Lyapunov stability of the parabolic p-Laplace equation is investigated. The nominal equation is shown to be asymptotically stable, while the stronger property of exponential stability is guaranteed by the presence of lower-order terms satisfying a suitable growth condition. Numerical simulations are provided to support and illustrate the theoretical results.
On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator
$\partial_t-\triangle+\int_0^t K(x,t,r)\triangle\ dr$
where the integral kernel has a behaviour like a weakly singular one.
In the proof we consider the integral term as a perturbation. The crucial point is
a special choice of the time factor of the weight function.
We consider the homogenization of a parabolic problem in a perforated domain with Robin–Neumann boundary conditions oscillating in time. Such oscillations must compensate the blow up of the boundary measure of the holes. We use the technique of time-periodic unfolding in order to obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition.
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