**Eurica Henriques**

(CMAT, UTAD)

*ANAP group seminar*

In this seminar we revisit several previous works concerning anisotropic differential equations (i.e. evolution equations in which the diffusion takes a different form within different space directions) whose prototypes are

$$ u_t- \mathrm {div} \left(u^{\gamma(x,t)} Du \right)=0 \ , \quad \gamma(x,t)>0 \ ;$$

$$u_t-\sum_{i=1}^{N}\left(u^{m_i}\right)_{x_i x_i}=0 \ , \quad m_i>0 \ ; $$

$$u_t-\sum_{i=1}^{N}\left(|u_{x_i}|^{p_i-2} u_{x_i}\right)_{x_i}=0 \ , \quad p_i>2 \ ,$$

given in $\Omega_T=\Omega \times (0,T]$, where is a bounded domain in ${\mathbb R}^{N+1}$ and $0<T<\infty$, for nonnegative bounded functions $u$.

Very recently, we started to work with the parabolic (singular) anisotropic $p_i$-Laplacian

$$u_t-\sum_{i=1}^{N}\left(|u_{x_i}|^{p_i-2} u_{x_i}\right)_{x_i}=0 \ , \quad 1<p_i<2 \ .$$

We'll discuss the topics we are addressing in this ongoing joint work, still at its early stage.

(joint work with S. Ciani)