We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution.

In this paper, a mixed tensor analysis for a two-dimensional (2D) manifold embedded into a three-dimensional (3D) Riemannian space is conducted and its applications to construct a dimensional splitting method for linear and nonlinear 3D elastic shells are provided. We establish a semi-geodesic coordinate system based on this 2D manifold, providing the relations between metrics tensors, Christoffel symbols, covariant derivatives and differential operators on the 2D manifold and 3D space, and establish the Gateaux derivatives of metric tensor, curvature tensor and normal vector and so on, with respect to the surface $\vec{\Theta}$ along any direction $\vec{\eta}$ when the deformation of the surface occurs. Under the assumption that the solution of 3D elastic equations can be expressed in a Taylor expansion with respect to transverse variable, the boundary value problems satisfied by the coefficients of the Taylor expansion are given.