Existence and attractors of solutions for nonlinear parabolic systems (Q2781080)

The authors consider the following doubly nonlinear system of two parabolic equations in a bounded domain \(\Omega\subset\mathbb R^N\): NEWLINE\[NEWLINE\frac{\partial}{\partial t}b_1(u_1)-\Delta_x u_1+f_1(x,u_1,u_2)=0,\quad \frac{\partial}{\partial t}b_2(u_2)-\Delta_x u_2+f_1(x,u_1,u_2)=0, NEWLINE\]NEWLINE endowed by Dirichlet boundary conditions and corresponding initial conditions. Under some assumptions on the nonlinear functions \(b_i\) and \(f_i\) (including the smoothness, nondegeneracy and sublinear growth at infinity of the functions \(b_i\)), the existence and uniqueness of a solution in the corresponding functional class is verified. Moreover, it is also proved that the semigroup generated by this equation in the appropriate phase space possesses a global attractor which has finite Hausdorff and fractal dimensions.