Well-posedness and attractor on the 2D Kirchhoff-Boussinesq models

The paper studies the well-posedness and the existence of attractors for a class of 2D Kirchhoff-Boussinesq models: u(tt) + ku(t) +Delta(2)u = gamma div{del u/root 1+vertical bar del u vertical bar(2)} + beta Delta g(u), with beta >= 0, gamma >= 0, beta + gamma > 0. We show that: (i) the IBVP of the equations is well-posed in natural energy space X-2 and strong solution space X-4, respectively, provided that vertical bar g ''(s)vertical bar <= C(1 + vertical bar s vertical bar(2)); (ii) the related solution semigroup has a global and an (generalized) exponential attractor in X-2 provided that the damping parameter k is suitably large and vertical bar g ''(s)vertical bar <= C; (iii) in particular when gamma = 0, the corresponding Boussinesq model has a subclass J of limit solutions and the subclass J has a weak global attractor in energy space X-1 without any upper bound restriction for the growth exponent of g(u); (iv) in the cases that either beta = 0 or gamma = 0, the corresponding model has a global attractor in X-4 provided that vertical bar g ''(s)vertical bar <= C(1 + vertical bar s vertical bar) and without any restriction for the damping parameter k > 0. Especially when gamma = 0, the corresponding results extend those in Grassell et al. (2009). (C) 2020 Elsevier Ltd. All rights reserved.