On a class of linear equations of parabolic type (Q1916326)

The paper is devoted to the existence of a solution for the parabolic system \(\partial_tu = A(t)u+f\), \(u: \mathbb{R}^n \to\mathbb{R}^N\), where \(A(t)\) is the differential operator \[ \bigl(A(t)u \bigr)_\alpha = \sum^N_{\beta, \gamma, \delta=1} \sum^n_{i,j=1} a^{\alpha\beta} \partial_{x_i} \bigl(b_{ij}^{\beta \gamma} \cdot \partial x_j (c^{\gamma \delta}u_\delta) \bigr), \quad \alpha=1, \dots, N \] and \(a^{\alpha\beta} = a^{\alpha\beta} (x,t)\). By means of variational methods (developed by J. L. Lions), a special class of operators \(A(t)\) is considered, for which a weak solution exists in a certain Hilbert space. To illustrate results, the particular operator \(A(t) = (1/ \rho (.,t)) (c_1 \Delta u+ c_2 \nabla (\nabla.u))\) is considered.