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A note on evolution equations with modified Hartree Nonlinearity

Khaldi Said

Abstract

We introduce a mathematical model in $\mathbb{R}^{n}$ for evolution equations with modified generalized Hartree nonlinearity given by $S_{α,p,q}(u)=I_α(|u|^{p+q}).$ One can see that this nonlinearity is not integrable due to the boundedness property of Riesz potential. In other words, we cannot deal with the Cauchy problem of semi-linear evolution equations with $S_{α,p,q}(u)$ and $L^{1}$-initial velocity. We will show that $S_{α,p,q}(u)$ produces the same semi-critical exponent that guarantees the global existence of small data solutions as in the well known generalized Hartree nonlinearity $H_{α,p,q}(u)=|u|^{p}I_α(|u|^{q})$ provided that the initial velocity belongs to $L^{m}(\mathbb{R}^{n})$, with $m>1$. We can expect a relation between some physical systems that are modeled and solved using Hartree nonlinearity and those in their modified form due to this coincidence property in the semi-critical exponent.

A note on evolution equations with modified Hartree Nonlinearity

Abstract

We introduce a mathematical model in for evolution equations with modified generalized Hartree nonlinearity given by One can see that this nonlinearity is not integrable due to the boundedness property of Riesz potential. In other words, we cannot deal with the Cauchy problem of semi-linear evolution equations with and -initial velocity. We will show that produces the same semi-critical exponent that guarantees the global existence of small data solutions as in the well known generalized Hartree nonlinearity provided that the initial velocity belongs to , with . We can expect a relation between some physical systems that are modeled and solved using Hartree nonlinearity and those in their modified form due to this coincidence property in the semi-critical exponent.
Paper Structure (4 sections, 5 theorems, 55 equations)

This paper contains 4 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Useful tools
  3. Main Result
  4. Conclusion

Key Result

Lemma 2.2

If $f \in L^{q}(\mathbb{R}^{n})$ for some $q\in(1,n/\alpha)$, then $I_{\alpha}f \in L^{r}(\mathbb{R}^{n})$ for some $q<r<\infty$ and satisfies the inequality Of course, $q<r$ due to the fact that $\alpha>0$.

Theorems & Definitions (10)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: pham reissig kainane
  • Lemma 2.4: pham reissig kainane
  • Lemma 2.5: ebert reissig
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • proof