We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller & Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

We consider a class of non-quasi-convex frame indifferent energy densities that includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem, we provide an explicit minimizer of the energy functional satisfying a non-trivial boundary condition. Other attainment results, both for the nonlinear and the linearised model, are obtained by using the theory of convex integration introduced by Müller and Šverák in the context of crystalline solids.