On the modified of the one-dimensional Cahn-Hilliard equation with a source term

Abstract

<abstract><p>We consider the modified Cahn-Hilliard equation that govern the relative concentration $ \phi $ of one component of a binary system. This equation is characterized by the presence of the additional inertial term $ \tau_{D}\frac{d^2\phi}{dt^2} $ which stands for the relaxation of the diffusion flux. This equation is associated with Dirichlet boundary conditions. We study the existence, uniqueness and regularity of solutions in one space dimension. We also prove the existence of the global attractor and exponential attractors.</p></abstract>

Hyperbolic relaxation of the viscous Cahn–Hilliard equation with a symport term for biological applications

“…The case 𝜏 D > 0 and 𝛿 = 0 seems out of reach in dimensions 2 and 3, where the standard energy estimates fail to hold (the problems arise concerning the uniqueness result). This obstacle is bypassed in one dimension, taking advantage of the continuous embedding H 1 (Ω) → C( Ω) (see previous research [9,10,12,13]). On account of this remark, we need to require that 𝜏 D is dominated by 𝛿.…”

“…The case $$$ {\tau}_D>0 $$$ and $$$ \delta =0 $$$ seems out of reach in dimensions 2 and 3, where the standard energy estimates fail to hold (the problems arise concerning the uniqueness result). This obstacle is bypassed in one dimension, taking advantage of the continuous embedding $$$ {H}^1\left(\Omega \right)\to C\left(\overline{\Omega}\right) $$$ (see previous research [9, 10, 12, 13]). On account of this remark, we need to require that $$$ {\tau}_D $$$ is dominated by $$$ \delta $$$.…”

“…Recently, researchers have devoted tremendous attention on the relaxed (CH) system, that is, the viscous CH system and its perturbed form with a hyperbolic relaxation effect. Both systems have been well-studied theoretically in papers which are mainly focused on the well-posedness and the global attractor, exponential attractors, and so on (see previous research [9][10][11][12][13]). Formally, the governing equation of the system is slightly different from the CH equation by incorporating two extra terms, including a strong damping (called "viscosity") term and a hyperbolic relaxation term (called "inertial").…”