Absolute stability of nonlinear nonstationary systems with distributed parameters (Q1070200)

Let \(L^ n_ 2(\Omega)\) be the space of square-integrable functions with values in \({\mathbb{R}}^ n\) which are defined on a closed subset \(\Omega \subset {\mathbb{R}}^ m\). Let further B, C be \(n\times n\) matrices and \(B\) be symmetric positive definite. Given a self-adjoint negative operator \(S: L^ n_ 2(\Omega)\to L^ n_ 2(\Omega)\), commuting with the operators which are determined by B, C on \(L^ n_ 2(\Omega)\), consider the following dynamic system on \(L^ n_ 2(\Omega)\) \[ (1)\quad du/dt=BSu+Cu+F(u,x,t),\quad t\geq 0, \] where \(F: {\mathbb{R}}^ n\times \Omega \times [0,\infty)\to {\mathbb{R}}^ n\) satisfies some Lipschitz-type conditions. Sufficient conditions for absolute exponential stability of the trivial solution to (1) are given. Several illustrative examples (predator-prey-type models, diffusion processes) are considered.