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Global existence and stability of solutions to the phase field equations. (English) Zbl 0733.35062
Free boundary value problems, Proc. Conf., Oberwolfach/FRG 1989, Int. Ser. Numer. Math. 95, 46-58 (1990).
[For the entire collection see Zbl 0702.00021.]
The existence of a (unique) global classical solution is obtained for the initial boundary value problems of the phase field equations: \[ \tau \phi_ t=\xi^ 2\Delta \phi +(1/2)(\phi -\phi^ 3)+2u,\quad u_ t+(\ell /2)\phi_ t=K\Delta u. \] Both Dirichlet and Neumann boundary conditions are considered. The results are proved without any restriction on the positive coefficients \(\tau\),\(\xi\),K. The asymptotic behaviour of solutions as \(t\to +\infty\) and the associated stationary problems are also studied.
Reviewer: C.Popa (Iaşi)

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)