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Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations

Changxing Miao, Zhiwen Zhao

Abstract

This paper aims to study the local behavior of solutions to a class of anisotropic weighted quasilinear degenerate parabolic equations with the weights comprising two power-type weights of different dimensions. We first capture the asymptotic behavior of the solution near the singular or degenerate point of the weights. In particular, we find an explicit upper bound on the decay rate exponent determined by the structures of the equations and weights, which can be achieved under certain condition and meanwhile reflects the damage effect of the weights on the regularity of the solution. Furthermore, we prove the local Hölder regularity of solutions to non-homogeneous parabolic $p$-Laplace equations with single power-type weights.

Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations

Abstract

This paper aims to study the local behavior of solutions to a class of anisotropic weighted quasilinear degenerate parabolic equations with the weights comprising two power-type weights of different dimensions. We first capture the asymptotic behavior of the solution near the singular or degenerate point of the weights. In particular, we find an explicit upper bound on the decay rate exponent determined by the structures of the equations and weights, which can be achieved under certain condition and meanwhile reflects the damage effect of the weights on the regularity of the solution. Furthermore, we prove the local Hölder regularity of solutions to non-homogeneous parabolic -Laplace equations with single power-type weights.
Paper Structure (7 sections, 18 theorems, 183 equations)

This paper contains 7 sections, 18 theorems, 183 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary
  3. Local energy estimates
  4. Local regularity for weak solutions
  5. The proof of Proposition \ref{['PRO01']}
  6. The proofs of Theorems \ref{['ZWTHM90']} and \ref{['THM060']}.
  7. Appendix: An alternative proof for the expansion of time

Key Result

Theorem 1.1

Assume that $p>2$, $n\geq2$, $\mathrm{(}\mathbf{H1}\mathrm{)}$--$\mathrm{(}\mathbf{H3}\mathrm{)}$, ZE90--E01 and $\mathrm{(}\mathbf{K1}\mathrm{)}$--$\mathrm{(}\mathbf{K2}\mathrm{)}$ hold. Let $u$ be a weak solution of problem PO001 with $\Omega\times(-T,0]=B_{1}\times(-1,0]$. Then there exists a sma where $\varepsilon_{0}$ and $\vartheta$ are defined by VAR01, $O(1)$ represents a quantity satisfyi

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 33 more