On some global well-posedness and asymptotic results for quasilinear parabolic equations (Q1906301)

The paper is concerned with the quasilinear parabolic initial boundary value problem \[ u_t- \sum^n_{i, j= 1} a_{ij}(\nabla u) \partial_i\partial_j u= f(x, t),\;u(x, 0)= u_0(x),\;u|_{\partial\Omega}= 0. \] The authors prove the global existence and uniqueness of solutions with values in some Sobolev space \(H^m(\Omega)\), where \(m\) is high enough that the solutions are also classical via the Sobolev imbedding theorem. The strong continuity with respect to the data and the asymptotic behavior of the solutions are also investigated.