Decay Estimates for α‎ and α‎ (Theorems M2, M3)

2020 ◽  
pp. 264-294
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
Parabolic Equation ◽  
Transport Equation ◽  
Decay Estimates ◽  
Prove Theorem

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To recover α‎ from q, the chapter derives a transport equation for α‎ where q is on the RHS. It then derives as Teukolsky–Starobinsky identity a parabolic equation for α‎. The chapter also improves bootstrap assumptions for α‎.

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

Initialization and Extension (Theorems M6, M7, M8)

2020 ◽  
pp. 372-485
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
Iterative Procedure ◽  
Higher Order ◽  
Boundary Term ◽  
Decay Estimates ◽  
Extension Theorems ◽  
Prove Theorem

This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first improves the bootstrap assumptions on decay estimates. The chapter then improves the bootstrap assumptions on energies and weighted energies for R and Γ‎ relying on an iterative procedure which recovers derivatives one by one. It also outlines the norms for measuring weighted energies for curvature components and Ricci coefficients. To prove Theorem M8, the chapter relies on Propositions 8.11, 8.12, and 8.13. Among these propositions, only the last two involve the dangerous boundary term.

scholarly journals Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Dengming Liu ◽  
Luo Yang
Keyword(s):  
Parabolic Equation ◽  
Decay Estimate ◽  
Energy Estimate ◽  
Quasilinear Parabolic Equation ◽  
Upper And Lower Solutions ◽  
Decay Estimates ◽  
Nonlinear Source ◽  
Quasilinear Parabolic

By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.

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.

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