Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems
In the article under review, the author studies uniform boundedness of solutions and gradients to the initial boundary value problem for the degenerate parabolic system: \[ \frac{\partial u_i}{\partial t} = \text{div}(|Du_i|^{p_i - 1}Du_i) + f_i(x,\vec u),\quad t>0,\;x\in\Omega, \] \[ u_i\frac{\partial u_i}{\partial n}\leq 0 \quad\text{on } \partial\Omega, \qquad u_i(0,x) = u_i^0\quad\text{in } \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(p_i\geq 1\), \(u_i^0\) are bounded functions given on \(\Omega\). It is assumed that there exists positive constants \(\alpha\), \(k_1\), \(k_2\) such that \[ \sum_{i=1}^r f_i(x,\vec u)|u_i|^{\lambda - 1}u_i\leq k_1\sum_{i=1}^r|u_i|^{\alpha + \lambda} + k_2 \] for all \(\lambda\geq 1\) and \((x,\vec u)\in\Omega\times \mathbb{R}^r\). The case when the diffusion terms are given by the \(p\)-Laplacians are also considered. By using the theory of monotone operators, the local existence and uniqueness results can be easily obtained for the problem. The author introduces the notions following: (\(A_q\)) Uniformly \(L^q\)- dissipative, (\(B_q\)) Pointwise \(L^q\)-dissipative and (\(C_q\)) Globally \(L^q\)- bounded. The studied question is whether one of these conditions holds for \(q=\infty\) if it is verified for some finite \(q\). The first results regards the boundedness of solutions and reads as: If there exists \(q=p_*>\max\{1, (n/(p+1))(\alpha - p)\}\) where \(p=\min_i p_i \geq 1\) and \(n\equiv N\) if \(N>p+1\), otherwise any fixed \(n>p+1\), such that: (i) the system satisfies (\(A_q\)), (resp. (\(B_q\))) then its also satisfies (\(A_q\)), (resp. (\(B_q\))) for \(q = \infty\); (ii) the solution of the problem satisfies (\(C_q\)) then it also satisfies (\(C_q\)) for \(q=\infty\). If the solution of the system satisfies Neumann boundary conditions, then the estimates of the gradients in the \(L^{\infty}\)-norm are uniformly with respect to the initial data. This type of result is important in establishing the compactness of the trajectories of a nonlinear continuous semigroup \(S(t)(u_1^0,\dots,u_r^0) = (u_1(t),\dots,u_r(t))\) in the Banach space \(X=\prod_1^rC(\overline\Omega)\). Applying the general theory for dissipative dynamical systems, the author proves that the semigroup \(\{S(t)\}_{t\geq 0}\) possesses a global attractor \(\mathcal A\).
scientific article; zbMATH DE number 1383337
scientific article