Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconditional uniqueness and non-uniqueness for Hardy-Hénon parabolic equations

Noboru Chikami, Masahiro Ikeda, Koichi Taniguchi, Slim Tayachi

Abstract

We study the problems of uniqueness for Hardy-Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy-Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

Unconditional uniqueness and non-uniqueness for Hardy-Hénon parabolic equations

Abstract

We study the problems of uniqueness for Hardy-Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy-Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.
Paper Structure (20 sections, 36 theorems, 262 equations, 3 figures)

This paper contains 20 sections, 36 theorems, 262 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction and our setting
  3. Statement of the results
  4. Weighted Lorentz spaces
  5. Linear estimates
  6. Smoothing and time decay estimates in weighted spaces
  7. Weighted Meyer inequality
  8. Unconditional uniqueness and uniqueness criterion
  9. Nonlinear estimates
  10. Proofs of Theorems \ref{['thm:unconditional1']}, \ref{['thm:unconditional2']}, \ref{['thm:uniqueness-criterion']} and Proposition \ref{['prop:uniqueness-sufficient']}
  11. Non-uniqueness
  12. Existence of the regular solution
  13. Existence of singular solution
  14. Scale-supercritical case
  15. Additional results and remarks
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $T>0$, and let $d,\gamma,\alpha,q,r,s$ be as in assum:main. Assume either $(1)$ or $(2)$: Then unconditional uniqueness holds for HH in $L^\infty(0,T ; L^{q,r}_s(\mathbb R^d))$.

Figures (3)

  • Figure 1: The figure shows the domain of $(\alpha,q)$ for $d\ge 3$ and $\gamma \le 0$, where $\alpha_0 := 1 + \frac{\gamma}{d}$, $\alpha_F := 1 + \frac{2+\gamma}{d}$ is the Fujita exponent, $\alpha^*:= \frac{d+\gamma}{d-2}$ is the Serrin exponent and $\alpha_{HS} := \frac{d+2+2\gamma}{d-2}$ is the Hardy-Sobolev exponent. Table 1 and Table 2 summarize the previous results on uniqueness for \ref{['HH']} with $\gamma \le 0$.
  • Figure 2: The figure shows the domain of $({s\over d},{1\over q})$ in the case $\gamma<0$ and $\min\{{1\over q_c},{1\over Q_c}\}<\max\{{1\over q_c},{1\over Q_c}\}$. (U.U.) and (N.U.) mean unconditional uniqueness and non-uniqueness, respectively. The cases $\gamma=0$ and $\gamma>0$ are deduced by moving the line ${s\over d}={\gamma\over d(\alpha-1)}$ to the right.
  • Figure 3: The figure shows the domain of $(\alpha,s)$ for $d\ge 3$ and $q >1$. Here, $\alpha_0 := \min\{1,1 + \frac{\gamma}{d}\}$, $\alpha_F, \alpha^*, \alpha_{HS}$ are given in Figure 1, $s_c, S_c$ are given in \ref{['def:s_c-S_c']}, and $s^* := d-2-\frac{d}{q}$. Table 3 and Table 4 summarize our results on uniqueness for \ref{['HH']}.

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1.2: Scale-subcritical case
  • Theorem 1.3: Scale-critical case
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6: Double critical case
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9: Scale-supercritical case
  • Definition 2.1
  • ...and 71 more