This paper studies the large time behavior of nonnegative solutions of $$ u_t = u^p \Delta u - u^{-q} \chi_{\{u>0\}} \qquad \mbox{in } \Omega \times (0,\infty), \qquad p \ge 1, q>-1, $$ with prescribed boundary values $u|_{{\partial\Omega}}=1$ in smoothly bounded domains $\Omega \subset \mathbb R^n$. The particular interest here is in the question of how the interplay of strongly degenerate diffusion and strong absorption influences the evolution of the dead core set $\{u(t)=0\}$ (which is enforced to be nontrivial for all time by the assumption that the initial data $u_0$ vanish in some ball). The main results state that there exist four parameter regimes in which, widely independent of $u_0$, all solutions undergo

(i)a complete extinction at time $t=0$ ($q>p-1$)

(ii) a total extinction in finite but positive time ($0 <q\le p-1$),

(iii) an extinction in infinite time ($1-p <q <0$), or

(iv) no extinction ($q <1-p$),

respectively. Specifically, the first two types of behavior are in sharp contrast to the non-degenerate and weakly degenerate cases $p=0$ and $p \in (0,1)$, respectively, where only (iv) occurs.

We study the instantaneous support shrinking phenomenon for a doubly nonlinear parabolic equation in the fast diffusion case. The initial data of the Cauchy problem are locally finite Radon measures. We obtain for nonnegative solutions necessary and sufficient condition for instantaneous support shrinking phenomenon in terms of local behavior of the initial data. In the same terms we express sharp with respect to rate bilateral estimates for the size of the support.

In this paper we consider a multi-dimensional damped semilinear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem. Secondly, the exponential growth of the energy and the $L^p$ norm of the solution is presented.

We consider an equation of the type $A(u+k*u)=f$, where $A$ is a linear second-order elliptic operator, $k$ is a scalar function depending on time only and $k*u$ denotes the standard time convolution of functions defined on ${{\bf R}}$ with their supports in $[0,T]$. The previous equation is endowed with dynamical boundary conditions. Assuming that the kernel $k$ is unknown and information is given, under suitable additional conditions $k$ can be recovered and global existence, uniqueness and continuous dependence results can be shown.

The aim of this paper is to study a nonlinear scalar field equation on a surface $\Sigma$ via a Morse-theoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], which used refined blow-up estimates from [34] and [17]. Related results are derived for the prescribed $Q$-curvature equation on four manifolds.

We find the rate of convergence of solutions of a semilinear parabolic equation to selfsimilar solutions of the linear heat equation. In particular, we show that the rate is not affected by the nonlinearity for some range of parameters, while in a complementary range the rate depends explicitly on the nonlinearity.

The paper is devoted to the study of a multi-dimensional semi-linear elliptic system of equations in an unbounded cylinder with a linear dependence of the components of the non-linearity vector. Problems of this type describe reaction-diffusion waves with the Lewis number different from $1$. Due to this property of non-linearity, the corresponding operator does not possess the Fredholm property. Therefore the usual solvability conditions and the conventional methods of non-linear analysis cannot be directly applied. We reduce the elliptic problem to an integro-differential system of equations and show how to apply the implicit function theorem to it. It allows us to prove existence of waves for the Lewis number different from $1$ and sufficiently close to it. Next we prove the Fredholm property of integro-differential operators, their properness, and construct the topological degree. The latter is used to study bifurcations of solutions.