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

$$left{begin{array}{ll}u_t-Delta _{p(x)}u = f(x,u)&quad text{in }quad Q_T stackrel{{rm{def}}}{=} (0,T)timesOmega,u = 0 & quadtext{on}quad Sigma_Tstackrel{{rm{def}}}{=} (0,T)timespartialOmega,u(0,x)=u_0(x)& quad text{in}quad Omega end{array}right.quadquad (P_{T})$$

involving the p(x)-laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.