The Osgood condition for stochastic partial differential equations
نویسندگان
چکیده
We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ assumed to be nonnegative continuous function. first problem on $[0,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together theorem of Bonder Groisman in (Phys. D 238 (2009) 209–215), our result shows that solution blows up finite time if only for $a>0$, \begin{equation*}\int _{a}^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,\end{equation*} which well-known Osgood condition. also consider same whole line show above sufficient nonexistence global solutions. Various other extensions are provided; we look at equations fractional Laplacian spatial colored noise $\mathbf{R}^{d}$.
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ژورنال
عنوان ژورنال: Bernoulli
سال: 2021
ISSN: ['1573-9759', '1350-7265']
DOI: https://doi.org/10.3150/20-bej1240