Upper Bound Of Blow-up Time Research Articles

Comparison theorem in blow-up problems for reaction diffusion equations and for their approximations

In this paper, blow-up sufficient conditions and upper bound of blow-up time for solution of Neumann and Dirichlet problems for reaction diffusion equations with non-linear gradient have been obtained. These equations have been found from the comparison of theorems, Jensen’s inequality and conservations laws. By using a similar proof approach for the finite-difference case, the finite-difference scheme was constructed, approximating the above-mentioned Neumann problem, for which sufficient conditions and upper bound of blow-up time, consistent with appropriate conditions and bound for the appropriate differential problem, have been obtained.

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})<0$$ J μ ( u 0 ) < 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.

Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

<p style='text-indent:20px;'>The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which has been studied in [<xref ref-type="bibr" rid="b5">5</xref>]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the <inline-formula><tex-math id="M1">\begin{document}$ L^{2}(\Omega) $\end{document}</tex-math></inline-formula> norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [<xref ref-type="bibr" rid="b5">5</xref>].

High initial energy finite time blowup with upper bound of blowup time of solution to semilinear parabolic equations

A sufficient condition on initial profile ensuring high initial energy finite time blowup along with the upper bound of the corresponding blowup time for the classical semilinear parabolic equations involving initial boundary value problem is established in this paper.

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 the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level

This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained. Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level. At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].

Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption

This paper deals with a quasilinear parabolic equation with nonlinear gradient absorption \begin{equation*} u_t-\Delta_{p}u=u^q-\mu u^{r}|\nabla u|^\delta, \ x\in\Omega, t>0. \end{equation*} Here, $\Delta_{p} u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplace operator, $\Omega \subset \mathbb{R}^{N}$ $(N \geq 1)$ is a bounded smooth domain. By a regularization approach, we first establish the local-in-time existence of its weak solutions. Then we prove the global existence by constructing a family of bounded super-solutions which technically depend on the inradius of $\Omega$. We also obtain an upper bound and a lower bound of the blowup time. We use a comparison with suitable self-similar sub-solutions to prove the blowup and an upper bound of blowup time. Finally, we derive a lower bound of the blowup time by using the differential inequality technique.

Blow-up of solutions to a quasilinear wave equation for high initial energy

This paper deals with blow-up solutions to a nonlinear hyperbolic equation with variable exponent of nonlinearities. By constructing a new control function and using energy inequalities, the authors obtain the lower bound estimate of the L2 norm of the solution. Furthermore, the concavity arguments are used to prove the nonexistence of solutions; at the same time, an estimate of the upper bound of blow-up time is also obtained. This result extends and improves those of [1,2].

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.

A lower bound for blow-up time in a fully parabolic Keller–Segel system

In this paper, an explicit lower bound for the blow-up time is obtained to a parabolic–parabolic Keller–Segel system, the blow-up conditions of which were established with an upper bound of blow-up time by Cieślak and Stinner [T. Cieślak, C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations 252 (2012) 5832–5851].

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)▽u)+ f (x;u; |▽u|2; t) under nonlinear boundary condition ¶u=¶n+g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f .

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations { u t t + u t + | u t | m − 1 u t = div ( ρ 1 ( | ∇ u | 2 ) ∇ u ) + f 1 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , v t t + v t + | v t | r − 1 v t = div ( ρ 2 ( | ∇ v | 2 ) ∇ v ) + f 2 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions f 1 , f 2 , ρ 1 , ρ 2 , parameters r , m and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.