Blow-up Rate Estimates Research Articles

Blow-up for a double degenerate quasilinear parabolic equation

Existence of a unique classical non-negative solution is established and the sufficient conditions for the solution that exists locally or blow-up in finite time are obtained for the double degenerate quasilinear parabolic equation with homogeneous Dirichlet boundary conditions. Furthermore, under certain conditions we obtain the estimates of blow-up rate of solution.

Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source

We prove universal bounds for nonnegative weak solutions of the porous medium equation with source $u_t-\Delta u^m=u^p$ where $1 < m < p$. These bounds imply initial and final blow-up rate estimates, as well as a~priori estimates or decay rates for global solutions. We consider both radial and nonradial solutions, and in the radial case we cover all Sobolev-subcritical values of $p/m$, which is the best possible range. Our bounds are proved as a consequence of Liouville-type theorems for entire solutions and doubling and rescaling arguments. In this connection, we use known Liouville-type theorems for radial solutions, along with some new Liouville-type theorems that are here established for nonradial solutions in RN and for solutions on a half-line.

Blow-up rate estimates for a parabolic system with multiple nonlinearities

In this paper, we study the blow-up behaviors for the solutions of parabolic systems ut=Δu+δ1e, vt=Δv+µ1u in ℝ×(0, T) with nonlinear boundary conditions Here δi⩾0, µj⩾0, pi⩾0, qj⩾0 and at least one of δiµjpiqj>0(i, j=1, 2). We prove that the solutions will blow up in finite time for suitable ‘large’ initial values. The exact blow-up rates are also obtained. Copyright © 2009 John Wiley & Sons, Ltd.

Blow-up properties for parabolic systems with localized nonlinear source

(Communicated by S. Cui) Abstract. This paper deals with blow-up properties of solutions to a semilinear parabolic system with nonlinear localized source involved a product with local terms ut = Δu+exp{mu(x,t)+nv(x0,t)}, vt = Δv+exp{pu(x0,t)+qv(x,t)} with homogeneous Dirichlet boundary conditions. We investigate the influence of localized sources and local terms on blow-up properties for this system, and prove that: (i )w henm, q 0 this system possesses uniform blow-up profiles, in other words, the localized terms play a leading role in the blow-up profile for this case; (ii )w henm, q > 0, this system presents single point blow-up patterns, or say that local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.

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.

Blowup rate estimates for the heat equation with a nonlinear gradient source term

The gradient blowup rate of the equation $u_t = \Delta u + |\nabla u|^p$, where $p>2$, is studied. It is shown that the blowup rate will never match that of the self-similar variables. In the one space dimensional case when assumptions are made on the initial data so that the solution is monotonically increasing in time, the exact blowup rate is found.

BLOW-UP RATE ESTIMATES FOR A SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH ABSORPTION

In this note, we consider a system of two reaction-diffusion equations with absorption, under homogeneous Dirichlet boundary. Using scaling methods, we establish the blow-up rate estimates.

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary ∂Ω. In addition, for a bounded Lipschitz domain Ω, we establish the blow-up rate estimates for the positive solution to this problem with a = 0.

Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations

In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with or without boundary conditions. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville-type theorems, which unifies and improves many results concerning a priori bounds, decay estimates and initial and final blow-up rates. For example, for the equation u t - Δu = u p on a domain Ω, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form u(x, t) ≤ C(Ω, p)(1 + t -1/(p-1) + (T - t) -1/(p-1) ). Our method is based on rescaling arguments combined with a key doubling property, and it is facilitated by parabolic Liouville-type theorems for the whole space or the half-space. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.

Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux

This paper deals with the blow-up properties of solutions to adegenerate parabolic system coupled via nonlinear boundary flux.Firstly, we construct the self-similar supersolution andsubsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.

Global existence and blow-up analysis to a degenerate reaction–diffusion system with nonlinear memory

In this paper, we consider a degenerate reaction–diffusion system coupled by nonlinear memory. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Furthermore, the blow-up rate estimates are obtained in some special case.

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.

A further blow-up analysis for a localized porous medium equation

This paper deals with the porous medium equation with a localized source v τ = Δ v m + av p 1 v q 1 ( x 0 , τ ) , x ∈ Ω , τ > 0 subject to homogeneous Dirichlet condition. We investigate the influence of the localized source and local term on blow-up properties for this system. It is proved that: (i) when p 1 ⩽ 1, the localized source plays a dominating role, i.e. the system has global blow-up and the uniformly blow-up profile is obtained. (ii) When p 1 > 1, we obtain the blow-up rate estimates under some appropriate hypotheses on initial datum. Moreover, if p 1 > m, this system presents single blow-up pattern. In other words, the local term dominates the localized term in the blow-up profile. This extends and generalizes a recent work of Chen and Xie [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. and Comput., 159 (2004) 79–93], which only considered the blow-up profile in the special case p 1 = 0.

Critical exponents and lower bounds of blow-up rate for a reaction–diffusion system

This paper is concerned with a reaction–diffusion system with absorption terms under Dirichlet boundary conditions, modelling the cooperative interaction of two diffusion biological species. By constructing blow-up sub-solutions and bounded super-solutions, we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions. Moreover, for a special case, we derive the lower bound estimates of blow-up rate.

Blow-up rate estimates for a doubly coupled reaction–diffusion system

This paper deals with a reaction–diffusion system with coupled nonlinear inner sources and a nonlinear boundary flux. Blow-up rates are determined for four different blow-up situations. The so-called characteristic algebraic system is introduced to get a very simple and clear description for the desired blow-up rate estimates. It is pointed out that one cannot directly use super and sub-solutions to establish blow-up rate estimates, since they do not share the same blow-up time in general.

Properties of blow-up solutions to a parabolic system with nonlinear localized terms

This paper deals with blow-up properties of the solution to a semi-linear parabolic system with nonlinear localized sources involved in a product with local terms, subject to the null Dirichlet boundary condition. We investigate the influence of localized sources and local terms on blow-up properties for this system. It will be proved that: (i) when $m, q\leq 1$ this system possesses uniform blow-up profiles. In other words, the localized terms play a leading role in the blow-up profile for this case. (ii) when $m, q>1$, this system presents single point blow-up patterns, or say that, in this time, local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.

Global existence and blow-up to a reaction-diffusion system with nonlinear memory

In this paper, we consider a reaction-diffusion system coupled by nonlinear memory. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Furthermore, the blow-up rate estimates are obtained.

The blow-up estimate for heat equations with non-linear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for a system of heat equations with non-linear boundary conditions. The upper and lower bounds of blow-up rate are obtained.

Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for semilinear parabolic systems with nonlinear boundary conditions. The upper and lower bounds of blow-up rates are obtained.

Blow-up and blow-up rate for a reaction–diffusion model with multiple nonlinearities

This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear diffusion, nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities. The necessary and sufficient blow-up conditions are established together with blow-up rate estimates for the positive solutions of the problem.