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Finite-dimensional attractors for reaction-diffusion equations in \(\mathbb{R}^n\) with a strong nonlinearity. (English) Zbl 0959.35025
The authors study the existence of absorbing sets, attractors and exponential attractors for a reaction-diffusion equation of the following type: \[ \partial_tu-\Delta u+\lambda_0u+f(u,\nabla u)+g(x)=0,\quad x\in\mathbb{R}^n,\quad \lambda_0>0.\tag{1} \] Here the basic difficulty is that the domain of definition, i.e. \(\mathbb{R}^n\), is unbounded. This calls for a number of changes in the functional frame, encountered in the bounded case. The most important is the introduction of the weighted Sobolev space \(H_{p\gamma}\) where \(p=0,1,2,\dots\) while \(\gamma\geq 0\) is a real parameter which is largely arbitrary. One sets: \[ \begin{aligned} & \|u\|^2_{0\gamma}=\int_{\mathbb{R}^n}(1+|\varepsilon x|^2)^\gamma)u(x)|^2dx^n,\\ & \|u\|^2_{p\gamma}=\sum_{|\alpha|\leq p}\|D^\alpha u\|^2_0\gamma\end{aligned} \] with \(\alpha=(\alpha_1,\dots,\alpha_s)\) a multiindes and \(\varepsilon\) an adhoc parameter, only required to be small. The nonlinearity \(f\) is subject to five conditions, three of which are as follows: \[ \begin{aligned} & |f(u,\nabla u)|\leq c|u|^\alpha (1+|\nabla u|)^\beta,\quad \alpha,\beta>0,\\ & f(u,\nabla u)u\geq 0,\quad f_u'(u,\nabla u)\geq -c_0.\end{aligned} \] In the course of the arguments various restrictions are imposed on \(\alpha\), \(\beta\). After some preliminary lemmas, providing some a priori estimates, the authors assert the existence and uniqueness of solutions in the space \(H_{1\gamma}\), \(H_{2\gamma}\) resp. (Proposition 2.1). This secures the existence of a solution semigroup \(S(t)\), \(t\geq 0\) for (1), whence concepts such as attractors ect. make sense. The authors then proceed successively to the construction of absorbing sets, of a global attractor and of an exponential attractor, and conclude the paper with further remarks on existence and uniqueness.

MSC:
35B41 Attractors
35K57 Reaction-diffusion equations
35K90 Abstract parabolic equations
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
47H20 Semigroups of nonlinear operators
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