arXiv 1706.033942017-06-11math.PR

关于时空分数阶热型非齐次时间分数阶泊松方程

On Space-Time Fractional Heat Type Non-Homogeneous Time-Fractional Poisson Equation

Ejighikeme McSylvester Omaba

备注：: 18 pages, British Journal of Mathematics and Computer Science, 2017

摘要：

考虑以下具有非齐次时间分数泊松过程的Riemann-Liouville导数的时空分数阶热方程\begin{eqnarray*} \partial^β_t u(x,t) =-κ(-Δ)^{α/2} u(x,t) + I_t^{1-β}[σ(u)D_t^\vartheta N^ν_λ(t)], \,\, t\geq 0, \,x \in \mathbb{R}^d, \end{eqnarray*}其中$κ>0, \,\,β,\,\vartheta\in(0,1), \,\,ν\in(0,1],\,α\in(0,2].$算子$D_t^\vartheta N^ν_λ(t) = \frac{\rm d}{\mathrm{d} t} I_t^{1-\vartheta} N_λ^ν(t) = \frac{\rm d}{\mathrm{d} t} \mathcal{N}_λ^{1-\vartheta,ν}(t)$，其中$\mathcal{N}_λ^{1-\vartheta,ν}(t)$是非齐次分数积分过程，$\partial^β_t$是Caputo分数阶导数，$-(-Δ)^{α/2}$是各向同性稳定过程的生成元，$I^β_t$是分数积分算子，并且$σ: \mathbb{R} \rightarrow \mathbb{R}$是Lipschitz连续的。上述时间分数阶随机热型方程可用于模拟具有热记忆的灾难事件序列。计算了某些特定速率函数的过程$\frac{\rm d}{\mathrm{d} t}\mathcal{N}^{1-\vartheta,ν}_λ(t)$的均值和方差。在此基础上，给出了非齐次分数时间Poisson过程扰动下热方程解的增长矩界，并证明了解在小时间区间$t\in [t_0,T], \,\,T<\infty$和$t_0>1$上指数增长;也就是说，该结果建立了对于初始数据的不同条件，解的能量至少增长为$ c_4(t+t_0)^{(\vartheta-aν)}\exp(c_5 t)$且至多增长为$c_1 t^{(\vartheta- aν)}\exp(c_3 t)$，其中$c_1,\,c_3,\,c_4$和$c_5$是取决于$T$的一些正常数。在线性增长条件下，利用$σ$给出了方程温和解的存在唯一性结果。

英文摘要：

Consider the following space-time fractional heat equation with Riemann-Liouville derivative of non-homogeneous time-fractional Poisson process \begin{eqnarray*} \partial^β_t u(x,t) =-κ(-Δ)^{α/2} u(x,t) + I_t^{1-β}[σ(u)D_t^\vartheta N^ν_λ(t)], \,\, t\geq 0, \,x \in \mathbb{R}^d, \end{eqnarray*} where $κ>0, \,\,β,\,\vartheta\in(0,1), \,\,ν\in(0,1],\,α\in(0,2].$ The operator $D_t^\vartheta N^ν_λ(t) = \frac{\rm d}{\mathrm{d} t} I_t^{1-\vartheta} N_λ^ν(t) = \frac{\rm d}{\mathrm{d} t} \mathcal{N}_λ^{1-\vartheta,ν}(t)$ with $\mathcal{N}_λ^{1-\vartheta,ν}(t)$ the Riemann-Liouville non-homogeneous fractional integral process, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2}$ is the generator of an isotropic stable process, $I^β_t$ is the fractional integral operator, and $σ: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz continuous. The above time fractional stochastic heat type equations may be used to model sequence of catastrophic events with thermal memory. The mean and variance for the process $\frac{\rm d}{\mathrm{d} t}\mathcal{N}^{1-\vartheta,ν}_λ(t)$ for some specific rate functions were computed. Consequently, the growth moment bounds for the class of heat equation perturbed with the non-homogeneous fractional time Poisson process were given and we show that the solution grows exponentially for some small time interval $t\in [t_0,T], \,\,T<\infty$ and $t_0>1$; that is, the result establishes that the energy of the solution grows atleast as $ c_4(t+t_0)^{(\vartheta-aν)}\exp(c_5 t)$ and at most as $c_1 t^{(\vartheta- aν)}\exp(c_3 t)$ for different conditions on the initial data, where $c_1,\,c_3,\,c_4$ and $c_5$ are some positive constants depending on $T$. Existence and uniqueness result for the mild solution to the equation was given under linear growth condition on $σ$.

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摘要：

英文摘要：

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