arXiv 2207.044642022-07math.AP

含分数阶$p$ -Laplacian算子的非局部时空反应扩散方程的Cauchy问题

The Cauchy problem of non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian

Fei Gao, Hui Zhan

备注：: 20 pages. arXiv admin note: text overlap with arXiv:2205.11040, arXiv:2112.11143, arXiv:2202.04928; text overlap with arXiv:2103.00552 by other authors

摘要：

对于含分数阶$p$ -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*} $μ>0 ,k>0,γ\geq 1,α\in(0,1),s\in(0,1),1<p$的非局部时空反应扩散方程，利用Gagliardo-Nirenberg不等式和分数阶Duhamel公式，考虑了$N\leq2$弱解的整体有界性问题.利用比较原理和局部李雅普诺夫型泛函证明了当$μ$值较小时，弱解指数收敛或局部一致收敛于$0$，如$t \rightarrow \infty$.在这些情况下，问题被降低到分数$p$ -拉普拉斯方程的非局部反应扩散范围内，这是处理的对称性和其他性质的核$(-Δ)_{p}^{s}$。最后，利用Moser迭代和分数阶微分不等式证明了具有分数阶非线性扩散项$(-Δ)_{p}^{s}u^{m}(2-\frac{2}{N}<m\leq 3,1<p<\frac{4}{3})$的整体有界弱解.

英文摘要：

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.

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摘要：

英文摘要：

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