scholarly journals Singularity and Decay Estimates for a Degenerate Parabolic Equation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Dongyan Li
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Decay Estimates ◽  
Nonnegative Solutions ◽  
A Priori Bound ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

In this paper, a degenerate parabolic equation u t − div x θ ∇ u = x a u p with p > 1 and θ < 2 , a ∈ ℝ , is considered. Based on rescaling arguments combined with a doubling property, the space-time singularity and decay estimates are established. Moreover, a universal and a priori bound of global nonnegative solutions for the corresponding initial boundary value problem is derived.

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.

scholarly journals REGULARITY AND FRACTAL DIMENSION OF PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMILINEAR DEGENERATE PARABOLIC EQUATION

2013 ◽  
Vol 55 (2) ◽  
pp. 431-448 ◽  
Author(s):  
CUNG THE ANH ◽  
TANG QUOC BAO ◽  
LE THI THUY
Keyword(s):  
Fractal Dimension ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Pullback Attractor ◽  
Formula Group ◽  
Degenerate Parabolic ◽  
Image Position ◽  
Initial Boundary Value

AbstractConsidered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation \begin{linenomath} u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t) \end{linenomath} in a bounded domain Ω in ℝN (N≥2). We prove the regularity in the space L2p−2(Ω)∩ $D_0^2(\Omega,\sigma)$, and estimate the fractal dimension of the pullback attractor in L2(Ω).

scholarly journals Parabolic Nonlinear Second Order Slip Reynolds Equation: Approximation and Existence

2007 ◽  
Vol 12 (1) ◽  
pp. 3-20
Author(s):  
K. Ait Hadi
Keyword(s):  
Reynolds Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Priori Estimates ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Model Existence ◽  
Nonlinear Degenerate

This work studies an initial boundary value problem for nonlinear degenerate parabolic equation issued from a lubrication slip model. Existence of solutions is established through a semi discrete scheme approximation combined with some a priori estimates.

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

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