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Nonlinear parabolic problem with time fractional derivative

Nikolai Kutev, Tsviatko Rangelov

Abstract

Time fractional parabolic problem for p-Laplacian with double singular Hardy-type potential is considered. Comparison principle and appriory estimates for the weak solutions are proved. Existence of global weak solutions and finite-time blow-up are investigated depending on the optimal Hardy constant.

Nonlinear parabolic problem with time fractional derivative

Abstract

Time fractional parabolic problem for p-Laplacian with double singular Hardy-type potential is considered. Comparison principle and appriory estimates for the weak solutions are proved. Existence of global weak solutions and finite-time blow-up are investigated depending on the optimal Hardy constant.
Paper Structure (8 sections, 8 theorems, 79 equations)

This paper contains 8 sections, 8 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Time fractional ordinary differential equations
  3. Comparison principle
  4. Blow-up solutions
  5. Time fractional parabolic problems
  6. Comparison principle
  7. Apriori estimates
  8. Proof of the main results

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$, $n>p$, $p>2$, $0\in\Omega$, be a star-shaped domain with respect to a small ball centered at the origin, and $\mu<\Lambda(n,p)$. If $u_0(x)\in W_0^{1,p}(\Omega)$ then every weak solution of the problem problem eq1 defined in the maximal existence interval $(0, T_m)$

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2: Lemma 2.1 in LV08
  • Theorem 2.3
  • proof
  • Theorem 3.1: Comparison principle
  • ...and 6 more