Attractors for a non-linear parabolic equation modelling suspension flows
The authors study the following scalar parabolic equation \[ \begin{aligned} & \partial_tp - D(p(t))\partial_{\sigma\sigma}^2p + \frac{1}{T_0}\mathbf 1_{\mathbb{R}\setminus [-1,1]}(\sigma)p = \frac{D(p(t))}{\alpha}\delta_0(\sigma),\\ & p(0,\sigma) = p_0(\sigma), \quad p \geq 0 \end{aligned} \] that describes the evolution of mesoscopic blocks in suspensions by means of finding the probability distribution density \(p(t,\sigma)\) which represents the distribution of stress in the assembly of blocks in time. Here \(\mathbf 1_{\mathbb{R}\setminus [-1,1]}\) denotes the characteristic function of the open set \(\mathbb{R}\setminus [-1,1]\), \(\sigma \in \mathbb{R}\) and \[ D(p,t) = \frac{\alpha}{T_0}\int_{|\sigma > 1}p(t,\sigma)d\sigma. \] The main aim is to prove the existence of a global attractor with respect to the weak topology of a suitable Banach space. From mathematical point of view, the above-mentioned problem presents several difficulties when studying the large time behavior. First, in view of the fact that the uniqueness theorem has not been obtained in a suitable phase space, the authors work with a multivalued semiflow rather than with a semigroup operators. Second, due to the terms \(D(p(t))\) and \(\mathbf 1_{\mathbb{R}\setminus [-1,1]}\) on the left-hand side of the equation under consideration, the dissipative mechanism of this equation is weakened, and it is not possible to obtain absorbing term. Nevertheless, the authors prove that the solutions starting at a bounded set of a suitable phase space remain uniformly bounded in time.
scientific article; zbMATH DE number 5503075