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This paper deals with the coupled fourth-order parabolic equations, subject to Dirichlet or Navier boundary conditions. We combine the generalized potential well method, the concave method, with the auxiliary methods to classify the initial energy about the existence of blow-up solutions and global solutions. Moreover, the decay estimate of global solutions and the bounds of blow-up time and rate are determined, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with the parabolic equations coupled via nonlocal multiple nonlinearity. We determine the critical Fujita exponents for blow-up solutions, which are influenced by the time coefficients. Moreover, we obtain the complete and optimal classification on simultaneous and non-simultaneous blow-up of solutions. Then simultaneous blow-up rates and estimates on blow-up time are studied according to the precise blow-up phenomena for all of the dimensions of the space domain. [ABSTRACT FROM AUTHOR]

This paper deals with qualitative properties of solutions in a parabolic equation with nonstandard growth conditions and singular medium void. For the sublinear source, we prove that all solutions are global. For the linear source, the solutions are global provided that the measure of domain is small. For the superlinear source, the solutions blow up at infinite time given the initial data belonging to the unstable set, and blow up in finite time with positive or negative energy, respectively. The exponential decay rates of global solutions are demonstrated as well and the bounds of blow-up time are determined for all dimensions of space domain. At last, we discuss the blow-up rates of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the reaction–diffusion equations with variable coefficients in some bounded domains. At least one of the components of solutions blows up for every initial data in some exponent regions, where the Fujita exponents are determined by the exponents of the sources and the coefficients and the dimension of the domain. We also show the classifications of simultaneous and nonsimultaneous blow-up of the components of solutions. The asymptotic properties are discussed including blow-up rates and sets. Moreover, the upper and the lower bounds of blow-up time are given for all dimensions of domains, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider the time-weighted parabolic equations in a bounded domain, subject to homogeneous Dirichlet boundary conditions. First, we determine the critical Fujita exponents of solutions prescribed by the coefficients of weighted functions and the first eigenvalue of Laplacian operator with zero Dirichlet boundary. Second, we distinguish completely simultaneous blow-up from non-simultaneous blow-up of two components of solutions. Third, blow-up rates and blow-up time estimates are studied according to different blow-up phenomena. It can be found out that the nonlinear functions of the components determine the orders of them. [ABSTRACT FROM AUTHOR]

This paper deals with the nonsimultaneous blow-up problems forwith coupled nonlinear boundary flux. The main results extend the existence of two components blowing up simultaneously in the paperJ. Math. Anal. Appl.2009;356:215–231 to the case ofcomponents blowing up simultaneously. We verify that blow-up phenomena are independent of the choosing of the initial data in some exponent regions, while, in some others, the blow-up phenomena depend sensitively on the choosing of initial data. Moreover, the blow-up rates and sets are obtained for blow-up components. [ABSTRACT FROM PUBLISHER]

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities. [ABSTRACT FROM PUBLISHER]

Abstract: This paper considers blow-up solutions for reaction–diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215–231]. [Copyright &y& Elsevier]

Abstract: This paper deals with u t =Δu + u m (x, t)e pv(0,t), v t =Δv + u q (0, t)e nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u 0(v 0) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m >1(n >0), while u(v) blows up everywhere for 0⩽ m ⩽1 (n =0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions. [Copyright &y& Elsevier]

Abstract: This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions. [Copyright &y& Elsevier]

Abstract: This paper deals with the non-simultaneous and simultaneous blow-up for some parabolic systems coupled via nonlinear boundary flux . For radially symmetric solutions, we obtain that one component can blow up by itself and may provide sufficient help to the blow-up of the other ones under suitable initial data. In particular, such phenomena happen for every initial data in some exponent regions. It is interesting that there exist initial data such that any two components blow up simultaneously, either of which blows up depending on itself and also can give sufficient help to the other components blowing up simultaneously. A necessary and sufficient condition is obtained on the simultaneous blow-up of at least two components for all initial data. Moreover, the non-simultaneous and simultaneous blow-up rates and sets are determined. [Copyright &y& Elsevier]

Abstract: This paper deals with simultaneous and non-simultaneous blow-up solutions for heat equations coupled via exponential sources, subject to null Dirichlet boundary conditions. The main results complete the previously known results on the optimal classification for simultaneous and non-simultaneous blow-up solutions by covering the whole ranges of exponents. Moreover, all kinds of simultaneous and non-simultaneous blow-up rates are obtained. [Copyright &y& Elsevier]

Abstract: This paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution to heat equations coupled via nonlinear boundary . It is proved that there exist suitable initial data such that blows up alone if and only if . All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k, i, n) of and leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained. [Copyright &y& Elsevier]

