Bound Of Blow-up Time Research Articles

Global existence and blow-up of solutions to a semilinear heat equation with logarithmic nonlinearity

ABSTRACT In this paper, we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using potential well method and energy method, we obtain the existence of global solutions and finite-time blow-up solutions. Moreover, the upper bound of blow-up time is discussed under suitable assumptions.

Bounds for Blow-up Time in Nonlinear Pseudo-parabolic Equations

The aim of this study is to analyze the initial boundary value problem for the following nonlinear pseudo-parabolic equations: $$\begin{aligned} u_t -\Delta u_t - \Delta u - u^mu_{x_{1}} + g(t,x,u,\nabla u)= |u |^{m_1}u, \quad x \in \Omega ,\quad \ t>0, \end{aligned}$$ where m, $$m_1$$ are positive numbers and $$g(t,x,u,\nabla u)$$ is a bounded function. We established sufficient conditions on initial data, m, $$m_1$$ and the function $$g(t,x,u,\nabla u)$$ for the nonblow-up case. Moreover, we obtained the lower and upper bounds for the blow-up time if blow-up happens.

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.

Bounds for blow-up time in a semilinear parabolic problem with viscoelastic term

We consider a semilinear parabolic equation with viscoelastic term ut−△u+∫0tg(t−s)△u(x,s)ds=|u|p−2u.By the means of differential inequality technique, we obtain a lower bound for blow-up time of the solution if blow-up occurs. At the same time, we establish a new blow-up criterion and give an upper bound for blow-up time of the solution under some conditions on p, g and u0.

Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source

This paper considers the following semilinear pseudo-parabolic equation with a nonlocal source: $$ u_{t}-\triangle u_{t}-\triangle u=u^{p}(x,t) \int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy, $$ and it explores the characters of blow-up time for solutions, obtaining a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, we investigate a nonblow-up criterion and compute an exact exponential decay.

Bounds for Blow-Up Time in Nonlinear Parabolic Systems Under Various Boundary Conditions

We consider blow-up solutions to parabolic systems, coupled through their nonlinearities under various boundary conditions with nonlinearities depending on the gradient solution. To obtain a lower bound to blow up time t* for the vector solution, Sobolev-type inequalities are introduced to make use of a differential inequality technique. In addition for Dirichlet systems sufficient conditions are introduced to derive an upper bound for t* and to have a criterion for the global existence of the vector solution.