Lyapunov exponents and asymptotic dynamics in random Kolmogorov models
Let \(((\Omega,{\mathcal F},P), \{\theta_{t}\}_{t\in R})\) be an ergodic metric dynamical system. The authors consider the pathwise parabolic equation \(u_{t}=\Delta u+f(\theta_{t}\omega,x,u)u\), \(x\in D\), \(Bu=0\), \(x\in\partial D\), where \(D\subset \mathbb R^{N}\) is bounded domain with smooth boundary \(\partial D\), \(f: \Omega\times\overline D\times \mathbb R_{+}\to \mathbb R\) is a measurable function satisfying some additional smoothness conditions, and \(B\) is a boundary operator of either the Dirichlet type, or Robin type. Let \(\lambda\) be the principal Lyapunov exponent of the linearization of considered random differential equation around the trivial equilibrium \(u=0\). One of the main results is following. Let \(X_{+}=\{u\in X: u(x)\geq0, x\in D\}\), where \(X\subset L^{p}(D)\), \(p>N\) is some fractional power space. If \(\lambda<0\), then for any nonzero \(u_0\in X_{+}\) and a.e. \(\omega\in\Omega\), \(u(t,\cdot;u_0,\omega)\to0\) as \(t\to\infty\). If \(\lambda>0\), then there is a unique positive random equilibrium \(\phi: \Omega\to \operatorname{Int} X_{+}\). Moreover, for any \(u_0>0\) and a.a. \(\omega\in \Omega\), \(u(t,\cdot; u_0,\omega)-\phi(\theta_{t}\omega)\to0\) and \(u(t,\cdot; u_0,\theta_{-t} \omega)-\phi(\omega)\to0\) as \(t\to\infty\). There are also results proved giving sufficient conditions for the unique random equilibrium to be positive uniformly in \(\omega\in \Omega\).
scientific article; zbMATH DE number 2133419