A new global Carleman estimate for Cahn-Hilliard type equation and its applications
In this article, the author considers a class of parabolic fourth-order problems of the type \[ \begin{alignedat}{2} \partial_t y + \partial_{xxxx} y & = f_0 + \partial_x f_1 + \partial_{xx} f_2 & \quad & \text{in } (0,1)\times (0,T),\\ y(0,t) & = 0 = y(1,t) & \quad & \text{in } (0,T),\\ \partial_x y(0,t) & = 0 = \partial_{x} y(1,t) & \quad & \text{in } (0,T),\\ y(x,0) & = y_0(x) & \quad & \text{in } (0,1). \end{alignedat} \] For this class of equations, he proves a Carleman inequality. In the case in which \[ f_0=a y, \;f_1 = by, \;f_2 = c y, \] with \(a,b,c\in L^{\infty}([0,1]\times[0,T])\), he deduces the weak unique continuation property for solutions to the problem at hand. Finally, the author derives an interior controllability result for a semilinear version of the equation.
scientific article; zbMATH DE number 6504559