论文标题
半线性SPDE的分数反应扩散系统的全球和爆炸性溶液具有分数噪声
Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise
论文作者
论文摘要
在本文中，我们研究了半线性随机偏微分方程（SPDE）的反应扩散系统的存在和有限的时间爆炸，该系统受到了二维分数布朗尼的运动。 \ begin {eqnarray*} du_ {1}（t，x）＆=＆\ left [δ_αu_{1}（t，x）+γ_{1} u_ {1}（t，t，x）+u^{1+β_{1}}} _ {2} _ {2}（t，x）\ right] dt ＆\ qquad \ \ +k_ {11} u_ {1}（t，x）db^{h} _ {1}（t）（t） +k_ {12} u_ {1}（t，x，x）db^{h} du_ {2}（t，x）＆=＆\ left [δ_αU_{2}（t，x）+γ_{2} u_ {2}（t，t，x）+u^{1+β_{2}}} _ {1} _ {1}（t，x）\ right] dt ＆\ qquad \ \ +k_ {21} u_ {2}（t，x）db^{h} _ {1} _ {1}（t） +k_ {22} u_ {2} u_ {2}（t，x）db^{h} \ end {eqnarray*} 对于$ x \ in \ mathbb {r}^{d}，\ t \ geq 0 $，以及 \ begin {equation*} \ begin {array} {ll} u_ {i}（0，x）= f_ {i}（x），＆x \ in \ mathbb {r}^{d}，\ nonumber \ end {array} \ end {equation*} where $Δ_α$ is the fractional power $-(-Δ)^{\fracα{2}}$ of the Laplacian, $0<α\leq 2$ and $β_{i}>0,\ γ_{i}>0$ and $k_{ij}\geq 0, i,j=1,2$ are constants.我们为存在全球弱解决方案提供了足够的条件。假设$β_{1} \ geqβ_{2}> 0 $带有hurst索引$ 1/2 \ leq h <1，$，我们根据棕色运动的指数功能的整体表示，获得了相关的随机部分微分方程系统的爆炸时间。此外，我们为上述SPDES系统的有限时间爆炸提供了下限和上限，并获得了上限，以使我们所考虑的系统非爆炸解决方案的概率。
In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction-diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by \begin{eqnarray*} du_{1}(t,x)&=&\left[ Δ_αu_{1}(t,x)+γ_{1}u_{1}(t,x)+u^{1+β_{1}}_{2}(t,x) \right]dt &\qquad \ \ +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t), du_{2}(t,x)&=&\left[ Δ_αu_{2}(t,x)+γ_{2}u_{2}(t,x)+u^{1+β_{2}}_{1}(t,x) \right]dt &\qquad \ \ +k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t), \end{eqnarray*} for $x \in \mathbb{R}^{d},\ t \geq 0$, along with \begin{equation*} \begin{array}{ll} u_{i}(0,x)=f_{i}(x), &x \in \mathbb{R}^{d}, \nonumber \end{array} \end{equation*} where $Δ_α$ is the fractional power $-(-Δ)^{\fracα{2}}$ of the Laplacian, $0<α\leq 2$ and $β_{i}>0,\ γ_{i}>0$ and $k_{ij}\geq 0, i,j=1,2$ are constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that $β_{1}\geq β_{2}>0$ with Hurst index $ 1/2 \leq H < 1,$ we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solutions to our considered system.
