scholarly journals Numerical simulations for initial value inversion problem in a two-dimensional degenerate parabolic equation

2021 ◽  
Vol 6 (4) ◽  
pp. 3080-3104
Author(s):  
Zui-Cha Deng ◽  
◽  
Fan-Li Liu ◽  
Liu Yang ◽  
Keyword(s):  
Parabolic Equation ◽  
Numerical Simulations ◽  
Degenerate Parabolic Equation ◽  
Inversion Problem ◽  
Two Dimensional ◽  
Initial Value ◽  
Degenerate Parabolic

scholarly journals On a weak solution matching up with the double degenerate parabolic equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Initial Value ◽  
Laplacian Equation ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Regularized Method ◽  
Increasing Function

Abstract The well-posedness of weak solutions to a double degenerate evolutionary $p(x)$ p ( x ) -Laplacian equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)\bigr), $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) ) , is studied. It is assumed that $b(x,t)| _{(x,t)\in \varOmega \times [0,T]}>0$ b ( x , t ) | ( x , t ) ∈ Ω × [ 0 , T ] > 0 but $b(x,t) | _{(x,t)\in \partial \varOmega \times [0,T]}=0$ b ( x , t ) | ( x , t ) ∈ ∂ Ω × [ 0 , T ] = 0 , $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , and $A(s)$ A ( s ) is a strictly monotone increasing function with $A(0)=0$ A ( 0 ) = 0 . A weak solution matching up with the double degenerate parabolic equation is introduced. The existence of weak solution is proved by a parabolically regularized method. The stability theorem of weak solutions is established independent of the boundary value condition. In particular, the initial value condition is satisfied in a wider generality.

scholarly journals Null controllability of a semilinear degenerate parabolic equation with a gradient term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Fengdan Xu ◽  
Qian Zhou ◽  
Yuanyuan Nie
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Null Controllability ◽  
Gradient Term ◽  
Degenerate Parabolic

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

Blow-up for a degenerate parabolic equation with a nonlocal source

2008 ◽  
Vol 201 (1-2) ◽  
pp. 250-259 ◽  
Author(s):  
Congming Peng ◽  
Zuodong Yang
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Nonlocal Source ◽  
Degenerate Parabolic
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