Existence and Non-existence of Global Solutions for a Nonlocal Choquard–Kirchhoff Diffusion Equations in $$\mathbb {R}^{N}$$

Qualitative analysis of solutions to a nonlocal Choquard–Kirchhoff diffusion equations in ℝN$$ {\mathbb{R}}^N $$

“…Finally, we introduce the basic space $$$ W $$$ used in this paper, which is defined as follows: $$$$ W=\left\{u\in {W}^{s,p}\left({\mathbb{R}}^N\right):{\int}_{{\mathbb{R}}^N}V(x){\left|u\right|}^p dx<\infty \right\}. $$$$ By Boudjeriou, 51, Lemmas 2.1 and 2.2 we get $$$ W $$$ is a separate and reflexive Banach space with the norm $$$ \left\Vert \cdotp \right\Vert $$$ defined in () and $$$ W\hookrightarrow {L}^{\nu}\left({\mathbb{R}}^N\right) $$$ is continuous for all $$$ \nu \in \left[2,{p}_s^{\ast}\right] $$$, and $$$ W\hookrightarrow \hookrightarrow {L}^{\nu}\left({\mathbb{R}}^N\right) $$$ for all $$$ \nu \in \left[2,{p}_s^{\ast}\right) $$$, where $$$ {p}_s^{\ast }=\frac{Np}{N- sp} $$$ and the notations $$…”

“…Remark Since $$$ W\hookrightarrow \hookrightarrow {L}^{\mu}\left({\mathbb{R}}^N\right) $$$ for $$$ \mu \in \left[2,{p}_s^{\ast}\right) $$$ (see Boudjeriou 51, Lemma 2.2 ), it follows from $$$ u\in {L}^{\infty}\left(0,T;W\right) $$$ and $$$ {u}_t\in {L}^2\left(0,T;{L}^2\left({\mathbb{R}}^N\right)\right) $$$ that $$$ u\in C\left(\left[0,T\right],{L}^{\nu}\left({\mathbb{R}}^N\right)\right) $$$ for any $$$ \mu \in \left[2,{p}_s^{\ast}\right) $$$ and $$$ T\in \left(0,{T}_{\mathrm{max}}\right) $$$, so the value of $$$ u $$$ at time $$$ t=0 $$$, that is, $$$ u(0) $$$ makes sense.…”

“…Then (see Boudjeriou 51, Lemma 3.3 ) $$$$ d\ge \left(\frac{2q-\theta p}{2\theta p q}\right){\left(\frac{\Lambda_{\frac{2 Nq}{2N-\mu}}^{2q}}{C_{\ast }}\right)}^{\frac{\theta p}{2q-\theta p}}, $$$$ where $$$ {\Lambda}_{\frac{2 Nq}{2N-\mu }} $$$ is the positive constant given in () and, by Lieb and Loss, 52, Theorem 4.3 $$$$ {C}_{\ast}:= {\pi}^{\frac{\mu }{2}}\frac{\varGamma \left(N/2-\mu /2\right)}{\varGamma \left(N-\mu /2\right)}{\left[\frac{\varGamma \left(N/2\right)}{\varGamma (N)}\right]}^{\frac{\mu }{N}-1}. $$$$ …”

“…Remark In the proof of the above theorem, some embedding constants were ignored in Boudjeriou, 51, (5.28) the right one is $$$$ {\left\Vert u(t)\right\Vert}_{W^{s-\epsilon, p}\left({\mathbb{R}}^N\right)}\le \zeta {\left\Vert u(t)\right\Vert}_{W^{s,p}\left({\mathbb{R}}^N\right)}^{\eta }{\left\Vert u(t)\right\Vert}_p^{1-\eta}\le \zeta {\varsigma}^{\eta }{\left\Vert u(t)\right\Vert}^{\eta }{\left\Vert u(t)\right\Vert}_p^{1-\eta }, $$$$ where $$$ \zeta >0 $$$ is given by the fractional Gagliardo–Nirenberg interpolation inequality (see Lemma 2.2 below), which depends only on $$$ \epsilon, s,\eta, p $$$, and $$$ \varsigma $$$ is the positive constant given in (). Moreover, the constant $$$ d $$$ was forgotten in the right‐hand side of the inequality (5.29) of Boudjeriou 51 and the right one is …”

“…Remark In the proof of the above theorem, one constant was ignored on the right‐hand side of the inequality (5.6) (see page s728) of Boudjeriou 51 and the right one is $$$ \frac{2\sigma }{\left(\theta p-4\right)}. $$$ So, $$$$ {T}_{\mathrm{max}}\le \frac{16{\left\Vert {u}_0\right\Vert}_2^2\left(\theta p-1\right)}{-\theta p E\left({u}_0\right){\left(\theta p-4\right)}^2}.$$…”