Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations
A random linear parabolic equation (1) \(\partial_{t}u=A(\theta_{t} \omega,x)u\) in \(D\), \(Bu=0\) on \(\partial D\), is studied, where \(D \subseteq \mathbb R^{N}\) is a bounded domain with a smooth boundary, \[ A(\omega,x) = \sum^{N}_{i,j=1} a_{ij}(\omega,x)\partial _{x_{i}x_{j}} + \sum^{N}_{i=1}a_{i}(\omega,x)\partial _{x_{i}} + a_0(\omega,x), \] the boundary operator \(B\) is either of the Dirichlet type or the Robin type, and \((\theta_{t}\), \(t\in\mathbb R)\) is a group of ergodic measure preserving transformations of a probability space \((\Omega,\mathfrak F,P)\). It is assumed that \(a_{ij}\), \(a_{i}\) \(a_0\) are bounded measurable functions satisfying some smoothness hypotheses, the matrix \((a_{ij}(\omega,x))\) being symmetric and positive definite uniformly on \(\Omega\times\overline D\). If the coefficients of the operator \(A\) depend only on \(x\) and the eigenvalue problem \(A(x)u=\lambda u\) in \(D\), \(Bu=0\) on \(\partial D\), is considered, then it is known that the eigenvalue with the largest real part is real, simple and the corresponding eigenfunction \(v\) may be chosen such that \(v>0\) in \(D\). The goal of the paper is to generalize such results to the random problem (1) and apply the obtained theorems to some deterministic but nonautonomous parabolic problems. Let \(X\subseteq L^{p}(D)\) be a suitable fractional power space of \(-\varDelta\) such that \(X\) is embedded into \(C^1(\overline D)\), let \(\Phi_{t}:\Omega\times X\to X\) be mappings such that \(\Phi(\cdot,\omega)u_0\) is the solution to (1) with the initial condition \(u_0\). The principal Lyapunov exponent \(\lambda\) of (1) is defined by \[ \lambda(\omega) = \limsup_{t\to\infty} t^{-1}\log\|\Phi(t,\omega)\|. \] The properties of \(\lambda\) are studied in detail, in particular it is shown that the principal Lyapunov exponent is deterministic and of simple multiplicity.
scientific article; zbMATH DE number 1915671
scientific article