论文标题
部分可观测时空混沌系统的无模型预测
The Cauchy problem of non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian
论文作者
论文摘要
对于非本地时空反应扩散方程，涉及分数$ -P $ -LAPLACIAN \ begin {等式*} \ begin {case} \ frac {\ partial^{α} u} {\ partial t^{α}}}}+（ - δ）_ {p}^{s}^{s} u =μu=μu^{2}（1-kj*u） u（x，0）= u_ {0}（x），＆x \ in \ mathbb {r}^{n} \end{cases} \end{equation*} $μ>0 ,k>0,γ\geq 1,α\in(0,1),s\in(0,1),1<p$, we consider for $N\leq2$ the problem of finding a global boundedness of the weak solution by virtue of Gagliardo-Nirenberg inequality and fractional Duhamel's formula.此外，我们证明了这种弱解决方案将$ 0 $指数或本地均匀地收敛为$ t \ rightarrow \ infty \ infty $，对于小$ $ $ $值，与比较原理和本地lyapunov型函数。在这种情况下，该问题将在非局部反应 - 扩散范围内的分数$ p $ -laplacian方程降低，该方程用$（ - δ）_ {p}^{s} $的对称性和其他特性处理。 Finally, a key element in our construction is a proof of global bounded weak solution with the fractional nonlinear diffusion terms $(-Δ)_{p}^{s}u^{m}(2-\frac{2}{N}<m\leq 3,1<p<\frac{4}{3})$ by using Moser iteration and fractional differential inequality.
For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{α}u}{\partial t^{α}}+(-Δ)_{p}^{s} u=μu^{2}(1-kJ*u)-γu,&(x,t)\in\mathbb{R}^{N}\times(0,T)\\ u(x,0)=u_{0}(x),& x\in\mathbb{R}^{N} \end{cases} \end{equation*} $μ>0 ,k>0,γ\geq 1,α\in(0,1),s\in(0,1),1<p$, we consider for $N\leq2$ the problem of finding a global boundedness of the weak solution by virtue of Gagliardo-Nirenberg inequality and fractional Duhamel's formula. Moreover, we prove such weak solution converge to $0$ exponentially or locally uniformly as $t \rightarrow \infty$ for small $μ$ values with the comparison principle and local Lyapunov type functional. In those cases the problem is reduced to fractional $p$-Laplacian equation in the non-local reaction-diffusion range which is treated with the symmetry and other properties of the kernel of $(-Δ)_{p}^{s}$. Finally, a key element in our construction is a proof of global bounded weak solution with the fractional nonlinear diffusion terms $(-Δ)_{p}^{s}u^{m}(2-\frac{2}{N}<m\leq 3,1<p<\frac{4}{3})$ by using Moser iteration and fractional differential inequality.
