Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911887551102976 |
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| author | Sankar, S. Mohan, Manil T. Karthikeyan, S. |
| author_facet | Sankar, S. Mohan, Manil T. Karthikeyan, S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03111 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise Sankar, S. Mohan, Manil T. Karthikeyan, S. Analysis of PDEs Probability 35R60, 60H15, 74H35 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. |
| title | Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise |
| topic | Analysis of PDEs Probability 35R60, 60H15, 74H35 |
| url | https://arxiv.org/abs/2211.03111 |