Blow-up Rate Estimates Research Articles

Blow-up of solutions to a scalar conservation law with nonlocal source arising in radiative gas

In this paper we consider a scalar conservation law with nonlocal source arising in radiative gas. We give a sufficient condition to assure that the solution blows up in a finite time. Moreover, the estimates of life span and blow-up rate are derived.

Upper and lower blow-up rate estimates of a semilinear heat equation with a nonlinear boundary condition

In this paper, we consider the blow-up solutions of a semi-linear heat equation with a Neumann boundary condition, where the nonlinear terms, appear in the differential equation and in the boundary conditions, are of exponential types. We study the effect of the nonlinear terms on the last blow-up behavior. Precisely, the upper (lower) blow-up rate estimates are established by using integral equation method and maximum principle techniques. The results show that the presentence of the reaction and boundary terms has important effects on the upper (lower) blow-up rate estimates forms.

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

On Blow-up Solutions of A Parabolic System Coupled in Both Equations and Boundary Conditions

This paper is concerned with the blow-up solutions of a system of two reaction-diffusion equations coupled in both equations and boundary conditions. In order to understand how the reaction terms and the boundary terms affect the blow-up properties, the lower and upper blow-up rate estimates are derived. Moreover, the blow-up set under some restricted assumptions is studied.

Sharp condition for the Liouville property in a class of nonlinear elliptic inequalities

We study a class of elliptic inequalities which arise in the study of blow-up rate estimates for parabolic problems, and obtain a sharp existence/nonexistence result. Namely, for any $p\ge 1$, we show that the inequality $\Delta u+ u^p \leq \varepsilon $

Global existence and blow-up of solutions of nonlocal diffusion problems with free boundaries

In this paper, we investigate some nonlocal diffusion problems with free boundaries. We first give the existence and uniqueness of local solution by the ODE basic theory and the contraction mapping principle. Then we provide a complete classification for the global existence and finite time blow-up of solutions. Moreover, estimates of blow-up rate and blow-up time are also obtained for the blow-up solution.

Some Results on Blow-up Phenomenon for Nonlinear Porous Medium Equations with Weighted Source

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

Blow-up Rate Estimates and Blow-up Set for a System of Two Heat Equations with Coupled Nonlinear Neumann Boundary Conditions

This paper deals with the blow-up properties of positive solutions to a parabolic system of two heat equations, defined on a ball in associated with coupled Neumann boundary conditions of exponential type. The upper bounds of blow-up rate estimates are derived. Moreover, it is proved that the blow-up in this problem can only occur on the boundary.

Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

Wave breaking analysis for the Fornberg–Whitham equation

In this paper, the wave breaking of the Fornberg–Whitham equation is studied. We first investigate the local well-posedness and show a blow-up scenario by local-in-time a priori estimates. Next, we present a sufficient condition on the initial data to ensure wave-breaking. Moreover, the estimates of life span and blow-up rate are obtained.

Blow-up phenomena for p-Laplacian parabolic problems with Neumann boundary conditions

In this paper, we deal with the blow-up and global solutions of the following p-Laplacian parabolic problems with Neumann boundary conditions: \t\t\t{(g(u))t=∇⋅(|∇u|p−2∇u)+k(t)f(u)in Ω×(0,T),∂u∂n=0on ∂Ω×(0,T),u(x,0)=u0(x)≥0in Ω‾,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (g(u) )_{t} =\\nabla\\cdot ( {|\\nabla u|^{p-2}}\\nabla u )+k(t)f(u) & \\mbox{in } \\Omega\\times(0,T), \\\\ \\frac{\\partial{u}}{\\partial n}=0 &\\mbox{on } \\partial\\Omega\\times (0,T), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{\\Omega}, \\end{cases} $$\\end{document} where p>2 and Ω is a bounded domain in mathbb{R}^{n} (ngeq 2) with smooth boundary ∂Ω. By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up time.

The Profile of Blow-Up for a Neumann Problem of Nonlocal Nonlinear Diffusion Equation with Reaction

In this note, we consider a Neumann problem for nonlocal nonlinear diffusion equation with reaction, which may be seen as a significant generalization of the usual Neumann problem for the heat equation. For the blow-up solutions, the blow-up rate estimates and spacial localization of blow-up set are studied.

