On singular perturbation of a semilinear hyperbolic equation
The author considers the semilinear hyperbolic equation: \[ u_{\tau\tau}+ \mu u_\tau+ \Delta u+ F(x, u, u_x, u_\tau)= 0,\tag{H\(_\mu\)} \] where \(F\) is a \(C^\infty\) function and \(\mu (\geq 0)\) represents resistance. With the change of variable \(\tau= \mu t\), and \(\lambda= \mu^{- 1}\), the above equation is reduced to \[ \lambda^2 u_{tt}+ u_t+ \Delta u+ F(x, u, u_x, \lambda u_t)= 0\tag{H\(_\lambda\)} \] which is a classical singular perturbation problem with unperturbed equation \[ u_t+ \Delta u+ F(x, u, u_x, 0)= 0.\tag{P} \] Using classical analysis the author proves that the solution to \((\text{H}_\lambda)\) converges uniformly to the solution of (P). It is also shown, by a counterexample, that if \(F\) is not linear in \(u_t\) then it must be very small. So the coefficient \(\lambda\), in front of \(u_t\) in the equation \((\text{H}_\lambda)\) is necessary for the convergence.
scientific article; zbMATH DE number 826772