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.

Complete blow-up for quasilinear degenerate parabolic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 877-908 ◽  
Author(s):  
R. Suzuki
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Image Position

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

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 On some degenerate parabolic equations II

1973 ◽  
Vol 52 ◽  
pp. 61-84 ◽  
Author(s):  
Tadato Matsuzawa
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Pseudodifferential Operators ◽  
Degenerate Parabolic Equations ◽  
Smooth Functions ◽  
Fundamental Assumption ◽  
Degenerate Parabolic ◽  
The Given ◽  
Complex Valued

In the article I: [8], we have proved the hypoellipticity of a degenerate parabolic equation of the form:where the coefficients a(x, t), b(x,t) and c(x, t) are complex valued smooth functions. The fundamental assumption on the coefficients is that Re a(x, t) satisfies the condition of Nirenberg and Treves ([8], (1.5)). To prove the hypoellipticity we have constructed recurcively the parametrices as pseudodifferential operators with parameter. This method may be viewed as an improvement of that of [9] and [7]. We have analyzed the properties of these parametrices by estimating the symbols with parameter associated with the given operator. We shall summerize these results in §3.

scholarly journals Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fahe Miao ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Degenerate Parabolic Equation ◽  
Pullback Attractors ◽  
Asymptotic Compactness ◽  
Absorbing Sets ◽  
Degenerate Parabolic ◽  
Standard Galerkin Method

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

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