Quasilinear parabolic problem with $p(x)$-Laplacian: existence, uniqueness of weak solutions and stabilization

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-{\mathrm{\Delta }}_{p(x)}u=f(x,u)$ in $(0,T)\times \mathrm{\Omega }$; $u=0$ on $(0,T)\times \mathrm{\partial }\mathrm{\Omega }$; $u(0,x)=u_0(x)$ in $\mathrm{\Omega }$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.