On the behaviour of stochastic heat equations on bounded domains
On the behaviour of stochastic heat equations on bounded domains
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发表时间：
2014-12
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通讯作者：
Mohammud Foondun;E. Nualart
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作者：
Mohammud Foondun;E. Nualart
考虑以下区间上的方程 $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$。在狄利克雷边界条件下，我们表明，从长远来看，如果 $\lambda$ 足够大，解的二阶矩会呈指数增长。但如果 $\lambda$ 很小，那么二阶矩最终会呈指数衰减。如果我们用诺依曼边界条件代替狄利克雷边界条件，那么无论 $\lambda$ 是多少，二阶矩都会以指数方式快速增长。我们还提供各种扩展。
Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $\lambda$ is large enough. But if $\lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $\lambda$ is. We also provide various extensions.