Asymptotic behaviour of solutions to a class of semilinear hyperbolic systems in arbitrary domains
The author considers a class of hyperbolic systems in \(\mathbb{R}^{1+ 3}\) of the form \[ \partial_0 U(t)= AU(t)+ F(t, U(t)) \] with the block matrix \(A:= \left(\begin{smallmatrix} E^{(1)} & 0\\ 0 & E^{(2)}\end{smallmatrix}\right) \left(\begin{smallmatrix} 0 & -H(\partial)^*\\ H(\partial) & 0\end{smallmatrix}\right)\), where \(H(\partial):= \sum^3_{k= 1} H_k\partial_k\), \(H_k\) constant, real \((N\times M)\)-matrices, and \(H(\partial)^*:= -\sum^3_{k=1} H^*_k\partial_k\) as its formal \(L_2\)-adjoint. The nonlinear term is of the block matrix form \[ F(t, U(t)):= G(t)- \begin{pmatrix} S(t,U(t))\\ 0\end{pmatrix}\in C([0, \infty[,\;L_2(\Omega, \mathbb{R}^M\times \mathbb{R}^N)) \] and the solution is sought in \(C([0, \infty[, L_2(\Omega, \mathbb{R}^M\times \mathbb{R}^N))\), with \(\Omega\) an arbitrary domain in \(\mathbb{R}^3\). The matrix-valued function \(E:= \left(\begin{smallmatrix} E^{(1)} & 0\\ 0 & E^{(2)}\end{smallmatrix}\right)\) is assumed to be symmetric, \(L_\infty\)-bounded and positive definite. It is also assumed that the spatial, linear operator part is given a suitable domain so that it is selfadjoint in the space \(L_2(\Omega, \mathbb{R}^M\times \mathbb{R}^N)\) equipped with a modified inner product weighted by \(E^{-1}\). The main result of the paper is the weak convergence of the solution \(U\) of the initial value problem \(U(t)\rightharpoonup U_\infty\) as \(t\to\infty\) in \(L_2(\Omega, \mathbb{R}^M\times \mathbb{R}^N)\), where \(U_\infty\) is in a certain subspace \({\mathcal N}\) of the nullspace of \(A\), together with the refinement, that \(U_\infty= 0\) if and only if the initial data \(U_0\) and the given right-hand side \(G\) are such, that \(\widetilde U_0:= U_0+ \int^\infty_0 G(t) dt\) is orthogonal to \({\mathcal N}\). The result is shown under suitable assumptions on the nonlinear damping terms \(S\). Maxwell's equations and the equations of acoustics (with an appropriate nonlinear damping term) are referred to as prominent applications.
scientific article; zbMATH DE number 1419589
scientific article