Longtime behavior of semilinear reaction-diffusion equations on the whole space
The paper deals with the following semilinear reaction-diffusion equation on the whole space \[ u_t - \Delta u + g(x,u) = f , \;\;(x,t) \in \mathbb{R}^3 \times (\tau, T],\qquad u(x,\tau)= u_0(x) \] where \(u= u(x,t)\), \(f=f(x,t)\). The authors prove existence, uniqueness and continuous dependence results. Moreover, they show the existence of absorbing sets and find, in the autonomous case, a universal attractor for the associated semigroup. All the stated results are also valid for systems in \(\mathbb{R}^n\), that is when \(u\), \(f\), \(g\) are \(m\)-dimensional vectors and \(x \in \mathbb{R}^n\). The authors develop a specific technique, in order to overcome the difficulty caused by the lack of compactness of the usual Sobolev embeddings in unbounded domains.
scientific article; zbMATH DE number 2035106
scientific article