Semilinear parabolic PDEs and unbounded attractors

Abstract

Despite being a stimulating and ongoing topic of research, the theory of attractors for infinite-dimensional dynamical systems has been relied, for a long time, on a crucial existence assumption of a compact absorbing set for the trajectories. Meanwhile, dynamical systems not having bounded attractors due to the existence of unbounded, and therefore not absorbed, trajectories are also of great interest. Motivated by that, in this research project we propose to investigate and develop a general theory of unbounded attractors, both in autonomous and nonautonomous settings. This direction of research is motivated, in particular, by the study of the attractors in evolution equations when the nonlinearity is not dissipative. With that in mind, we plan to consider a scalar nonautonomous parabolic equation, with no dissipativity properties, as our toy model. Since, trajectories might become unbounded in finite or infinite time, it is inevitable not addressing blow-up phenomena for evolution equations. In this line, we aim at exploring the possible asymptotic regimes and profiles appearing on semilinear nondissipative heat equations, defined on $n$-dimensional domains. (AU)