Uniform Blow-up Research Articles

Global Existence and Uniform Blow-Up to a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising in a Thermal Explosion Theory

This paper deals with a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which arise in the thermal explosion model. Firstly, we prove a comparison principle for some kinds of parabolic systems under nonlinear boundary conditions. Using this, we improve a new theorem of the sub-and-super solution. Secondly, based on the new sub-and-super solution theorem, the sufficient conditions that the solution exists and blows up uniformly in finite time are presented. Then, we generalize some of the lemmas related to uniform blow-up solutions, which are used to introduce the uniform blow-up profiles of solutions. Finally, we give several numerical simulations to illustrate the existence and uniform blow-up of solutions.

Blow-Up Profile of Solutions in Parabolic Equations with Nonlocal Dirichlet Conditions

This paper deals with parabolic equations with nonlocal Dirichlet boundary conditions. Critical exponent and simultaneous blow-up criteria are studied for blow-up solutions. Moreover, we determine uniform blow-up profiles and blow-up set to all kinds of simultaneous blow-up solutions. It is interesting that large weighted functions could provide sufficient help to the occurring of blow-up of solutions without conditions on initial data.

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.

Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

The profile and boundary layer for parabolic system with critical simultaneous blow-up exponent

This paper deals with simultaneous blow-up solutions to a Dirichlet initial–boundary problem of the parabolic equations u t = div ( a ( x ) ∇ u ) + ∫ Ω u m v s d x and v t = div ( b ( x ) ∇ v ) + ∫ Ω u q v p d x in Ω × [ 0 , T ) . We complete the previous known results by covering the whole range of possible exponents. Then uniform blow-up profile is obtained for all simultaneous blow-up solutions through proving new rules for some auxiliary systems. At last, boundary layer is studied.

Blow-up analysis in degenerate parabolic systems coupled via norm-type reactions

This paper deals with a class of degenerate parabolic equations coupled via nonlinear reactions in -norm type, subject to null Dirichlet boundary conditions. Firstly, we give the existence and uniqueness of local classical solutions and the comparison principle. Secondly, we determine the critical exponents for the blow-up solutions. Thirdly, all of the uniform blow-up profiles are obtained for simultaneous blow-up solutions.

Uniform Blow-Up Rate for Diffusion Equation with a Weighted Localized Source

In this paper,we investigate the Dirichlet problem for a localized nonlinear equation with a weighted localized source term.Under certain conditions, the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.Moreover,we obtain the blow-up rate on maximal value point under some appropriate hypotheses.

Global existence and blowup for a degenerate and singular parabolic system with nonlocal source and absorptions

The aim of this paper is to investigate the asymptotic behavior of positive solutions to the following degenerate and singular parabolic system $$\begin{array}{lll}{u_t}=(x^ \alpha u_x)_x+ \int \limits_{0}^{a} v^{p_1}{\rm d} x - {k_1} u^{q_1}, \quad 0 0, \\ \upsilon_{t}=(x^{\beta} v_{x})_{x} + \int \limits_{0}^{a} u^{p_2}{\rm d}x - k_{2} \upsilon^{q_2}, \, \, \, \, \, 0 0,\\ u(0,t) = u(a,t) = \upsilon(0,t) = \upsilon(a,t) = 0, \, \, \, \, \, t > 0,\\ u(x,0) = u_{0}(x) \geq 0, \upsilon(x,0) = \upsilon_{0}(x) \geq 0,\, \, \, \, \, 0 \leq x \leq a,\end{array}$$ where the constants $${0 \leq \alpha, \beta 0}$$ . Under appropriate hypotheses, we first prove a local existence of classical solution by a regularization method. Then, we discuss the global existence and blowup of positive solutions by using a comparison principle. Finally, we give the precise blowup rate estimates and the uniform blowup profiles by using the method developed by Souplet [19].

A reaction–diffusion system with mixed-type coupling

This paper deals with asymptotic behavior of solutions to a reaction–diffusion system coupled via localized and local sources: ut=Δu+vp(x∗(t),t),vt=Δv+uq. Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the two kinds of sources. For the initial-boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow-up profiles. In addition, it is interesting to observe that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq>1. All these imply that the blow-up behavior of solutions is governed by the localized source for the two problems with mixed-type coupling.

