Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity $u_t?{△}_{X}u=u\log ?|u|$, where $X=(X_1,X_2,?,X_m)$ is an infinitely degenerate system of vector fields, and ${△}_X:={\sum }_{j=1}^m{X}_j^2$ is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions.     

Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|^p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].     

Sharp criteria of global existence and blow-up for a type of nonlinear parabolic equations

We study a type of nonlinear parabolic equations. In terms of the variational characterization of the corresponding nonlinear elliptic equations and the invariant flow arguments, we establish the sharp criteria for global existence and blow-up. Furthermore, we also get the instability of the steady states and the global existence with small initial data.     

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

This paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions.     

Life span of solutions with large initial data for a semilinear parabolic system

We investigate the initial-boundary value problem

     

Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level

We consider a nonlocal parabolic equation. By exploiting the boundary condition and the variational structure of the equation, we prove finite time blow-up of the solution for initial data at arbitrary energy level. We also obtain the lifespan of the blow-up solution. The results generalize the former studies on this equation.     

On blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy

This article is concerned with a class of semilinear parabolic equations with variable reaction $u_t=\Delta u+u^{p\left(x\right)}$ with homogeneous Dirichlet boundary conditions. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result is established with positive initial energy.     

A note on blow-up of solution for a class of semilinear pseudo-parabolic equations

In this short note, we revisit the blow-up of solution for the initial boundary value problem of semilinear pseudo-parabolic equations with low/critical initial energy stated in Xu and Su (2013) [4], and amend the proofs of the original paper.     

Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systems

with null Dirichlet boundary conditions in a smooth bounded domain ΩR^N, where p,q>1, , and a,b>0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=B_R={xR^N:|x|<R}(R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.     

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Blow-up and global existence for a nonlocal degenerate parabolic system

This paper investigates the blow-up and global existence of nonnegative solutions of the system

     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Fujita type exponent for fully nonlinear parabolic equations and existence results

In this paper, we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a Fujita type exponent. These exponents are related with an eigenvalue problem in all RN and play the same role in blow-up theorems as the classical p^?=1+2/N introduced by Fujita for the Laplacian. We also obtain some associated existence results.     

Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is under proper conditions on initial data. Furthermore, for N=1, more complete conclusions about such two topics are given.     

Determining nodes for semilinear parabolic equations

We are concerned with the determination of the asymptotic behavior of strong solutions to the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set. More precisely, if the asymptotic behavior of the strong solution is known on a suitable finite set which is called determining nodes, then the asymptotic behavior of the strong solution itself is entirely determined. We prove the above property by the energy method.