A parabolic free boundary problem with double pinning
For each \(\epsilon\in (0,1)\) the initial value problem \[ \partial_t u_{\epsilon} -\Delta u_{\epsilon}=-\beta_{\epsilon}(u_{\epsilon})\,\,\, {\text{in}}\,\,\,(0,\infty)\times\mathbb{R}^n \] \[ u_{\epsilon}(0,\cdot)=u^0_{\epsilon}\,\,\,\text{in }\mathbb{R}^n \] is considered with \(\beta_{\epsilon}(z)=\epsilon^{-1}\beta(z/\epsilon)\) and \(\beta\in C'_0([a,b])\) having the following property: it possesses only a finite number of zeroes in \((a,b)\) and not more than one in which it changes from negative to positive. In addition it is assumed that for each \(\epsilon\) the difference \((u^0_{\epsilon}-\epsilon a)\) is bounded in \(C^{(0,1)}(\mathbb{R}^M)\) with uniformly bounded support, and it has a limit \(u^0\leftarrow H^{1,2}(\mathbb{R}^n)\) as \(\epsilon\rightarrow 0\). The author proves that (i) the set of the solutions \(u_{\epsilon}\) is compact in the sense that \(u_{\epsilon}-\epsilon a\) is bounded in \(L^{\infty}[(0,\infty);H^{1,2}_{loc}(\mathbb{R}^n)]\) and \(\partial_t u_{\epsilon}\) is bounded in \(L^2_{loc}([0,\infty)\times\mathbb{R}^n)\) and that \(u_{\epsilon}\) is also bounded in \(C^{1/2}([0,T]\times\mathbb{R}^n)\) for all bounded \(T\), (ii) each limit of \(u_{\epsilon}\) is (in a suitable sense) a solution of the free boundary problem \[ \partial_t u-\Delta u=0\,\,\, \text{in}\,\,\,(\{u>0\}\cup\{u<0\})\cap ((0,\infty)\times\mathbb{R}^n), \] \[ | \nabla u^+| ^2-| \nabla u^-| ^2=2\int^b_a \beta (s)\,ds \] on the interface.
scientific article; zbMATH DE number 2102563
scientific article