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This paper deals with the nonlinear phase field system  \begin{equation*}  \begin{cases}  \partial_t (\theta +\ell \varphi) - \Delta\theta = f  & \mbox{in}\ \Omega\times(0, T),  \\[1mm]  \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi)  = \ell \theta,\ \xi\in\beta(\varphi)  & \mbox{in}\ \Omega\times(0, T)  \end{cases}  \end{equation*} in a general domain $\Omega\subseteq\mathbb{R}^N$. Here $N\in \mathbb{N}$, $T>0$, $\ell>0$, $f$ is a source term, $\beta$ is a maximalmonotone graph and $\pi$ is a Lipschitz continuous function. We note that inthe above system the nonlinearity $\beta+\pi$ replaces the derivative of apotential of double well type. Thus it turns out that the system is ageneralization of the Caginalp phase field model and it has been studied bymany authors in the case that $\Omega$ is a bounded domain. However, forunbounded domains the analysis of the system seems to be at an early stage. Inthis paper we study the existence of solutions by employing a timediscretization scheme and passing to the limit as the time step $h$ goes to$0$. In the limit procedure we face with the difficulty that the embedding$H^1(\Omega) \hookrightarrow L^2(\Omega)$ is not compact in the case ofunbounded domains. Moreover, we can prove an interesting error estimate oforder $h^{1/2}$ for the difference between continuous and discrete solutions.

Time discretization of a nonlinear phase field system in general domains