Global existence and non-existence analyses for a semilinear edge degenerate parabolic equation with singular potential term

2022 ◽  
Vol 309 ◽  
pp. 508-557
Author(s):  
Guangyu Xu ◽  
Chunlai Mu ◽  
Yafeng Li
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Degenerate Parabolic Equation ◽  
Singular Potential ◽  
Potential Term ◽  
Degenerate Parabolic

scholarly journals Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
K. Atifi ◽  
Y. Balouki ◽  
El-H. Essoufi ◽  
B. Khouiti
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Singular Potential ◽  
Heat Conduction Problem ◽  
Hybrid Genetic Algorithm ◽  
Inverse Heat Conduction Problem ◽  
Descent Method ◽  
Initial Temperature Distribution ◽  
Degenerate Parabolic ◽  
The Difference

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.

scholarly journals Attractors for Nonautonomous Parabolic Equations without Uniqueness

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Cung The Anh ◽  
Nguyen Dinh Binh
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Global Existence ◽  
Global Attractor ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Global Attractors ◽  
Degenerate Parabolic ◽  
Uniform Global Attractor

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

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.

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