论文标题
时间空间分数非局部反应扩散方程的全局界限和渐近行为
Global boundedness and asymptotic behavior of time-space fractional nonlocal reaction-diffusion equation
论文作者
论文摘要
全局界限和渐近行为是针对时间空间分数非本地反应扩散方程（TSFNRDE）的解决方案的研究 $ \ frac {\ partial^{α} u} {\ partial t^{α}}} = - （ - Δ）其中$ s \ in（0,1），α\ in（0,1），n \ leq 2 $。操作员$ \ partial_ {t}^{α} $是caputo分数衍生物，$ - （ - δ）^{s} $是分数laplacian运算符。对于$ j $的适当假设，事实证明，对于均质的dirichlet边界条件，此问题承认了$ n = 1 $的全球有限弱解决方案，而对于$ n = 2 $，Gagliardo-Nirenberg不平等的大$ K $值存在全球有限的弱解决方案，并且分数差异不平等。对于初始基准的进一步假设，对于小$μ$值，该解决方案被证明将$ 0 $指数或本地均匀地收敛为$ t \ rightarrow \ infty $。此外，在$ j \ equiv 1 $的条件下，事实证明，非线性tsfnrde具有独特的弱解决方案，该解决方案在分数sobolev空间中具有非线性分数扩散项$ - （ - δ）^{s}^{s}^{m} u^{m} {m} \，（2- \，（2- \ frac freac）。
The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=-(-Δ)^{s} u+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where $s\in(0,1),α\in(0,1), N \leq 2$. The operator $\partial_{t}^{α}$ is the Caputo fractional derivative, which $-(-Δ)^{s}$ is the fractional Laplacian operator. For appropriate assumptions on $J$, it is proved that for homogeneous Dirichlet boundary condition, this problem admits a global bounded weak solution for $N=1$, while for $N=2$, global bounded weak solution exists for large $k$ values by Gagliardo-Nirenberg inequality and fractional differential inequality. With further assumptions on the initial datum, for small $μ$ values, the solution is shown to converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$. Furthermore, under the condition of $J \equiv 1$, it is proved that the nonlinear TSFNRDE has a unique weak solution which is global bounded in fractional Sobolev space with the nonlinear fractional diffusion terms $-(-Δ)^{s} u^{m}\, (2-\frac{2}{N}<m<1)$.
