| Abstract: |
| In this talk, we will present some recent results concerning on the decay rate of solutions to the following Cauchy-Dirichlet problem
\begin{equation}\label{PCD}\tag{$P$}
\left\{
\begin{aligned}
&u_t=\sum_{i=1}^N(u^{m_i})_{x_ix_i}& \quad &\text{in }\Omega\times (0,T), \
&u(x,0)=u_0(x)& \quad & \text{on }\Omega, \
&u(x,t)=0 &\quad & \text{on }\partial \Omega\times (0,T),
\end{aligned}
\right.
\end{equation}
where $\Omega$ is a bounded open set of $\mathbb{R}^N$ with smooth boundary, $N\geq 2$,$00$, for all $i=1, \dots, N$, and $0\leq u_0\in L^{r_0}(\Omega)$ with $r_0\geq 1$.
Our approach relies on showing that the solution $u$ satisfies some suitable integral inequalities, which allow us to derive quantitative estimates for the decay rate of solutions as $t \rightarrow+\infty$. We distinguish different regimes depending, in particular, on the relation between 1 and the minimum or the maximum of the exponents $m_i$. Moreover, by an appropriate choice of parameters in these inequalities, we prove that finite-time extinction occurs in certain cases.
These results are based on recent joint work with A. G. Grimaldi and M. M. Porzio. |
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