scholarly journals On the initial-boundary value problems for a degenerate parabolic equation

2008 ◽  
Author(s):  
Jan Goncerzewicz
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problems ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

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.

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.

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