Global Sobolev solutions of quasilinear parabolic equations
Global existence, uniqueness and a priori estimates of solutions to the initial and homogeneous Dirichlet boundary value problem for the equation \[ u_t - \sum _{i,j=1}^{n} a_{i,j}(\nabla u) \partial _i \partial _j u = f(x,t)\quad\text{on} \Omega \times (0,T) \] is proved in Sobolev spaces \(X_{s+2}(T)\) for sufficiently large \(s.\) Here \[ X_m(T) = \{ u \in L^2(0,T;H^{m}(\Omega)),\;\partial ^{m'}_t u \in L^2(0,T;H^{m - 2m'}(\Omega))\} \] with \(m' = [\frac {m+1}{2}]\). The data of the problem satisfy natural smoothness assumptions. The proof uses as a starting point classical existence results of O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva combined with higher-order estimates obtained with the help of Gagliardo-Nirenberg interpolation inequalities.
homogeneous Dirichlet boundary value problem
Gagliardo-Nirenberg interpolation inequalities
scientific article; zbMATH DE number 1453647