The Critical Exponent for Evolution Models with Power Non-linearity

Abstract

In this note we derive L ^r − L ^q estimates for the solutions to the Cauchy problem

$$\displaystyle u_{tt} +(-\varDelta )^{\sigma } u = 0\,, \qquad t\geq 0, \ x\in {\mathbb {R}}^n, \qquad u(0,x)=0, \;\; u_t(0,x)=g(x), $$

with σ > 1. Moreover, we derived the critical index p _c(n) for the existence of global in time small data solutions to the associated semilinear Cauchy problem with power nonlinearity |u|^p, p > 1.

Appendix

In the Appendix we list results of Harmonic Analysis.

The first ingredient is the celebrated Mikhlin-Hörmander multiplier theorem:

Theorem A.1

Let 1 < p < ∞ and $k=\max { \left \{[n(1/p-1/2)]+1,[n/2]+1 \right \}}$ . Suppose that $m \in \mathcal C^{k}({\mathbb {R}}^{n}\backslash \left \{ 0 \right \})$ and

$$\displaystyle \begin{aligned} \left| \partial_\xi^{\beta}m(\xi)\right| \leq C\, |\xi|{}^{-|\beta|}, \quad |\beta|\leq k. \end{aligned} $$

Then .

The next result is about translation invariant operators in $L^p=L^p(\mathbb {R}^n)$ spaces (see [6]).

Theorem A.2

Let f be a measurable function. Moreover, we suppose the following relation with suitable positive constants C and b ∈ (1, ∞):

$$\displaystyle \begin{aligned} \mathit{\mbox{meas}}\, \{ \xi \in \mathbb{R}^n : |f(\xi)| \geq l \} \leq C l^{-b}.\end{aligned} $$

Then $f \in M_p^q$ if 1 < p ≤ 2 ≤ q < ∞ and $\frac {1}{p}-\frac {1}{q} = \frac {1}{b}$.

In [10] one can find the following result, that is useful tool to derive L ^q − L ^q estimates.

Proposition A.1 (Berstein’s inequality)

Let $N>\frac {n}{2}$ . If f, D ^N f ∈ L ² , then there exists a constant C > 0 such that

$$\displaystyle \begin{aligned} \|f\|{}_{M_1}\leq C \|f\|{}_{L^2}^{1-\frac{n}{2N}}\|D^Nf\|{}_{L^2}^{\frac{n}{2N}}. \end{aligned} $$

In [8] one can find the following result, well known as Littman’s lemma, that is a very useful tool to derive L ^r − L ^q estimates on the dual line.

Theorem A.3

Let us consider for τ ≥ τ ₀ , τ ₀ is a large positive number, the oscillating integral

$$\displaystyle \begin{aligned} F^{-1}_{\eta\rightarrow x}\big(e^{-i\tau p(\eta)} v(\eta)\big).\end{aligned}$$

The amplitude function v = v(η) is supposed to belong to $C_0^\infty (\mathbb {R}^n)$ with support in $\{\eta \in \mathbb {R}^n: |\eta | \in [\frac {1}{2},2]\}$ . The function p = p(η) is C ^∞ in a neighborhood of the support of v. Moreover, the rank of the Hessian H _p(η) is supposed to satisfy the assumption rank H _p(η) ≥ k on the support of v. Then the following L ^∞− L ^∞ estimate holds:

$$\displaystyle \begin{aligned} \big\|F^{-1}_{\eta\rightarrow x}\big(e^{-i\tau p(\eta)} v(\eta)\big)\|{}_{L^\infty(\mathbb{R}^n_x)} \leq C(1+\tau)^{-\frac{k}{2}} \sum_{|\alpha| \leq L} \|D^\alpha_\eta v(\eta)\|{}_{L^\infty(\mathbb{R}^n_\eta)}, \end{aligned}$$

where L is a suitable entire number.

The next result about singular Fourier multipliers is due to Miyachi (see Theorem 4.1 in [7]):

Theorem A.4

Let us consider Fourier multiplier

$$\displaystyle \begin{aligned} m_{a,b}(\xi)= \frac{(1-\chi(\xi))e^{ i|\xi|{}^{a}}}{|\xi|{}^b}, \qquad \xi \in \mathbb{R}^n, \qquad a>0, a \neq 1, \qquad b\in \mathbb{R}, \end{aligned}$$

where χ is as in Notation 2 . If 1 < p ≤ q, then $m \in M_p^q$ if, and only if, $\frac 1{p}+ \frac 1{q}\leq 1$ and $\frac {1-a}{p}- \frac 1{q}\leq \frac {b}{n}-\frac {a}{2}$ or $\frac 1{p}+ \frac 1{q}\geq 1$ and $\frac 1{p}-\frac {1-a}{q}\leq \frac {b}{n}+\frac {a}{2}$ . Moreover, $m \in M_1^q$ if, and only if, $1-\frac {1-a}{q}< \frac {b}{n}+\frac {a}{2}$.