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

Abstract

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $$ \left\{ \begin{array}{ll} u_t-\Delta _{p(x)}u = f(x,u)&\quad \text{in } \quad Q_T \stackrel{{\rm{def}}}{=} (0,T)\times\Omega, u = 0 & \quad\text{on} \quad \Sigma_T\stackrel{{\rm{def}}}{=} (0,T)\times\partial\Omega, u(0,x)=u_0(x)& \quad \text{in} \quad \Omega \end{array} \right. \quad\quad (P_{T}) $$ involving the p(x)-laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.