Nonlinear Differential Equations and Applications (NoDEA) ( IF 1.1 ) Pub Date : 2020-05-06 , DOI: 10.1007/s00030-020-00629-9 Sunday A. Asogwa , Mohammud Foondun , Jebessa B. Mijena , Erkan Nane
We will look at reaction–diffusion type equations of the following type,
$$\begin{aligned} \partial ^\beta _tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }]. \end{aligned}$$We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when \(0<\eta \leqslant \eta _c\), there is no global solution other than the trivial one while for \(\eta >\eta _c\), non-trivial global solutions do exist. The critical parameter \(\eta _c\) is shown to be \(\frac{1}{\eta ^*}\) where
$$\begin{aligned} \eta ^*:=\sup _{a>0}\left\{ \sup _{t\in (0,\,\infty ),x\in \mathbb {R}^d}t^a\int _{\mathbb {R}^d}G(t,\,x-y)V_0(y)\,\mathrm{d}y<\infty \right\} \end{aligned}$$and \(G(t,\,x)\) is the heat kernel of the corresponding unforced operator. \(V_0\) is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.
中文翻译：
时空分数导数的反应扩散方程的关键参数
我们将研究以下类型的反应扩散型方程，
$$ \ begin {aligned} \ partial ^ \ beta _tV（t，x）=-（-\ Delta）^ {\ alpha / 2} V（t，x）+ I ^ {1- \ beta} _t [V （t，x）^ {1+ \ eta}]。\ end {aligned} $$我们首先通过积分方程来研究整个空间上的方程。粗略地讲，我们将证明当\（0 <\ eta \ leqslant \ eta _c \）时，除了琐碎的解决方案外，没有其他全局解决方案，而对于\（\ eta> \ eta _c \）来说，则是非琐碎的全局解决方案确实存在。关键参数\（\ eta _c \）显示为\（\ frac {1} {\ eta ^ *} \），其中
$$ \ begin {aligned} \ eta ^ *：= \ sup _ {a> 0} \ left \ {\ sup _ {t \ in（0，\，\ infty），x \ in \ mathbb {R} ^ d} t ^ a \ int _ {\ mathbb {R} ^ d} G（t，\，xy）V_0（y）\，\ mathrm {d} y <\ infty \ right \} \ end {aligned} $ $和\（G（T，\，X）\）是相应的非受迫性操作的热内核。\（V_0 \）是非负初始函数。我们还研究了具有Dirichlet边界条件的有界域上的方程，并表明分数时间导数的存在会引起溶液行为的显着变化。
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