Blow-up For Heat Equation Research Articles

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.

Nonexistence of Type II Blowup for Heat Equation with Exponential Nonlinearity

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = Δu + eu in ℝN. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9, and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen (2008) in a finite ball to the whole space.

Non-simultaneous blowup in heat equations with nonstandard growth conditions

This paper cares about blowup solutions for a system of n-componential heat equations coupled via localized reactions and with variable exponents. The criteria for non-simultaneous and simultaneous blowup are established for radial solutions with or without assumptions on initial data, including the existence of non-simultaneous blowup for n components; any blowup must be simultaneous or non-simultaneous.

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u t = Δ u + u m ( x, t) e pv(0, t) , v t = Δ v + u q (0, t) e nv( x, t) , subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u 0( v 0) leads to the blow-up of u( v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u( v) blows up only at a single point if m > 1( n > 0), while u( v) blows up everywhere for 0 ⩽ m ⩽ 1 ( n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.

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.

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.