Simultaneous Blow-up Rates Research Articles

Asymptotic results and critical Fujita exponent in parabolic equations with nonlocal nonlinearity

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.

A Cauchy problem of parabolic equations with time-dependent coefficients

ABSTRACT 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.

Thermal avalanche after non-simultaneous blow-up in heat equations coupled via nonlinear boundary

In this paper, we study a parabolic problem defined in the half line coupled via exponential-type boundary flux. Firstly, we prove the optimal classification of the two components of blow-up solutions when time reaches the blow-up time from left. Blow-up takes place only at the origin, and simultaneous blow-up rates are determined as well. Secondly, we study the weak extension after the blow-up time. Complete blow-up always occurs whether simultaneous blow-up arises or not. Moreover, an instantaneous propagation of the blow-up singularity to the whole spatial domain occurs at the blow-up time, which is the so-called thermal avalanche phenomenon. Finally, we use the evolution of the $k$-level set of solutions in the approximations to characterize the propagation of the singularity.

Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux

This paper deals with a reaction–diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green’s identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.

Asymptotic analysis to a parabolic system with weighted localized sources and inner absorptions

In this paper we consider blow-up solutions for a parabolic model with inner absorptions and coupled nonlinear weighted and localized sources. Three simultaneous blow-up rates are established under different dominations of nonlinearities and simply represented via a characteristic algebraic system. In particular, for the case of weak absorptions, a uniform blow-up profile is established, while for the case of unbalanced absorptions, unlike the existing results in literature, the multiple blow-up rates are shown to be related to the absorptions.

Uniform blow-up profiles for heat equations with coupling nonlocal sources of asymmetric mixed type nonlinearities

This article deals with a nonlocal heat system subject to null Dirichlet boundary conditions, where the coupling nonlocal sources consist of mixed type asymmetric nonlinearities. We at first give the criterion for simultaneous blow-up of solutions, and then establish the uniform blow-up profiles of solutions near the blow-up time. It is observed that not only the simultaneous blow-up rates of the two components u and v are asymmetric, but also the blow-up rates of the same component u (or v) may be in different levels under different dominations.

Multiple absorption-related blow-up rates to a coupled heat system

This paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi-nonlinearities. It is known that although absorption mechanisms may affect such as blow-up criteria, blow-up time, and initial data required for blow-up solutions, they cannot change blow-up rates of solutions in general. It has been reported in the current literature that blow-up rates for scalar equations with absorptions are all absorption-independent. In a previous paper of the authors, four absorption-independent simultaneous blow-up rates were obtained already for the same problem under weak absorptions. The present paper will furthermore prove that if the absorptions are unbalanced in the model (i.e., the absorption is stronger for one component and weaker for another), then there are in addition eight possible absorption-related blow-up rates for the model, besides the four absorption-independent ones. This exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions.

Non-simultaneous blow-up for [formula omitted]-componential parabolic systems

This paper deals with the non-simultaneous and simultaneous blow-up for some parabolic systems ( u i ) t = Δ u i + u i p i coupled via nonlinear boundary flux ∂ u i ∂ η = u i + 1 q i + 1 ( i = 1 , 2 , … , n ) . 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 k ( ∈ { 0 , 1 , … , n − 2 } ) 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.

Asymptotic estimates to non-global solutions of a multi-coupled parabolic system

This paper deals with asymptotic behavior of solutions to a parabolic system, where two heat equations with inner absorptions are multi-coupled via inner sources and boundary flux. We determine four kinds of simultaneous blow-up rates under different dominations of nonlinearities in the model. Two characteristic algebraic systems associated with the problem are introduced to get very simple descriptions for the four simultaneous blow-up rates as well as the known critical exponents, respectively. It is observed that the blow-up rates are independent of the nonlinear inner absorptions.

Blow-up properties for heat equations coupled via different nonlinearities

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.

Non-simultaneous blow-up in a parabolic system with three components

This paper deals with heat equations coupled via nonlinear boundary flux ∂ u 1 ∂ η = u 1 p 11 + u 2 p 12 , ∂ u 2 ∂ η = u 2 p 22 + u 3 p 23 , ∂ u 3 ∂ η = u 3 p 33 + u 1 p 31 . 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 ( 1 2 ( p 22 − 1 ) , 1 2 ( p 33 − 1 ) ) and ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) ; (iii) e.g., u 1 remains bounded and u 2 , u 3 blow up simultaneously with blow-up rate ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) for every initial data. Moreover, the eight kinds of simultaneous blow-up rates and blow-up sets are discussed.

Asymptotic analysis of a coupled nonlinear parabolic system

This paper deals with asymptotic analysis of a parabolic system with inner absorptions and coupled nonlinear boundary fluxes. Three simultaneous blow-up rates are established under different dominations of nonlinearities, and simply represented in a characteristic algebraic system introduced for the problem. In particular, it is observed that two of the multiple blow-up rates are absorption-related. This is substantially different from those for nonlinear parabolic problems with absorptions in all the previous literature, where the blow-up rates were known as absorptionindependent. The results of the paper rely on the scaling method with a complete classification for the nonlinear parameters of the model. The first example of absorption-related blow-up rates was recently proposed by the authors for a coupled parabolic system with mixed type nonlinearities. The present paper shows that the newly observed phenomena of absorptionrelated blow-up rates should be due to the coupling mechanism, rather than the mixed type nonlinearities.

Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes $$\frac{\partial u}{\partial\eta} = u^{m} + v^{p}, \frac{\partial v}{\partial\eta} = u^{q} + v^{n}$$ . It is proved that, if m q + 1 or n > p + 1. We find three regions: (i) q + 1 q+1 and n > p+1, where both simultaneous and non-simultaneous blow-up are possible. Four different simultaneous blow-up rates are obtained under different conditions. It is interesting that different initial data may lead to different simultaneous blow-up rates even for the same values of the exponent parameters.

Blow-up properties of solutions for a multi-coupled parabolic system

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.

Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system

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.

A complete classification of simultaneous blow-up rates

We study the simultaneous blow-up rates of a system of two heat equations coupled through the boundary in a nonlinear way. We complete the previous known results by covering the whole range of possible parameters.