Non-simultaneous Blow-up Research Articles

Optimal classification for blow-up phenomena in heat equations coupled via exponential sources

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.

Non-simultaneous blow-up in coupled heat equations with multi-nonlinearities

This paper studies coupled heat equations with multi-nonlinearities of six nonlinear parameters. The critical blow-up exponent is established via a complete classification for all the six nonlinear parameters, where a precise analysis on the geometry of Ω and the absorption coefficients is given for the balanced interaction situation among the multi-nonlinearities. The main attention is contributed to non-simultaneous phenomena in the model to determine the necessary and sufficient conditions of non-simultaneous blow-up with suitable initial data, as well as the conditions under which any blow-up must be non-simultaneous. Finally, the model of the paper is characterized by a comparison with the known results for other models.

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.

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of u t = Δ u + u m ( x , t ) v n ( 0 , t ) , v t = Δ v + u p ( 0 , t ) v q ( x , t ) , subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u ( v ) blows up alone if and only if m > p + 1 ( q > n + 1 ), which means that any blow-up is simultaneous if and only if m ≤ p + 1 , q ≤ n + 1 . (ii) Any blow-up is u ( v ) blowing up with v ( u ) remaining bounded if and only if m > p + 1 , q ≤ n + 1 ( m ≤ p + 1 , q > n + 1 ). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m > p + 1 , q > n + 1 . Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.

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.

Non-simultaneous versus simultaneous blow-up for coupled heat equations with positive–negative sources

This is a successive work to the previous paper [S.N. Zheng, L. Qiao, Non-simultaneous blow-up for heat equations with positive–negative sources and coupled boundary flux, Commun. Pure Appl. Anal. 6 (4) (2007), 1113–1129] of the authors. We continue considering non-simultaneous versus simultaneous blow-up for a heat equation model with positive–negative sources and coupled non-linear boundary flux. Owing to a complete classification for all the non-linear parameters of the model, we arrive at a criterion for distinguishing between non-simultaneous and simultaneous blow-up in the coupled system.

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.

Note on non-simultaneous blow-up for a reaction–diffusion system

In this note, we study the non-simultaneous and simultaneous blow-up for a reaction–diffusion system with inner absorptions and coupled through the boundary in a nonlinear way. We characterize completely the set of parameters for non-simultaneous blow-up and simultaneous blow-up, and the optimal conditions are obtained.

Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux

This paper deals with non-simultaneous blow-up forheat equations with positive-negative sources coupled via nonlinearboundary flux. At first, we establish the necessary-sufficientconditions for non-simultaneous blow-up of solutions under suitableinitial data. Furthermore, the sufficient conditions are determinedunder which any blow-up of solutions to the model would benon-simultaneous. Comparing with those for a similar systempossessing negative-negative sources obtained in a previous paper [19] we show clearly the contribution of the positivesource in the present model to the non-simultaneous blowing upbehavior of solutions.

Simultaneous versus Nonsimultaneous Blowup for a System of Heat Equations Coupled Boundary Flux

This paper deals with a semilinear parabolic system in a bounded interval, completely coupled at the boundary with exponential type. We characterize completely the range of parameters for which nonsimultaneous and simultaneous blowup occur.

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.

Blowup properties for several diffusion systems with localised sources

AbstractThis paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

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.

Non-simultaneous blow-up in a reaction-diffusion system

This paper deals with non-simultaneous blow-up for a reaction-diffusion system with absorptions and coupled non-linear boundary flux. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data. Moreover, we determine the conditions under which any blow-up would be non-simultaneous.

Non-simultaneous blow-up for a multi-coupled reaction-diffusion system

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.

The role of non-linear diffusion in non-simultaneous blow-up

We study a parabolic system of two non-linear reaction–diffusion equations completely coupled through source terms and with power-like diffusivity. Under adequate hypotheses on the initial data, we prove that non-simultaneous blow-up is sometimes possible; i.e., one of the components blows up while the other remains bounded. The conditions for non-simultaneous blow-up rely strongly on the diffusivity parameters and significant differences appear between the fast-diffusion and the porous medium case. Surprisingly, flat (homogeneous in space) solutions are not always a good guide to determine whether non-simultaneous blow-up is possible.

Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system

We study simultaneous and nonsimultaneous blow-up for solutions of the following system $$\left\{\begin{array}{l} u_t = \Delta u + u^r + v^p, \\ v_t = \Delta v + v^s + u^q, \end{array} \quad\mbox{ in } \Omega \times (0,T), \right. $$ with Dirichlet boundary conditions. We show that, in the range of exponents where either component may blow up alone, there also exist initial data for which both components blow up simultaneously. The proof is based on a continuity argument, which requires upper and lower blow-up estimates, independent of initial data, and continuous dependence of the existence time. In turn, we prove a result of continuous dependence of the existence time, under the assumption of uniform upper blow-up estimates, in the framework of general abstract semiflows.

Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary

We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.

Optimal condition for non-simultaneous blow-up in a reaction-diffusion system

We study the positive blowing-up solutions of the semilinear parabolic system: ut-Δu=vp+ur,vt-Δv=uq+vs, where t∊(0,T),x∊RN and p,q,r,s>1. We prove that if r>q+1 or s>p+1 then one component of a blowing-up solution may stay bounded until the blow-up time, while if r<q+1 and s<p+1 this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters p,q,r and s, solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.

Non-simultaneous Blow-up in a Nonlinear Parabolic System

Abstract We prove the existence of a nontrivially coupled parabolic system such that one of its components becomes unbounded at a finite time while the other remains bounded, a situation that we denote as non-simultaneous blow-up. Our system consists of two porous medium equations with coupled nonlinear flux boundary conditions. As a preliminary step, we will obtain a necessary and sufficient condition for blow-up. Next we characterize completely, in the case of increasing in time solutions, the set of parameters appearing in the system for which nonsimultaneous blow-up indeed occurs. In the course of our proofs we will obtain a necessary and sufficient condition for the blow-up of solutions to general porous medium type equations on the half-line with a prescribed flux at the boundary blowing up at a finite time, a result of independent interest.