The main part of the paper deals with local existence and global existence versus blow-up for solutions of the Laplace equation in bounded domains with a non-linear dynamical boundary condition. More precisely, we study the problem consisting in: (1) the Laplace equation in $(0, \infty) \times \Omega$; (2) a homogeneous Dirichlet condition $(0, \infty) \times \Gamma_0$; (3) the dynamical boundary condition $ \frac {\partial u}{\partial \nu} = - |u_t|^{m-2} u_t + |u|^{p - 2} u$ on $(0, \infty) \times \Gamma_1$; (4) the initial condition $u(0, x) = u_0 (x)$ on $\partial \Omega$. Here $\Omega$ is a regular and bounded domain in $\mathbb{R}^n$, with $n \ge 1$, and $\Gamma_0$ and $\Gamma_1$ endow a measurable partition of $\partial \Omega$. Moreover, $m>1$, $2 \le p < r$, where $r = 2 (n - 1) / (n - 2)$ when $n \ge 3$, $r = \infty$ when $n = 1,2$, and $u_0 \in H^{1/2} (\partial \Omega)$, $u_0 = 0$ on $\Gamma_0$.

The final part of the paper deals with a refinement of a global non-existence result by Levine, Park and Serrin, which is applied to the previous problem.

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian \[ \left\{ \begin{array}{@{}r@{\,}c@{\,}l@{\qquad}l} u_{t}+( -\Delta ) ^{{\alpha}/{2}}u+u^{1+\sigma } &=&0, & x\in {\mathbf{R}}^{n},\text{ }t>0, \\[4pt] u( 0,x) &=& u_{0} ( x), &x\in {\mathbf{R}}^{n}, \end{array} \right. \label{A} \] where $\alpha \in ( 0,2)$, with critical $\sigma ={\alpha }/{ n}$ and sub-critical $\sigma \in ( 0,{\alpha }/{n}) $ powers of the nonlinearity. Let $u_{0}\,{\in}\, \mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }\cap \mathbf{C}, $$u_{0}( x) \,{\geq}\, 0$ in $\mathbf{R}^{n},$$\theta \,{=}\,\int_{ \mathbf{R}^{n}}u_{0}( x) \,dx\,{>}\,0.$ The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution $u \in \mathbf{C}( [0,\infty); \mathbf{L}^{\infty }\cap \mathbf{L}^{1,a}\cap \mathbf{C})$ and the large time asymptotics are obtained.