Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains
In this well-written paper, the existence of a global attractor for nonlinear reaction-diffusion equations in \({\mathbb R}^n\) \((n\geq 3)\) of the form \[ u_t=\Delta u-\lambda u-f(u)+f_{x_i}^i+g(x) \quad\text{in }{\mathbb R}^+\times{\mathbb R}^n \tag{*} \] with initial data \[ u(0,x)=u_0(x) \quad\text{in }{\mathbb R}^n \tag{**} \] is shown. Here, the nonlinearity \(f\) is allowed to have polynomial growth of arbitrary order \(p-1\) \((p\geq 2)\) and in the inhomogeneous term, \(f_{x_i}^i\), \(i=1,\ldots,n\), are distributional derivatives of \(f\in L^2({\mathbb R}^n)\), \(g\in L^2({\mathbb R}^n)\). In order to obtain this for the problem \((\ast)\), \((\ast\ast)\) two difficulties appear: (1) The regularity of its solutions is not sufficiently high to apply appropriate embedding theorems. (2) It is hard to get continuity of the associated semigroup in the \(L^p({\mathbb R}^n)\)-topology without restriction on \(p\). Thus, for abstract semigroups in \(L^2({\mathbb R}^n)\) the authors derive a sufficient criterion that a global attractor in \(L^2({\mathbb R}^n)\) also attracts bounded sets of \(L^2({\mathbb R}^n)\) w.r.t.~the \(L^p({\mathbb R}^n)\)-norm. Using a new method based on a priori estimates, this criterion applies to show that the semigroup in \(L^2({\mathbb R}^2)\) associated with \((\ast)\), \((\ast\ast)\) possesses a \((L^2({\mathbb R}^n),L^p({\mathbb R}^n))\)-global attractor \(A\) in the sense that \(A\) is nonempty, compact, invariant in \(L^p({\mathbb R}^n)\) and attracts every bounded subset of \(L^2({\mathbb R}^n)\) in the \(L^p({\mathbb R}^n)\)-norm.
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