Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions in a suitable Banach space. The second one considers well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the potential and $\pi\textrm{–Camassa–Holm}$; generalised Boussinesq equations; and the modified Euler–Poisson system.
We study an initial-boundary-value problem for three-dimensional nonhomogeneous magneto-micropolar fluid equations without angular viscosity. Using linearization and Banach’s fixed point theorem, we prove the local existence and uniqueness of strong solutions. Moreover, a regularity criterion is also obtained.
We consider the nonlinear Schrödinger equation with $L^2$-supercritical and $H^1$-subcritical power type nonlinearity. Duyckaerts and Roudenko [8] and Campos, Farah, and Roudenko [3] studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.
Based on the recent surprising work on the symmetry breaking phenomenon of the Allen–Cahn equation [11, 12], we consider the one-dimensional parabolic sine‑Gordon equation with periodic boundary conditions. Particularly, we derive a strong dependence of the non-trivial steady states on the diffusion coefficient $\kappa$ and provide some description on them for $0 \lt \kappa \lt 1$. To further investigate the property of energy associated to the steady states, we give a complete classification and prove the monotonicity of the ground state energy with respect to the diffusion constant $\kappa$. Finally, we identify the exact decay rate of the solution to the parabolic equation together with the explicit leading term for $\kappa \geq 1$.
We consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the “interior” wave equation on the “boundary” wave equation and show that it generates a structural damping.