Abstract: The paper deals with the radially symmetric solutions of , , subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that () blows up alone if and only if (), which means that any blow-up is simultaneous if and only if , . (ii) Any blow-up is () blowing up with () remaining bounded if and only if , (, ). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if , . Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model. [Copyright &y& Elsevier]

Abstract: This work deals with heat equations coupled via nonlinear boundary flux which obey different laws. We give a complete classification for non-simultaneous and simultaneous blow-up by covering all of the possible exponents. [Copyright &y& Elsevier]

Abstract: This paper deals with heat equations coupled via nonlinear boundary flux , , . A necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions. Three kinds of exponent regions are obtained as follows, (i) only one component blows up for every initial data; (ii) the existence of two components blowing up simultaneously while the third one remains bounded, also with two kinds of blow-up rates and ; (iii) e.g.,  remains bounded and blow up simultaneously with blow-up rate for every initial data. Moreover, the eight kinds of simultaneous blow-up rates and blow-up sets are discussed. [Copyright &y& Elsevier]

Abstract: This paper deals with heat equations coupled via exponential and power nonlinearities, subject to null Dirichlet boundary conditions. The complete and optimal classification on non-simultaneous and simultaneous blow-ups is proposed by four sufficient and necessary conditions. We find out that, in some exponent region, the blow-up properties of the solutions depend much on the choosing of initial data. Moreover, all kinds of non-simultaneous and simultaneous blow-up rates are obtained. [Copyright &y& Elsevier]

Abstract: This paper deals with the blow-up behavior of radial solutions to a parabolic system multi-coupled via inner sources and boundary flux. We first obtain a necessary and sufficient condition for the existence of non-simultaneous blow-up, and then find five regions of exponent parameters where both non-simultaneous and simultaneous blow-up may happen. In particular, nine simultaneous blow-up rates are established for different regions of parameters. It is interesting to observe that different initial data may lead to different simultaneous blow-up rates even with the same exponent parameters. [Copyright &y& Elsevier]

Abstract: This paper deals with a reaction-diffusion equation with inner absorption and boundary flux of exponential forms. The blow-up rate is determined with the blow-up set, and the blow-up profile near the blow-up time is obtained by the Giga–Kohn method. It is observed that the blow-up rate and profile are independent of the nonlinear absorption term. [Copyright &y& Elsevier]

In this paper, we study a Cauchy problem of parabolic equations with time-dependent coefficients. The critical Fujita exponents are determined by the exponents of components and dimension of domain and the coefficients. Moreover, we give the optimal classification of blow-up solutions which discriminates simultaneous and non-simultaneous blow-up of two components. At last, simultaneous blow-up rates are obtained. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a parabolic system, cross-coupled via a nonlinear source and a nonlinear boundary flux. We get a necessary and sufficient condition for the existence of non-simultaneous blow-up. In particular, four different simultaneous blow-up rates are obtained in different regions of parameters, described by an introduced characteristic algebraic system. It is observed that different initial data may result in different simultaneous blow-up rates even in the same region of parameters. [Copyright &y& Elsevier]

Abstract: This paper deals with a reaction–diffusion system with coupled nonlinear inner sources and a nonlinear boundary flux. Blow-up rates are determined for four different blow-up situations. The so-called characteristic algebraic system is introduced to get a very simple and clear description for the desired blow-up rate estimates. It is pointed out that one cannot directly use super and sub-solutions to establish blow-up rate estimates, since they do not share the same blow-up time in general. [Copyright &y& Elsevier]

In this paper, we study a reaction–diffusion system involving coupled sources and nonlinear nonlocal boundary flux. By using the comparison principle, we give the criteria on blow-up and global solutions. There exist no nontrivial global solutions under some conditions on the coefficients. After classifying simultaneous and non-simultaneous blow-up of the components of solutions, we obtain blow-up rates and sets. Lower and upper bounds of blow-up time are determined quantificationally for all dimensions of the space domains. [ABSTRACT FROM AUTHOR]

This paper deals with a reaction–diffusion equation with nonlocal inner source and nonlocal boundary flux. Besides the influence of exponents, we find out that the weighted functions and the measure of the domain could lead to the existence of global solutions for any initial data. Larger weighted functions require smaller initial data in guaranteeing the existence of global solutions to the problem. Blow-up rates are obtained under different dominations of nonlinearities in one dimensional space. The upper and lower bounds of blow-up time are determined for all spatial dimensions, respectively. Moreover, the solutions blow up only on the boundary when the nonlocal flux dominates the blow-up. [ABSTRACT FROM AUTHOR]