## 1 Introduction  
Much interest has been shown in the solution of the Allen-Cahn (AC) equation, the primary purpose of which has been depiction of the motion of antiphase boundaries in crystalline solids and phase separation in binary alloys [1]. This equation is now utilized in many moving boundary issues involving thermodynamic driving forces in microstructure evolution, domain evolution in thin films, image processing, fluid dynamics, and materials science through a phase-field approach [2; 3; 4; 5; 6; 7]. The Allen-Cahn equation has been employed to model diverse phenomena in nature, for example, [PERSON] et al. [8] proposed a method of pattern recovery (image segmentation) based on solution of the AC equation. The method is often realized as a regularization of the level-set motion by mean curvature where a particular forcing term is added that allows the initial level set to nearly encompass the pattern in question. This demonstrates convergence of the numerical approach and display function of the method for several artificial and real instances. [PERSON] and [PERSON] [9] constructed some useful a priori error estimates for proposed numerical schemes. In addition, the optimal order and quasi-optimal order error bounds were displayed for semi-discrete and fully discrete schemes under different constraints of mesh and time step size and with different regularity assumptions about the initial datum function. Later, [PERSON] et al. [10] identified three stages of temporal evolution for AC equations: the first corresponds to interfacial genesis, which happens very fast; the second is interfacial motion controlled by diffusion and the local energy difference across the interface; the time scale of the last stage was longer and here curvature effects were pivotal. After this, [PERSON] et al. [11] employed the mathematical formulation of tumor hypoxia-targeting by presenting the decay parameter of oxygen in a model [12; 13]. For numerical calculation the conforming Q1 finite element scheme for space discretization, as well as the second-order diagonally implicit fractional step \(\theta\)-method for temporal discretization, were used. Note that the distribution of nutrients in tissues has substantial influence on tumor structure and growth rate. Later, [PERSON] [14] developed a numerical solution based on the nonpolynomial B-spline (trigonometric B-spline) collocation method for solving the AC equation. This algorithm combined the trigonometric B-spline interpolant and the \(\theta\)-weighted scheme for space and time discretization. Von Neumann stability analysis showed the proposed technique to be unconditionally stable. [PERSON] et al. [15] investigated the numerical solution of the AC equation with constant and degenerate mobility, as well as with polynomial and logarithmic energy functionals. They discretized the model equation using the symmetric interior penalty Galerkin method in space, and by the average vector field (AVF) method in time. They also showed that the energy stable AVF method as the time integrator for gradient systems, like the AC equation, satisfies the energy decreasing property for a fully discrete scheme. They found that the discrete energy decreased monotonically, phase separation and metastability phenomena could be observed, and the ripening time was detected correctly. After that, [PERSON] et al. [16] addressed the uniform bounds associated with the AC equation and its numerical discretization schemes. These uniform bounds were different from, and weaker than, the conventional energy dissipation and the maximum principle. However, they can be helpful in the analysis of the numerical approach. Moreover, fully discretized schemes on the basis of the Fourier collocation method for spatial discretization and the Strang splitting method for time discretization also preserved the uniform \(L^{2}\)-bound unconditionally. In all the references above discussions were about time-dependent AC equations and regular domains. There was no mention of time-independent AC equations and irregular domains.