We study the asymptotic behaviour of the solution of nonlinear monotone elliptic problem \begin{equation*} -{\operatorname{div}}[a(x,Du_l)] = \mu \mbox{ on } \ ( -l,l)^q \times {\omega} \end{equation*} with homogeneous Cauchy-Dirichlet boundary conditions, where ${\omega}$ is a bounded, open, connected subset of $\mathbb R^{N-q}$ with second member in $L^{1}{\omega} +W^{-1,p'}(\omega)$, using the framework of renormalized solutions. Assuming specific dependence of the operator $a$ with respect to the variable $(x_1,x_2) \in ( -l,l ) ^q \times {\omega}$ and that $\mu=\mu(x_2)$, we show the convergence of $u_l$, in an appropriate sense, toward the solution of the same problem posed in $\omega$.

We consider the initial-value problem of abstract wave equations with weak dissipation. We show that under conditions on the dissipation coefficient and its derivative the solutions to the abstract dissipative equation are closely related to solutions of the free problem multiplied by a decay function. This paper gives the counterpart to a recent paper of T. Yamazaki [Adv. Differential Equ., 11(4):419--456, 2006], where effective dissipation terms and the relation to the corresponding abstract parabolic problem are considered.

We consider the Cauchy-Dirichlet problem of the heat equation in the exterior domain of a ball, and study the movement of hot spots $H(t)$ as $t\to\infty$. In particular, we give a rate for the hot spots to run away from the boundary of the domain as $t\to\infty$. Furthermore we give a sufficient condition for the hot spots to consist of only one point after a finite time.

We study the nonstationary boundary-value problem for the Poisson equation in a plane corner with a dynamic boundary condition on the one part of the corner and the Dirichlet condition on the other part. We prove one-to-one solvability of the problem in weighted Hölder spaces and obtain the corresponding coercive estimates. These estimates will be useful to solve a free boundary problem.