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

We prove a Liouville-type theorem for semilinear parabolic systems of the form $$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$ in the whole space \({\mathbb R}^N\times {\mathbb R}\). Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for \(m=2\) in dimension \(N\le 2\), and partial results in higher dimensions in the range \(p< N/(N-2)\). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Veron, we partially improve the results of Quittner in dimensions \(N\ge 3\). In particular, our results solve the important case of the parabolic Gross–Pitaevskii system—i.e. the cubic case \(r=1\)—in space dimension \(N=3\), for any symmetric (m, m)-matrix \((\beta _{ij})\) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space \({\mathbb R}^N_+\times {\mathbb R}\). As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

Bounds for the Blow-up Time and Blow-up Rate Estimates for Nonlinear Parabolic Equations with Dirichlet or Neumann Boundary Conditions

This paper is concerned with the blow-up phenomena for a type of parabolic equations with weighted nonlinear source  (b(u))t = div(|∇u|p−2∇u) + f(x, u), x ∈ Ω, t > 0, u(x, t) = 0 or ∂u ∂n = 0, x ∈ ∂Ω, t > 0, u(x, 0) = g(x) ≥ 0, x ∈ Ω, where Ω ⊂ R (N ≥ 3) is a smooth bounded domain. Through constructing some suitable auxiliary functions and using the first-order differential inequality technique, we obtain the bounds for the blow-up time and the estimates of the blow-up rate of the solution to the problem.

Supersolutions for a class of nonlinear parabolic systems

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including{∂tu=Δu+vp,x∈Ω,t>0,∂tv=Δv+uq,x∈Ω,t>0,u=v=0,x∈∂Ω,t>0,(u(x,0),v(x,0))=(u0(x),v0(x)),x∈Ω, where p≥0, q≥0, Ω is a (possibly unbounded) smooth domain in RN and both u0 and v0 are nonnegative and locally integrable functions in Ω. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions.

Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka–Volterra systems

We consider positive solutions of cooperative parabolic Lotka–Volterra systems with equal diffusion coefficients, in bounded and unbounded domains. The systems are complemented by the Dirichlet or Neumann boundary conditions. Under suitable assumptions on the coefficients of the reaction terms, these problems possess both global solutions and solutions which blow up in finite time. We show that any solution (u,v) defined on the time interval (0,T) satisfies a universal estimate of the formu(x,t)+v(x,t)≤C(1+t−1+(T−t)−1), where C does not depend on x, t, u, v, T. In particular, this bound guarantees global existence and boundedness for threshold solutions lying on the borderline between blow-up and global existence. Moreover, this bound yields optimal blow-up rate estimates for solutions which blow up in finite time. Our estimates are based on new Liouville-type theorems for the corresponding scaling invariant parabolic system and require an optimal restriction on the space dimension n: n≤5. As an application we also prove the existence of time-periodic positive solutions if the coefficients are time-periodic. Our approach can also be used for more general parabolic systems.

Blow-Up Rate Estimates and Liouville Type Theorems for a Semilinear Heat Equation with Weighted Source

We study the Liouville-type theorem for the semilinear parabolic equation $$u_t-\Delta u =|x|^a u^p$$ with $$p>1$$ and $$a\in {\mathbb R}$$ . Relying on the recent result of Quittner (Math Ann, doi: 10.1007/s00208-015-1219-7 , 2015), we establish the optimal Liouville-type theorem in dimension $$N=2$$ , in the class of nonnegative bounded solutions. We also provide a partial result in dimension $$N\ge 3$$ . As applications of Liouville-type theorems, we derive the blow-up rate estimates for the corresponding Cauchy problem.

Global unsolvability of one-dimensional problems for Burgers-type equations

In this paper, we study the global solvability of well-known equations used to describe nonlinear processes with dissipation, namely, the Burgers equation, the Korteweg–de Vries–Burgers equation, and the modified Korteweg–de Vries–Burgers equation. Using a method due to Pokhozhaev, we obtain necessary conditions for the blow-up of global solutions and estimates of the blow-up time and blow-up rate in bounded and unbounded domains. We also study the effect of linear and nonlinear viscosity on the occurrence of a gradient catastrophe in finite time.

Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation

In this paper, we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as a solution is bounded above by the critical singularity 1x (the equilibrium solutions behave like this power law near the origin), it remains bounded in the uniform norm. In Sec. III of the paper, we prove a local existence result for a class of measure-valued mild solutions, which is of independent interest and which allows us to solve the Boltzmann-Nordheim equation for some classes of unbounded densities.