We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation $u_t=\mathrm{div}(u^{m-1}{\left|Du\right|}^{p-2}Du)-u^q$ with an initial condition $u(x,0)=u_0\left(x\right)$. Here the exponents $m$, $p$ and $q$ satisfy $m+p\ge 3$, $p>1$ and $q>m+p-2$.

Let $\Omega$ be a bounded open subset of ${ℝ}^n$, $n>2$. In $\Omega$ we deduce the global differentiability result $$u\in H^2(\Omega ,{ℝ}^N)$$ for the solutions $u\in H^1(\Omega ,{ℝ}^n)$ of the Dirichlet problem $$u-g\in H_0^1(\Omega ,{ℝ}^N),-\sum _i{D}_i{a}^i(x,u,Du)=B_0(x,u,Du)$$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

Let $\Omega$ be a bounded open subset of ${ℝ}^n$, let $X=(x,t)$ be a point of ${ℝ}^n\times {ℝ}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result $$u\in L^2(-a,0,H^2(B\left(\sigma \right),{ℝ}^N))\cap H^1(-a,0,L^2(B\left(\sigma \right),{ℝ}^N))$$ for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega ,{ℝ}^N))\cap C^{0,\lambda }(\overline{Q},{ℝ}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system $$-\sum _{i=1}^n{D}_i{a}^i(X,u,Du)+{\displaystyle \frac{\partial u}{\partial t}}=B^0(X,u,Du)$$ with quadratic growth and nonlinearity $q\ge 2$. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1,q}\cap C^{0,\lambda }$.

We study the realization $A$ of the operator $\mathcal{A}=\frac{1}{2}\mathrm{△}-(DU,D\cdot )$ in $L^2\left(\mathrm{\Omega },\mu \right)$, where $\mathrm{\Omega }$ is a possibly unbounded convex open set in ${\mathbb{R}}^N$, $U$ is a convex unbounded function such that ${lim}_{x\to \partial \mathrm{\Omega },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$ and ${lim}_{\left|x\right|\to +\mathrm{\infty },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$, $DU\left(x\right)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu \left(dx\right)=c\exp \left(-2U\left(x\right)\right)dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D\left(A\right)=\left\{u\in H^2\left(\mathrm{\Omega },\mu \right):\left(DU,Du\right)\in L^2\left(\mathrm{\Omega },\mu \right)\right\}$ is a dissipative self-adjoint operator in $L^2\left(\mathrm{\Omega },\mu \right)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities...