Attractors for parabolic problems with nonlinear boundary conditions
The authors investigate the long time behaviour of solutions of reaction diffusion systems of the following structure: \[ u_t-\text{div}(a(x)\nabla u)+ \sum B_j(x)\partial_ju+ \lambda u+ f(u)=0\text{ in }\Omega,\quad {\partial u\over\partial n_a}+ g(u)=0\text{ on }\partial\Omega.\tag{1} \] Here \(u=(u_1,\dots,u_m)\), \(m\geq 1\), \(a(x)=\text{diag}(a_1(x),\dots,a_m(x))\), \(a_i\in C^1(\overline\Omega)\), \(a_i(x)\geq m_0>0\). Moreover, \(\partial u/\partial n_a=\langle a\nabla u,n\rangle\) with \(n\) the outward unit normal to \(\partial\Omega\) and \(B_j=\text{diag}(b^1_j,\dots,b^m_j)\) continuous on \(\overline\Omega\) \((j=1,\dots,N)\). Moreover, \(\Omega\subset\mathbb{R}^N\) is a smooth, bounded domain, \(\lambda\in\mathbb{R}\), and \(f=(f_1,\dots,f_m)\), \(g=(g_1,\dots,g_m)\) are nonlinearities subject to suitable growth conditions, stated in the course of the arguments. In case \(N=3\), \(f\), \(g\) have to satisfy among others \[ |f(u)- f(v)|\leq L(1+|u|^2+ |v|^2)|u-v|,\;|g(u)- g(v)|\leq L(1+|u|^{1+p}+ |v|^{1+p})|u-v|,\;u,v\in\mathbb{R}^m, \] supplied by a similar condition for the derivative \(Dg(u)\); here \(p\in [0,1]\). In addition, certain sign conditions are imposed on \(f,g\) and their introduction justified. In order to handle equation (1), i.e. to take care of the nonlinear boundary conditions, the authors treat (1) not as an equation on \(L^2(\Omega)\) but, after suitable reinterpretation, on \(H^{-1}(\Omega)\), what allows them to subsume the nonlinear boundary conditions into an abstract parabolic evolution equation of type: \[ u_t+ Au+ h(u)=0, \qquad u(0)= u_0\in X^\alpha.\tag{2} \] Here \(X\) is a Hilbert space, \(A\) a sectorial operator and \(h(\cdot)\) a mapping from the fractional power space \(X^\alpha\) to \(X^\beta\) which is assumed to be locally Lipschitz; thereby \(0\leq\alpha-\beta< 1\). Section 3 contains an extended discussion of the conditions which have to be satisfied so that system (1) can be cast into the form (2). The advantage of this is that local existence and uniqueness results are available for (2). While this highly technical digression takes place in Section 3, Section 4 is devoted to the proof that (1), under the stated conditions, admits a global attractor. In addition, it is shown that this result can be obtained for other functional settings and assumptions. A series of further results in this direction is obtained in Section 5.
scientific article; zbMATH DE number 1005688
scientific article