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Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. (English) Zbl 1147.35121

Summary: We study a multidimensional moving boundary problem modeling the growth of tumor cord. This problem contains two coupled elliptic equations defined in a bounded domain in \(\mathbb R^2\) whose boundary consists of two disjoint closed curves, one fixed and the other moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation. By using the functional analysis method based on applications of the theory of analytic semigroups, we prove that (1) this problem is locally well-posed in Hölder spaces, (2) it has a unique radially symmetric stationary solution, and (3) this radially symmetric stationary solution is asymptotically stable for arbitrary sufficiently small perturbations in these Hölder spaces.

MSC:

35R35 Free boundary problems for PDEs
35B35 Stability in context of PDEs
76D27 Other free boundary flows; Hele-Shaw flows
76Z05 Physiological flows
92C35 Physiological flow
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