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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 work deals with non-simultaneous and simultaneous blow-up solutions for , , subject to homogeneous Dirichlet boundary conditions. We obtain the complete results of non-simultaneous and simultaneous blow-up solutions for any fixed point in any general bounded domain. The critical exponents of non-simultaneous blow-up are proposed. [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 localized parabolic equations , with homogeneous Dirichlet boundary conditions, where is any fixed point in a bounded domain of . The optimal classification of non-simultaneous and simultaneous blow-up phenomena is proposed for all of the nonnegative exponents. Moreover, uniform blow-up profiles are obtained for all kinds of 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 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 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 system multi-coupled via inner sources and nonlinear boundary flux. The necessary and sufficient condition for the existence of non-simultaneous blow-up is proposed. Some regions of parameters are determined where both simultaneous and non-simultaneous blow-up can occur. In addition, sufficient conditions for non-simultaneous blow-up under any positive initial data are obtained. [Copyright &y& Elsevier]