Applied Mathematics and Optimization ( IF 1.6 ) Pub Date : 2021-05-09 , DOI: 10.1007/s00245-021-09783-7 Tahir Boudjeriou
In this paper, we investigate the local existence, global existence, and blow-up of solutions to the Cauchy problem for Choquard–Kirchhoff-type equations involving the fractional p-Laplacian. As a particular case, we study the following initial value problem
$$\begin{aligned}\left\{ \begin{array}{llc} u_{t}+M\left( \Vert u\Vert ^{p}\right) [(-\Delta )^{s}_{p}u+V(x)|u|^{p-2}u]=\left( \int _{\mathbb {R}^{N}}\frac{|u|^{q}}{|x-y|^{\mu }}\,dy\right) |u|^{q-2} u&{} \text {in}\ &{}\mathbb {R}^{N}\times (0, +\infty ), \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\mathbb {R}^{N} , \end{array}\right. \end{aligned}$$where
$$\begin{aligned} \Vert u\Vert =\left( [u]^{p}_{s,p}+\int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx\right) ^{1/p}, \end{aligned}$$\(s\in (0,1)\), \(N>ps\), \(p,q> 2\), \((-\Delta )^{s}_{p}\) is the fractional p-Laplacian, \(u_{0} :\mathbb {R}^{N}\rightarrow [0, +\infty )\) is the initial function, \(M :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}\) is a continuous function given by \(M(\sigma )=\sigma ^{\theta -1}\), \(\theta \in [1, N/(N-sp))\) and \(V :\mathbb {R}^{N}\rightarrow \mathbb {R}^{+}\) is the potential function. Under some appropriate conditions, the well-posedness of nonnegative solutions for the above Cauchy problem is established by employing the Galerkin method. Moreover, the asymptotic behavior of global solutions is investigated under some assumptions on the initial data. We also establish upper and lower bounds for the blow-up time.
中文翻译：
$$ \ mathbb {R} ^ {N} $$ RN中非局部Choquard–Kirchhoff扩散方程的整体解的存在和不存在
在本文中，我们研究了涉及分数p -Laplacian的Choquard-Kirchhoff型方程的柯西问题的局部存在，整体存在和解的爆炸性。作为一个特例，我们研究以下初始值问题
$$ \ begin {aligned} \ left \ {\ begin {array} {llc} u_ {t} + M \ left（\ Vert u \ Vert ^ {p} \ right）[（-\ Delta）^ {s} _ {p} u + V（x）| u | ^ {p-2} u] = \ left（\ int _ {\ mathbb {R} ^ {N}} \ frac {| u | ^ {q}} {| xy | ^ {\ mu}} \，dy \ right）| u | ^ {q-2} u＆{} \ text {in} \＆{} \ mathbb {R} ^ {N} \ times（0 ，+ \ infty），\\ u（x，0）= u_ {0}（x），＆{} \ text {in}＆{} \ mathbb {R} ^ {N}，\ end {array} \正确的。\ end {aligned} $$在哪里
$$ \ begin {aligned} \ Vert u \ Vert = \ left（[u] ^ {p} _ {s，p} + \ int _ {\ mathbb {R} ^ {N}} V（x）| u | ^ {p} \，dx \ right）^ {1 / p}，\ end {aligned} $$\（s \ in（0,1）\），\（N> ps \），\（p，q> 2 \），\（（-\ Delta）^ {s} _ {p} \）是分数p -Laplacian，\（u_ {0}：\ mathbb {R} ^ {N} \ rightarrow [0，+ \ infty）\）是初始函数\（M：\ mathbb {R} ^ {+} \ rightarrow \ mathbb {R} ^ {+} \）是一个连续函数，由\（M（\ sigma）= \ sigma ^ {\ theta -1} \），\（\ theta \ in [1，N / （N-sp））\）和\（V：\ mathbb {R} ^ {N} \ rightarrow \ mathbb {R} ^ {+} \）是潜在的功能。在某些适当的条件下，通过使用Galerkin方法建立了上述柯西问题的非负解的适定性。此外，在一些关于初始数据的假设下，研究了整体解的渐近行为。我们还为爆炸时间设定了上限和下限。
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