Explicit decay estimates for solutions to nonlinear parabolic systems
The author investigates the following nonlinear parabolic system with initial and boundary values of Dirichlet type: \[ \begin{cases} \Delta u+f\left( \left| \nabla u\right| ^2+\left| \nabla v\right| ^2\right) = \frac{\partial_tu}{\partial t} \quad \text{ in }\,\Omega, \\ \Delta v+g\left( \left| \nabla u\right| ^2+\left| \nabla v\right| ^2\right) = \frac{\partial_tv}{\partial t}\quad \text{ in }\Omega, \\ u\left( x,t\right) =0\quad \text{on}\,\partial \Omega, \\ u\left( x,t\right) =0\quad \text{ on }\partial \Omega, \\ u\left( x,0\right) =u_0\left( x\right) \quad \text{ on }\Omega, \\ v\left( x,0\right) =v_0\left( x\right) \quad \text{ on }\Omega, \end{cases}\tag{1} \] where \(\Omega \) is a bounded convex domain in \(\mathbb{R}^N\), \(N\geq 2,\) with \( \partial \Omega \in C^{2+\varepsilon },\varepsilon \in ( 0,1)\). The functions \(f,g\in C^1\) are nonnegative, and the nonnegative functions \(u_0,v_0\) in \(\Omega \) have bounded first derivatives with \(u_0\left( x\right) =v_0\left( x\right) =0\) on \(\partial \Omega\). The authors is interested in establishing sufficient conditions on the data in order to preclude blow up and to deduce that the solution decays exponentially in time. The system (1) is considered first in \(\Omega \times \left[ 0,T\right] \), where \(T\) is any time below the possible blow-up time \(T^\ast\). After this, by introducing conditions to avoid blow up of the solution, the system is considered for all \(t>0\). In Section 2 of the paper the author proves that under suitable conditions on \(f\) and \(g\) the function \(U( x,t) :=u^2\left( x,t\right) +v^2\left( x,t\right)\), satisfies a parabolic inequality. In Section 3, the author derives an upper bound for the gradient of the solution that allows to introduce a sufficient condition on the data in order to prove that \(U\) decays exponentially in time for all \(t>0\).
scientific article; zbMATH DE number 6134331