## Singular limit of a fourth-order problem arising in the microphase separation of diblock copolymers.(English)Zbl 1010.35003

The author deals with a model for microphase separation of diblock copolymers and considers $\begin{cases} u^\varepsilon_t=\Delta w_\varepsilon \quad & \text{in }\Omega \times(0,T),\\ w^\varepsilon= -\left(\varepsilon \Delta u^\varepsilon+ {1\over\varepsilon} f(u^\varepsilon)- v^\varepsilon\right) \quad & \text{in }\Omega\times (0,T),\\ -\Delta v^\varepsilon= u^\varepsilon-\int -_\omega u^\varepsilon dx\quad & \text{in }\Omega\times (0,T),\\ \int -_\Omega v^\varepsilon dx=0\quad & \text{for }t\in(0,T),\\ {\partial u^\varepsilon \over \partial n}= {\partial v^\varepsilon \over\partial n}= {\partial w^\varepsilon \over \partial n}=0 \quad & \text{in }\partial \Omega \times(0,T),\\ u^\varepsilon (x,0)=u^\varepsilon_0 (x)\quad & \text{for }x\in \Omega.\end{cases}$ The author considers the case of spherical symmetry, and shows the convergence to a free-boundary Hele-Shaw type problem.

### MSC:

 35B25 Singular perturbations in context of PDEs 35R35 Free boundary problems for PDEs 35K30 Initial value problems for higher-order parabolic equations

### Keywords:

spherical symmetry; Hele-Shaw type problem