Regional blow-up of solutions to the initial boundary value problem for ut = uδ(Δu + u)

1997 ◽  
Vol 127 (4) ◽  
pp. 871-887 ◽  
Author(s):  
Masayoshi Tsutsumi ◽  
Tetsuya Ishiwata
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

Non-negative solutions of the initial boundary value problem for a degenerate parabolic equation are investigated. It is shown that solutions blow up regionally in finite tine. The size of blow-up sets is determined for radially symmetric cases.

Existence and Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

2013 ◽  
Vol 785-786 ◽  
pp. 1454-1458
Author(s):  
Yan Ping Ran ◽  
Cong Ming Peng
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Equation ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

This article considers the following degenerate semilinear parabolic initial-boundary value problem,where be constants. We obtained the conditions of global existence and blow-up.

scholarly journals On a degenerate parabolic equation from double phase convection

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Double Phase ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$ u t ∈ L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ b ( x ) . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x ∈ ∂ Ω = 0 is found for the first time.

Single Point Blow-Up and Regional Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

2013 ◽  
Vol 760-762 ◽  
pp. 1777-1781
Author(s):  
Yong Ding Meng ◽  
Cong Ming Peng
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Single Point ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Equation ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

Let. This article considers the following degenerate semilinear parabolic initial-boundary value problem, where be constants. We obtained the blow up set and find the conditions of single point blow-up.

scholarly journals Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Fosheng Wang ◽  
Chengqiang Wang
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Viscoelastic Wave Equation ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Initial Boundary Value

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.

Sign in / Sign up

Export Citation Format

Share Document