In this paper, we analyse a family of stationary nonlinear equations with $p\& q$- Laplacian $-\Delta_p u -\Delta_q u=\lambda c(x,u)$ which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter $\lambda$ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.

In this paper, we are interested in the study of the asymptotic behavior of a generalization of the Cahn-Hilliard equation with a proliferation term and endowed with Neumann boundary conditions. Such a model has, in particular, applications in biology. We show that either the average of the local density of cells is bounded, in which case we have a global in time solution, or the solution blows up in finite time. We further prove that the relevant, from a biological point of view, solutions converge to $1$ as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

We consider a finite element space semi-discretization of the Cahn-Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming enough regularity on the solution. When the solution is less regular, we prove a convergence result in some weak topology. We also prove the stability of a fully discrete problem based on the backward Euler scheme for the time discretization. Some numerical results show the applicability of the method.

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.