Attractors of reaction-diffusion equations in Banach spaces (Q2756147)

This paper deals with the existence of global attractors for differential inclusions of the type NEWLINE\[NEWLINE\begin{cases} x'(t)\in Ax(t),\\ x(0)= x_0,\end{cases}NEWLINE\]NEWLINE where \(A\) is a multivalued \(\omega\)-dissipative operator. The author applies the obtained results for proving the existence of a global attractor for the reaction-diffusion equation NEWLINE\[NEWLINE\begin{cases} {\partial u\over\partial t}- \Delta u+ f(u)\ni\omega u+h,\quad &\text{on }(0,T)\times\Omega,\\ u=0,\quad &\text{on }(0,T)\times \Gamma,\\ u(0,x)= u_0(x)\in L_p(\Omega),\end{cases}NEWLINE\]NEWLINE where \(f: \mathbb{R}\to 2^{\mathbb{R}}\) is a multivalued maximal monotone map and \(2\leq p<\infty\). Note that the author allows to be discontinuity for \(f\). In the case \(p=2\) he obtains some estimates of the fractal dimensions of the global attractor.