We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$. A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$, whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$. In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.