Bingchen Liu

Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux ☆

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.

Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

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.

Non-simultaneous blow-up in parabolic equations coupled via localized sources☆

Abstract This work deals with non-simultaneous and simultaneous blow-up solutions for u t = Δ u + u m ( x 0 , t ) v p ( x 0 , t ) , v t = Δ v + u q ( x 0 , t ) v n ( x 0 , t ) , subject to homogeneous Dirichlet boundary conditions. We obtain the complete results of non-simultaneous and simultaneous blow-up solutions for any fixed point x 0 in any general bounded domain. The critical exponents of non-simultaneous blow-up are proposed.

The profile and boundary layer for parabolic system with critical simultaneous blow-up exponent

Abstract This paper deals with simultaneous blow-up solutions to a Dirichlet initial–boundary problem of the parabolic equations u t = div ( a ( x ) ∇ u ) + ∫ Ω u m v s d x and v t = div ( b ( x ) ∇ v ) + ∫ Ω u q v p d x in Ω × [ 0 , T ) . We complete the previous known results by covering the whole range of possible exponents. Then uniform blow-up profile is obtained for all simultaneous blow-up solutions through proving new rules for some auxiliary systems. At last, boundary layer is studied.

Blow-up time and boundary layer for solutions in parabolic equations with different diffusion

Abstract This paper deals with parabolic equations with different diffusion coefficients and coupled nonlinear sources, subject to homogeneous Dirichlet boundary conditions. We give many results about blow-up solutions, including blow-up time estimates for all of the spatial dimensions, the critical non-simultaneous blow-up exponents, uniform blow-up profiles, blow-up sets, and boundary layer with or without standard conditions on nonlocal sources. The conditions are much weaker than the ones for the corresponding results in the previous papers.

A complete classification for non-simultaneous blow-up

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.

Blow-up of multi-componential solutions in heat equations with exponential boundary flux

This paper deals with heat equations coupled via exponential boundary flux, where the solution is made up of n components. Under certain monotone assumptions, necessary and sufficient conditions are obtained for simultaneous blow-up of at least two components for each initial datum. As for two components blowing up simultaneously, it is interesting that the representations of blow-up rates are quite different with respect to the different blow-up mechanisms and positions between the two components.

Critical singular exponent and asymptotic estimates in the parabolic equations

Abstract This paper considers both quenching and blowup phenomena to the coupled parabolic equations with zero Dirichlet boundary. Here, one component of the solution represents the density of some chemical and the other denotes its temperature in some ignition process. All the singular phenomena have been obtained including simultaneous and nonsimultaneous blow-up or quenching, which are classified completely by the exponents. The results extend the ones in the previous paper (Liu and Chan, 2011).

Blow-up analysis in degenerate parabolic systems coupled via norm-type reactions

This paper deals with a class of degenerate parabolic equations coupled via nonlinear reactions in -norm type, subject to null Dirichlet boundary conditions. Firstly, we give the existence and uniqueness of local classical solutions and the comparison principle. Secondly, we determine the critical exponents for the blow-up solutions. Thirdly, all of the uniform blow-up profiles are obtained for simultaneous blow-up solutions.

Note on nonsimultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with the nonsimultaneous blow-up problems for with coupled nonlinear boundary flux . The main results extend the existence of two components blowing up simultaneously in the paper J. Math. Anal. Appl. 2009;356:215–231 to the case of components 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.