of semi-linear reaction-diffusion equations with a more complicated source term, which is a product of nonlocal (or localized) source and weight function a(x), is investigated. It is proved that the solutions have global blow-up, and that the rates of blow-up are uniform in all compact subsets of the domain. Furthermore, the blow-up rate of _∞|u(t)| is precisely determined.     

Blow-up phenomena for a class of fourth order parabolic equation

A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).     

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.