Global solution to a phase transition model based on a microforce balance
The paper deals with the global existence of solutions to the system of nonlinear equations of the parabolic type: \[ (\vartheta^2)_t-\Delta\vartheta=h+\vartheta\chi\chi_t+\chi_t^2, \qquad \chi_t-\Delta\chi + W'(\chi)=-\chi(\vartheta -\vartheta_c). \] This system, which is called an Allen-Cahn system, or, more generally, may be regarded as a specific example of Ginzburg-Landau systems, is considered as a model of a nonisothermal phase transition in the multidimensional space. The phase state is described by the real order parameter \(\chi\), \(\vartheta\) is the temperature, the potential function \(W(\chi)\) drives the phase transition, the constant \(h\) is a heat-source term, \(\vartheta_c\) is the critical temperature at which the phase transition takes place. The equations are supplemented by initial conditions, and by the boundary conditions in the form of \(\partial_n\vartheta +n_0(\vartheta-\vartheta_{\Gamma})=0\), where \(\vartheta_{\Gamma}\) is a given temperature distribution at the boundary \(\Gamma\) of the solution domain, and \(\partial_n\vartheta\) is the respective normal derivative. The subject of the paper is to prove the existence and uniqueness of a solution to the system in the global sense, valid for all \(t>0\). To this end, the existence and uniqueness are first proven locally in \(t\) by means of a fixed-point theorem. A crucial important role in the proof is played by a lemma which states that \(\chi\) takes values within a confined interval, between some minimum and maximum values. Then, the confinement of values of \(\chi\) is proven to be time independent, holding for all \(t>0\), which makes it possible to prove the main results in the global (in time) form.
scientific article