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Longtime behavior of semilinear multi-term fractional in time diffusion

Nataliya Vasylyeva

Abstract

In the paper, the initial-boundary value problems to a semilinear integro-differential equation with multi-term fractional Caputo derivatives are analyzed. A particular case of this equation models oxygen diffusion through capillaries. Under proper assumptions on the coefficients and a nonlinearity, the longtime behavior (as $t\to+\infty$) of a solution is discussed. In particular, the existence of absorbing sets in suitable functional spaces is established.

Longtime behavior of semilinear multi-term fractional in time diffusion

Abstract

In the paper, the initial-boundary value problems to a semilinear integro-differential equation with multi-term fractional Caputo derivatives are analyzed. A particular case of this equation models oxygen diffusion through capillaries. Under proper assumptions on the coefficients and a nonlinearity, the longtime behavior (as ) of a solution is discussed. In particular, the existence of absorbing sets in suitable functional spaces is established.
Paper Structure (6 sections, 13 theorems, 167 equations)

This paper contains 6 sections, 13 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Functional Spaces and Notation
  3. Main Results
  4. Technical Results
  5. Proof of Theorem \ref{['t3.1']}
  6. Proof of Theorem \ref{['t3.2']}

Key Result

Theorem 3.1

Let $n=1$ and conditions h1-h5 hold. If $\frac{\partial a_{1}}{\partial x}\not\equiv 0$, then we additionally assume that there is a positive value $\delta^{*}\in(0,\delta_{1})$ such that Then a solution of i.1-i.3 satisfies the estimates Here the positive constants $C_{0}$ and $C_{1}$ depend only on the structural parameters in the model, the Lebesgue measure of $\Omega$, and the corresponding

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 3.1
  • Example 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Example 3.2
  • Definition 3.1
  • Definition 3.2
  • ...and 16 more