If $p>1+2/n$, then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.

We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine equation in dimension $1$, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. More precisely, the dimensionless equations depend strongly on three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations, respectively. For the Boussinesq-Peregrine model, one has small amplitude variations ($\epsilon = O(\mu)$). We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq ([9], [8]) and Peregrine ([22]) in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a local existence result on a time interval of size $\frac{1}{\epsilon}$ in dimension $1$ for this new equation, without any assumption on the smallness of the bathymetry $\beta$, which is an improvement of the long time existence result for the Boussinesq systems in the case of flat bottom ($\beta=0$) by [24].

We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups.

We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper, we treat center, center-stable and center-unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the two-phase Stefan problem with surface tension.