Weak solution to hyperbolic Stefan problems (Q676741)

Let \(\Omega \subset \mathbb{R}^N\) be a bounded domain and at \(H= L^2(\Omega)\). Let \(V\) be a subspace of \(H^1 (\Omega)\) containing \(H_0^1 (\Omega)\). The scalar product \(\langle\cdot,\cdot\rangle\) between elements of \(V'\) and \(V\) can be defined as \(\langle u,v \rangle =(u,v)_H\). The problem considered in this paper is the following: Look for a pair \((u,\chi)\) such that \(u\in C^0([0,T];V) \cap C^1([0,T];H)\), \(\chi\in L^\infty (\Omega\times (0,T))\), \[ {d\over dt} \langle(u_t +\psi*\chi)) (t),v\rangle +\int_\Omega \nabla (k*u_t)(t) \cdot\nabla v =\langle f(t),v\rangle,\;\forall v\in V\text{ in } {\mathcal D}'(0,T), \] with \(u(0)=0\), \(u_t(0)= \theta_0\). Here \(\psi,k\) are the so-called memory kernels (which may depend on space), the symbol * denotes the normal time convolution. The assumptions on the data are \(\psi\in W^{2,1} (0,T)\), \(\psi(0)>0\), \(k\in W^{2,1} (0,T)\), \(k(0)>0\), \(f\in L^1(0,T;H) +W^{1,1} (0,T;V')\), \(\theta_0\in H\). The physical meaning of the scheme above is a phase change model in which \(\chi\) is the phase fraction and \(u_t\) is the temperature (i.e. \(u\) is the so-called freezing index). The choice of the space \(V\) determines the boundary conditions. In a previous paper P. Colli and M. Grasselli have proved uniqueness under the assumptions listed above, while the existence theorem required more regularity on the data. In the present paper existence is shown under the same minimal assumptions needed for uniqueness. The proof is achieved by means of compactness arguments: the data \(f\), \(\theta_0\) are approximated by sequences \(\{f_0^\varepsilon\}\), \(\{\theta_0^\varepsilon\}\), converging in the approximate way, and suitable estimates are obtained for the corresponding solutions \((u^\varepsilon, \chi^\varepsilon)\) which allow to establish the strong convergence of a subsequence to a pair \((u,\chi)\) solving the given problem.