Non-simultaneous Blow-up Research Articles

Non-simultaneous blow-up for a system with local and non-local diffusion

Non-simultaneous blow-up for a system with local and non-local diffusion

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.

Blow-up for a semilinear non-local diffusion system

In this paper we study the blow-up phenomena for the system ut=∫ΩJ(x−y)u(y)dy−u+uαvp(x,t)∈Ω×(0,T)vt=∫ΩJ(x−y)v(y)dy−v+uqvβ(x,t)∈Ω×(0,T) Under certain hypothesis we prove that the solution blows up infinite time and we classify in terms of the parameters when both components blow up together (simultaneous blow-up), or when only one component blows up (non-simultaneous blow-up). We also prove the blow-up rate for a particular case.

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.

Blow-up analyses in reaction–diffusion equations with nonlinear nonlocal boundary flux

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.

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.

Blow-up analyses in reaction–diffusion equations with Fujita exponents

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.

Non-simultaneous blow-up profile and boundary layer estimate in nonlinear parabolic problems

ABSTRACT This paper deals with homogeneous Dirichlet problems of localized parabolic equations with different diffusion, where reaction terms are the product of the three components of solutions. According to the nontrivial coupled relationship, different non-simultaneous and simultaneous blow-up criteria are proposed for the three components. Then we show non-simultaneous blow-up profile and boundary layer estimates of solutions with or without standard growth conditions on the localized sources near blow-up time.

Asymptotic Estimates and Fujita Exponents of Blow-Up Solutions in Weighted Parabolic Systems

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.

Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent

In this paper, we deal with a time-weighted parabolic system coupled via nonlocal sources. Firstly, we give the critical Fujita exponent of solutions. Secondly, we propose the time-weighted blow-up profiles of solutions. Thirdly, the simultaneous and non-simultaneous blow-up of solutions are discussed under some conditions on the initial data. Fourthly, besides blow-up rates, we classify total blow-up and single point blow-up of solutions. At last, upper and lower bounds of blow-up time are studied according to precise blow-up phenomena.

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.

Nonsimultaneous blowup for a complex valued semilinear heat equation

This paper is concerned with blow-up solutions for a complex valued semilinear heat equation. Nonsimultaneous blow-up solutions predicted in our previous work are constructed by a fixed point argument.

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.

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

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.

Critical singular exponent and asymptotic estimates in the parabolic equations

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).

Blowup profile for a complex valued semilinear heat equation

This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We classify blow-up solutions and derive their blow-up profiles under some assumptions. In particular, we discuss the possibility of a nonsimultaneous blow-up.

Estimates on the blowup time for a quasilinear parabolic system

In this paper, we will establish the lower and upper bounds for the blowup time of the solution to the following systemsubject to on and nonnegative initial data under some assumptions. Here is a smooth bounded open domain, and . We also deal with the non-simultaneous blowup phenomenon of the solution.

Estimates for the blowup time of a combustion model with nonlocal heat sources

In this paper, we establish the lower and upper bounds for the blowup time of the solution to a system of parabolic equations with nonlocal nonlinearities which appears in a model of combustion. Different techniques are applied to deal with the problems in different space dimensions. We also discuss non-simultaneous blowup phenomenon of the solution.

Non-simultaneous blow-up of a reaction-diffusion system with inner absorption and coupled via nonlinear boundary flux

This paper deals with a parabolic reaction-diffusion system with a nonlinear absorption term, meanwhile the two equations of the system coupled via nonlinear boundary flux which obey different laws. Under the hypothesis condition of the initial data, we get the sufficient and necessary conditions under which there exist initial data such that non-simultaneous blow-up occurs.

Blow-up properties for a degenerate parabolic system coupled via nonlinear boundary flux

In this paper, we study the simultaneous and non-simultaneous blow-up problem for a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain hypotheses on the initial data and parameters, we prove that non-simultaneous blow-up is possible. Moreover, we get some conditions on which simultaneous blow-up must occur, as well as the non-simultaneous blow-up conditions for every initial data. Furthermore, we get a result on the coexistence of both simultaneous and non-simultaneous blow-ups.