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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]

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 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 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]

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]