Together with Klemens Fellner, Stefanie Sonner, and Do Duc Thuan, I have uploaded a paper on arXiv, which deals with stabilising effect of noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions.
Certain dissipative PDEs have either stable or unstable steady states depending on the magnitude of parameters. Consider for example the heat equation in a domain with Dirichlet boundary condition
Multiplying the equation by in and using the Poincare inequality we get
Here is the first eigenvalue of Laplace operator with homogeneous Dirichlet boundary condition.
By Gronwall’s inequality, one has
which means that if , then the zero steady state is exponentially stable.
When , this stability might be lost. (In fact, one can prove that if , then the zero steady state is unstable). The stability can be regained when, surprisingly, some multiplicative noise is added. More precisely, consider the stochastic equation
where is a real-valued scalar Wiener process defined in an appropriate probability space, and denotes the Ito differential. It was proved that the zero steady state is exponentially stable provided
- When this condition is obviously satisfied, therefore the stability of zero steady state is preserved with the addition of noise.
- When , the zero steady state is exponentially stable for all .
Roughly speaking, the noise shifts all eigenvalues to the right with the distance , thus helps to stabilise the equation as soon as the first eigenvalue (which is now ) is larger than . Remark that the larger is, the better the zero steady state can be stabilised. In particular, one can stabilise the equation with an infinite range of noise intensities.
This phenomenon of stabilisation by noise has been studied extensively in the last decades. However, it seems not known if one can also stabilise the equation using the noise only on the boundary. Our paper fills in this gap in the context of a nonlinear Chafee-Infante equation with dynamical boundary conditions. More precisely,
Our analysis uses in a crucial way the Poincare-Trace inequality: for each there exists such that
Here the dependence of on is of importance. Note that even in the limit , the constant remains bounded, for instance (by choosing . An optimal expression of is, up to our knowledge, unknown.
Using this Poincare-Trace inequality, we show that one can stabilise the equation with a finite range of noise intensities . This differs strongly from the case described above. It is an interesting open problem to decide whether this finite range is due to the boundary noise or merely a technical limitation. However, due to the upper bound of , we conjecture it to be the latter case.