Global attractors for degenerate parabolic equations on unbounded domains
The authors study the global solution structure of the scalar nonlinear parabolic problem \(u_t-\Delta\phi(u)+f(u)=g(x)\); \(x\in\mathbb{R}^N\), \(t\geq 0\); \(u(.,0)=u_0\), by means of methods of nonlinear dynamics. The assumptions on \(\phi\) and \(f\) are: \(\phi\in C^1(\mathbb{R})\), \(\phi(0)=0\), \(\phi\) strictly increasing in \(\mathbb{R}\), \(f\in C^1(\mathbb{R})\), \(f(0)=0\), \(\text{sgn}(r)f(r)\geq c_1|r|- c_2\) and \(0\leq \phi'(r)\leq c_3+c_4f'(r)\) for all \(r\in\mathbb{R}\) and for some constants \(c_1\), \(c_2\), \(c_3\), \(c_4>0\). In bounded domains the existence of global attractors was investigated by A. Eden, B. Michaux, J. M. Rakotoson, and F. Simondon, compactness follows from Rellich-Kondrachov's theorem. The authors obtain \(u(x,t)\) as the (unique) semigroup solution in \(X=L^1_\rho(\mathbb{R}^N)\) for a suitable weight \(\rho(x)\) and study the associated dynamical system in \(X\). By means of a splitting technique frequently used in hyperbolic problems they write \(u=v+w\), with \(v\) asymptotically small and \(w\) having asymptotically compact trajectories, which yields the existence of a global compact attractor in \(X\). The paper encloses an appendix devoted to the discussion of the case \(f=0\).
scientific article; zbMATH DE number 933275