and $u\in \widetilde{L}^{\rho}_{T}(B^{s}_{p,r})\cap L^{\infty}_{T}(L^{\infty})$.

1.  Let $F\in W_{loc}^{[s]+2,\infty}({\mathbb{R}}^{N})$ such that $F(0)=0$. Then $F(u)\in \widetilde{L}^{\rho}_{T}(B^{s}_{p,r})$. More precisely there exists a function $C$ depending only on $s$, $p$, $r$, $N$ and $F$ such that: $$\|F(u)\|_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}\leq C(\|u\|_{L^{\infty}_{T}(L^{\infty})}\|u\|_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}.$$

2.  Let $F\in W_{loc}^{[s]+3,\infty}({\mathbb{R}}^{N})$ such that $F(0)=0$. Then $F(u)-F^{'}(0)u\in \widetilde{L}^{\rho}_{T}(B^{s}_{p,r})$. More precisely there exists a function $C$ depending only on $s$, $p$, $r$, $N$ and $F$ such that: $$\|F(u)-F^{'}(0)u\|_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}\leq C(\|u\|_{L^{\infty}_{T}(L^{\infty})}\|u\|^{2}_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}.$$

Here we recall a result of interpolation which explains the link of the space $B^{s}_{p,1}$ with the space $B^{s}_{p,\infty}$, see [@DFourier].

\[interpolationlog\] There exists a constant $C$ such that for all $s\in{\mathbb{R}}$, ${\varepsilon}>0$ and $1\leq p<+\infty$, $$\|u\|_{\widetilde{L}_{T}^{\rho}(B^{s}_{p,1})}\leq C\frac{1+{\varepsilon}}{{\varepsilon}}\|u\|_{\widetilde{L}_{T}^{\rho}(B^{s}_{p,\infty})}
\biggl(1+\log\frac{\|u\|_{\widetilde{L}_{T}^{\rho}(B^{s+{\varepsilon}}_{p,\infty})}}
{\|u\|_{\widetilde{L}_{T}^{\rho}(B^{s}_{p,\infty})}}\biggl).$$ \[5Yudov\]

Now we give some result on the behavior of the Besov spaces via some pseudodifferential operator (see [@DFourier]).

Let $m\in{\mathbb{R}}$. A smooth function function $f:{\mathbb{R}}^{N}\rightarrow{\mathbb{R}}$ is said to be a ${\cal S}^{m}$ multiplier if for all muti-index $\alpha$, there exists a constant $C_{\alpha}$ such that: $$\forall\xi\in{\mathbb{R}}^{N},\;\;|{\partial}^{\alpha}f(\xi)|\leq C_{\alpha}(1+|\xi|)^{m-|\alpha|}.$$ \[smoothf\]

Let $m\in{\mathbb{R}}$ and $f$ be a ${\cal S}^{m}$ multiplier. Then for all $s\in{\mathbb{R}}$ and $1\leq p,r\leq+\infty$ the operator $f(D)$ is continuous from $B^{s}_{p,r}$ to $B^{s-m}_{p,r}$. \[singuliere\]

Let us now give some estimates for the heat equation:

\[5chaleur\] Let $s\in{\mathbb{R}}$, $(p,r)\in[1,+\infty]^{2}$ and $1\leq\rho_{2}\leq\rho_{1}\leq+\infty$. Assume that $u_{0}\in B^{s}_{p,r}$ and 