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Non-homogeneous problem for the fractional wave equation with irregular coefficients and data

Manel Bouguenna, Mohammed Elamine Sebih

Abstract

In this paper, we consider the Cauchy problem for a non-homogeneous wave equation generated by the fractional Laplacian and involving different kinds of lower order terms. We allow the equation coefficients and data to be of distributional type or less regular, having in mind the Dirac delta function and its powers, and we prove that the problem is well-posed in the sense of the concept of very weak solutions. Moreover, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory.

Non-homogeneous problem for the fractional wave equation with irregular coefficients and data

Abstract

In this paper, we consider the Cauchy problem for a non-homogeneous wave equation generated by the fractional Laplacian and involving different kinds of lower order terms. We allow the equation coefficients and data to be of distributional type or less regular, having in mind the Dirac delta function and its powers, and we prove that the problem is well-posed in the sense of the concept of very weak solutions. Moreover, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory.

Paper Structure

This paper contains 17 sections, 18 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Hölder's inequality
  5. The fractional Sobolev space $H^s$ and the fractional Laplacian
  6. Fractional Leibniz rule and the Kato-Ponce inequality
  7. Duhamel's principle
  8. Energy estimates for the classical solution
  9. Auxiliary result
  10. Homogeneous equation case
  11. Non-homogeneous equation case
  12. Very weak well-posedness
  13. Existence of very weak solutions
  14. Existence of a very weak solution: general case
  15. Existence of a very weak solution: case $g\equiv 1$
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $r,p,q \geq 1$, such that: $\frac{1}{r}=\frac{1}{p} + \frac{1}{q}$. Assume that $f\in L^{p}(\mathbb{R}^d)$ and $g\in L^{q}(\mathbb{R}^d)$, then, $fg\in L^{r}(\mathbb{R}^d)$ and we have

Theorems & Definitions (44)

  • Proposition 2.1
  • Definition 1: Fractional Sobolev space
  • Definition 2: Fractional Laplacian
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3: Fractional Sobolev inequality, Theorem 1.1. CT04
  • Proposition 2.4: The Kato-Ponce inequality, Theorem 1. GO14
  • Theorem 2.5
  • proof
  • ...and 34 more