A generalized phase-field system
The authors consider the problem \((\Omega\) bounded domain in \(\mathbb{R}^N\) with \(C^2\)-boundary) \[ \begin{aligned} {\partial u\over\partial t}+l {\partial \varphi\over\partial t}=\Delta u+f(x,t), \quad & (x,t)\in Q:=\Omega \times(0,T),\\ {\partial\varphi \over\partial t}=\Delta\varphi+ F(x,t,\varphi) +u,\quad & (x,t)\in Q,\\ {\partial u\over\partial v}={\partial \varphi\over \partial v}=0,\quad & \text{on }\partial \Omega\times (0,T),\end{aligned} \] \(u(x,0)=u_0(x)\), \(\varphi(x,0)=\varphi_0(x)\), \(x\in\Omega\), representing a general class of phase-field models, where \(u\) is temperature, and \(\varphi\) is the phase function \((\varphi=-1\) being associated to the solid and \(\varphi=1\) to the liquid). Here \(f\in L^q(Q)\), \(q\geq 2\), \(u_0\in W^2_\infty (\Omega)\), \(\varphi_0 \in W^2_\infty (\Omega)\) (satisfying compatibility conditions on \(\partial \Omega)\) and \(l\) is a positive constant. The assumptions on \(F(x,t,\varphi)\) are \((F(x,t,z_1)- F(x,t,z_2)) (z_1-z_2)\leq a_0(z_1-z_2)^2\), with \(a_0\) constant, \((F(x,t,z_1)- F(x,t,z_2))^2 \leq\overline F(x,t,z_1,z_2)(z_1-z_2)^2\) with \(\overline F\) satisfying growth conditions w.r.t. \(|z_1|\), \(|z_2 |\) with exponent \(2r-2\), with \(r\geq 1\) and \(r<{N+2\over N-2}\) if \(N>2\), and \(F=k(x,t,z)-h(z)\) with \(h\in C^1\), \(k\) measurable and continuous in \(\varphi\), \(k(\cdot,0,0)\in L^\infty(Q)\). The main result of the paper is an existence and uniqueness theorem with \(u\in W_q^{2,1}(Q)\), \(\varphi\in W^{2,1}_\mu(Q)\), \(\mu\) depending on \(N,q\). Continuous dependence on the data and on \(f\) is also proved.
compatibility conditions on \(\partial\Omega\)
scientific article; zbMATH DE number 1346375