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The Stefan problem for the Fisher-KPP equation. (English) Zbl 1257.35110

The authors investigate as an analog of the one-phase Stefan problem with a logistic type nonlinear source term (Fisher-KPP equation) the following free boundary problem
\[ \begin{cases} u_{t}-d\Delta u=a(x)u-b(x)u^{2}\,\,\,\,\, \text{for}\,\,\, x\in\Omega(t),\,\,\,t>0,\\ u=0,\,\,\, u_{t}=\mu|\nabla_{x}u|^{2}\,\,\,\,\text{for}\,\,\, x\in \Gamma(t),\,\,\,t>0,\\ u(0,x)=u_{0}(x)\,\,\,\text{for}\,\,\,x\in\Omega_{0},\\ \end{cases} \]
where \(\Omega(t)\subset \mathbb{R}^{N}(N\geq 2)\) is bounded by the free boundary \(\Gamma(t),\) with \(\Omega(0)=\Omega_{0},\) \(\mu\) and \(d\) are given positive constants, \(a,\) \(b\) are positive functions in \(C(\mathbb{R}^{N}),\) and \(u_{0}>0\) in \(\Omega_{0}.\) The existence and uniqueness of the weak solution result at the usage of suitable comparison arguments. It is shown that the classical Aronson-Weinberger result on the spreading speed obtained through the travelling wave solution approach is a limiting case of the considered free boundary problem.

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35K58 Semilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R35 Free boundary problems for PDEs
35J60 Nonlinear elliptic equations
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