Blow-up For Reaction Research Articles

Improved conditions for single-point blow-up in reaction–diffusion systems

We study positive blowing-up solutions of the system:ut−δΔu=vp,vt−Δv=uq, as well as of some more general systems. For any p,q>1, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions:(i) no type I blow-up assumption is made (and it is known that this property may fail);(ii) no equidiffusivity is assumed, i.e. any δ>0 is allowed;(iii) a large class of nonlinearities F(u,v), G(u,v) can be handled, which need not follow a precise power behavior.As a side result, we also obtain lower pointwise estimates for the final blow-up profiles.

A numerical investigation of blow-up in reaction–diffusion problems with traveling heat sources

This paper studies the numerical solution of a reaction–diffusion differential equation with traveling heat sources. According to the fact that the locations of heat sources are known, we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem. Several examples are provided to demonstrate the efficiency of the new moving mesh method, especially in the case of two or three traveling heat sources. Moreover, numerical results illustrate that the speed of the movement of the heat source is critical for blow-up when there is only one traveling heat source. For the case of two traveling heat sources, blow-up depends not only on the speed but also on the distance between the two traveling heat sources.

Moving mesh methods for blowup in reaction–diffusion equations with traveling heat source

In this paper moving mesh methods are used to simulate the blowup in a reaction–diffusion equation with traveling heat source. The finite-time blowup occurs if the speed of the movement of the heat source remains sufficiently low, and the blowup procedure is not fixed at one point not like that for stationary heat source. As time goes to the blowup time, the blowup profile converges to a stationary state. In the simulation a new moving mesh algorithm is designed to deal with the difficulty caused by the delta function in the traveling heat source. The convergence rates are verified and new blowup figures are generated from the numerical experiments.

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.