论文标题
乘法噪声对非线性分数随机微分方程爆炸的积极影响
Positive effects of multiplicative noise on the explosion of nonlinear fractional stochastic differential equations
论文作者
论文摘要
对于非线性随机偏微分方程，由form \ [d_t^βU= \ left [{ - { - {{ - { - { - δ} \ right）}^s}}}}} u + e +ζ\ left（u \ right（u \ right）} dt + a sum \ sum d z sum d + sus z y z y z^sum pros \ [d_t^βu= \ left [{ - { - { - { - { - { - { - { - { - { - { - { - { {\ sum \ limits_ {j = 1}^{d - 1} {{θ_m} {σ_{m，j}}} \ left（x \ right）}}}} \ circ dw_t^{m，j} {m，j} s \ ge 1，\; \; \ frac {1} {2} <β<1，\]其中$ d_ {t}^ββ$表示caputo衍生物，$ a> 0 $是恒定的，具体取决于噪声强度，$ \ circ $代表Stratonovich-type the Stratonovich-type the Stratonovich-type senterivition senteripation，我们考虑了blig的blig-time blime blime blime。我们发现，当上述方程式中的$ζ$满足某些假设时，噪声的引入可以有效地延迟确定性微分方程的爆炸时间。我们构造中的一个关键要素是使用盖金近似值，先验估计方法证明了上述随机方程的解决方案的存在和唯一性，可以将其视为\ cite {flandoli2021delayed}中结论的分数阶阶扩展。我们还验证了时间分数凯勒 - 塞格和时间分数fisher-kpp方程的验证。
For the nonlinear stochastic partial differential equation which is driven by multiplicative noise of the form \[D_t^βu = \left[ { - {{\left( { - Δ} \right)}^s}u + ζ\left( u \right)} \right]dt + A\sum\limits_{m \in Z_0^d} {\sum\limits_{j = 1}^{d - 1} {{θ_m}{σ_{m,j}}\left( x \right)} } \circ dW_t^{m,j},\;\; s \ge 1,\;\;\frac{1}{2} < β< 1,\] where $D_{t}^β$ denotes the Caputo derivative, $A>0$ is a constant depending on the noise intensity, $\circ$ represent the Stratonovich-type stochastic differential, we consider the blow-up time of its solutions. We find that the introduction of noise can effectively delay the blow-up time of the solution to the deterministic differential equation when $ζ$ in the above equation satisfies some assumptions. A key element in our construction is using the Galerkin approximation and a priori estimates methods to prove the existence and uniqueness of the solutions to the above stochastic equations, which can be regarded as the fractional order extension of the conclusions in \cite{flandoli2021delayed}. We also verify the validation of hypotheses in the time fractional Keller-Segel and time fractional Fisher-KPP equations in 3D case.
