Cauchy problem with subcritical nonlinearity (Q1363589)

The authors consider the asymptotic behaviour of solutions of parabolic systems \[ u_t=- Au+f(u_1,\dots,u_d,\dots, \partial^\alpha_iu_j,\dots),\;i\leq n,\;j\leq d,\;(x,t)\in\mathbb{R}^n\times \mathbb{R}_+,\tag{1} \] with \(u(0,x)= u_0(x)\). In (1), \(A\) is a linear matrix operator of uniformly elliptic type of order \(2m\) while the order \(\alpha\) of the derivatives in (1) does not exceed \(2m-k\) for some fixed \(k\leq 2m-1\). Finally, \(f=(f_1,\dots,f_d)\) is a nonlinearity whose components \(f_i\) are locally Lipschitz with respect to each argument separately and such that \(f(0)= 0\). System (1) is cast into an abstract functional analytic frame as follows. With \(d\), \(k\), \(m\), \(n\) as above a \(p\) such that \(2m-k> np^{-1}\) is fixed; \(-A\) restricted to \(\text{dom}(A)= W^{2m,p}(\mathbb{R}^n)^d\) then becomes the generator of a holomorphic semigroup with spectrum lying strictly in the left halfplane. By these stipulations one considers (1) as an abstract evolution equation on \(X= L^p(\mathbb{R}^n)^d\): \[ u_t= Au+F(u),\quad t\in\mathbb{R}_+,\quad u(0),\quad u\in X.\tag{2} \] By established theory it holds that if \(\alpha\in(0,1)\) satisfies \(2m\alpha- k>np^{-1}\) then \(F: D(A^\alpha)\), \(|\;|_\alpha\to X\) is Lipschitz continuous on bounded subsets of \(D(A^\alpha)\), \(|\;|_\alpha\); here \(F\) is induced by \(f\) in the obvious way. The authors impose two technical conditions on equation (2) which suffice to ensure that all solutions of (2) are global or, more explicitely, that orbits of bounded sets in \(D(A^\alpha)\), \(|\;|_\alpha\) remain bounded. The authors then proceed to bring attractors into play. To this effect, a further condition is needed which guarantees the existence of absorbing sets. In order to handle compactness a further condition (\(D(A^\alpha)\)-assumption) is introduced which roughly speaking guarantees that bounded sets \(B\) of \(D(A^\alpha)\), \(|\;|_\alpha\) which are precompact in \(X\) have an image \(T(t)B\) which is precompact in \(D(A^\alpha)\), \(|\;|_\alpha\); \(T(t)\) is the semiflow generated on \(D(A^\alpha)\), \(|\;|_\alpha\). A last technical assumption is needed, which allows the introduction of weighted spaces such as \(L^p(\mathbb{R}^n,(1+|x|^2)^\nu)^d\), which helps to restore compactness and ultimately leads to the existence of global attractors. In fact, the last abstract theorem asserts that under all these assumptions equation (2) has a global attractor. The paper concludes with a number of applications.