On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem
The authors start by considering the reaction-diffusion system \[ \partial_t u-\nabla \cdot (a(\theta)\nabla u)=f\quad\text{ in }\{u > 0\} \] \[ \tau \partial_t \theta - \sigma \Delta \theta +\theta =\chi_{\{u>0\}}\quad\text{ in }Q=\Omega \times (0,T) \] with \(\Omega\) a bounded domain in \(\mathbb R^n\), where, to fix ideas, \(u\) is a concentration and \(\theta\) is temperature. The diffusivity \(a(\theta)\) is a strictly positive continuous function. The unknown \(u(t)\) is sought in the convex set \(K=\{v\in H^1_0(\Omega): v>0\) in \(\Omega\}\) and \(\theta\) satisfies a homogeneous Neumann condition. The initial data \(u_0,\theta_0\) are such that \(u_0\in K\), \(\theta_0\in W^{2,\infty}(\Omega)\) and \(\partial_{\nu}\theta_0=0\quad\text{ on } \partial \Omega\). Under the assumptions \(\partial \Omega \in C^2\), \(f \in L^2(Q), a(\theta)\) Lipschitz, \(f\neq 0\) a.e. in \(Q\), existence is proved for any pair \((\tau, \sigma)\) of positive constants, with the property \(u_{\tau \sigma}\in W^{2,1}_2 (Q)\), \(\theta_{\tau \sigma}\in W^{2,1}_q(Q)\), \(\nabla q<\infty\). The problem is formulated in a variational form and a fixed point argument is used. Next the authors investigate the limit \(\sigma \rightarrow +\infty\) (Problem \(P_{\tau}\)) and the further limit \(\tau \rightarrow 0\). In \(P_{\tau}\) the equation for the temperature is replaced by \[ \tau \dfrac{d\xi}{d\tau}+\xi =\langle u^+ \rangle, \] \(\langle u^+\rangle\) representing the volume fraction of the set \(\{u>0\}\). A solution \(u_\tau \in W^{2,1}_2 (Q)\), \(\xi_\tau \in W^{1,\infty}(0,T)\) is obtained as an appropriate limit of \(u_{\tau \sigma}\theta _{\tau \sigma}\). A discussion is performed about the assumption \(f\neq 0\) in \(Q\) and on how to reformulate the problem in its absence, still preserving existence. Finally, letting \(\tau \rightarrow 0\), basically \(\xi\) is replaced by \(\langle u^+\rangle\) and Problem P consists in just one variational inequality, which again is shown to have a solution \(U\in W^{2,1}_2 (Q)\), if \(f\neq 0\) a.e. in \(\Omega\), obtained as the limit \(u_\tau \rightarrow u\), \(\xi_\tau \rightarrow \langle u^+\rangle\) in appropriate spaces. The question of uniqueness is left open.
scientific article; zbMATH DE number 2066963