Asymptotic analysis to a parabolic system with weighted localized sources and inner absorptions

In this paper we consider blow-up solutions for a parabolic model with inner absorptions and coupled nonlinear weighted and localized sources. Three simultaneous blow-up rates are established under different dominations of nonlinearities and simply represented via a characteristic algebraic system. In particular, for the case of weak absorptions, a uniform blow-up profile is established, while for the case of unbalanced absorptions, unlike the existing results in literature, the multiple blow-up rates are shown to be related to the absorptions.

Uniform blow-up profiles for heat equations with coupling nonlocal sources of asymmetric mixed type nonlinearities

This article deals with a nonlocal heat system subject to null Dirichlet boundary conditions, where the coupling nonlocal sources consist of mixed type asymmetric nonlinearities. We at first give the criterion for simultaneous blow-up of solutions, and then establish the uniform blow-up profiles of solutions near the blow-up time. It is observed that not only the simultaneous blow-up rates of the two components u and v are asymmetric, but also the blow-up rates of the same component u (or v) may be in different levels under different dominations.

Uniform blow-up rate for compressible reactive gas model

The Dirichlet initial-boundary value problem of a compressible reactive gas model equation with a nonlocal nonlinear source term is investigated. Under certain conditions, it can be proven that the blow-up rate is uniform in all compact subsets of the domain, and the blow-up rate is irrelative to the exponent of the diffusion term, however, relative to the exponent of the nonlocal nonlinear source.

Uniform blow-up rate for a porous medium equation with a weighted localized source

Abstract In this article, we investigate the Dirichlet problem for a porous medium equation with a more complicated source term. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, in each case, the blow-up rate of |u(t)|∞ is precisely determined.

Global existence and blow-up of solutions to reaction-diffusion system with a weighted nonlocal source

In this paper, we study a semi-linear reaction-diffusion system with a weighted nonlocal source, subject to the null Dirichlet boundary condition. Under certain conditions, we prove that the classical solution exists globally and blows up in finite time respectively, and then obtain the uniform blow-up rate in the interior.

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

In this paper, we investigate the initial-boundary problem of a degenerate parabolic system with nonlinear localized sources. We classify the blow-up solutions into global blow-up cases and single-point blow-up cases according to the values of m , n , p i , q i . Furthermore, we obtain the uniform blow-up profiles of solutions for the global blow-up case. Finally, we give some numerical examples to verify the results. These extend and generalize a recent work of one of the authors [L. Du, Blow-up for a degenerate reaction–diffusion systems with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304–320], which only considered uniform blow-up profiles under the special case p 1 = p 2 = 0 .

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

In this paper, the blow-up rate of solutions of semi-linear reaction–diffusion equations with a more complicated source term, which is a product of nonlocal (or localized) source and weight function a ( x ) , is investigated. It is proved that the solutions have global blow-up, and that the rates of blow-up are uniform in all compact subsets of the domain. Furthermore, the blow-up rate of | u ( t ) | ∞ is precisely determined.

Optimal conditions of non-simultaneous blow-up and uniform blow-up profiles in localized parabolic equations

This paper deals with localized parabolic equations u t = Δ u + u m ( x 0 , t ) e p v ( x 0 , t ) , v t = Δ v + u q ( x 0 , t ) e n v ( x 0 , t ) with homogeneous Dirichlet boundary conditions, where x 0 is any fixed point in a bounded domain of R N . The optimal classification of non-simultaneous and simultaneous blow-up phenomena is proposed for all of the nonnegative exponents. Moreover, uniform blow-up profiles are obtained for all kinds of simultaneous blow-up solutions.

Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources

This paper investigates the global existence of the nonnegative solution and the finite time blow-up of solutions of nonlinear parabolic equation with a more complicated source term, which is a product of localized source, local source, and weight function; we also study the blow-up rate of solution to this problem.