Caustics for dissipative semilinear oscillations
Jean-Luc Joly, Guy Métivier, Jeffrey B. Rauch · Birkhäuser Boston eBooks · 1997
Consider in ℝ1+d the semilinear wave equation (1.1) $$ \square u + f\left( {\partial u} \right) = 0 $$ where ∂u:= (∂t u,∂x u) and f is a smooth function from ℝ1+d into ℝ1+d. Consider oscillatory Cauchy data (1.2) $$ \left\{ {\begin{array}{*{20}c} {u_{\left| {t = 0} \right.}^\varepsilon = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} _0 \left( x \right) + \varepsilon \,{\text{U}}_{\text{0}} \left( {x,\psi \left( x \right)/\varepsilon } \right),} \\ {\partial _t u_{\left| {t = 0} \right.}^\varepsilon = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} _1 \left( x \right) + {\text{U}}_{\text{1}} \left( {x,\psi \left( x \right)/\varepsilon } \right),} \\ \end{array} } \right. $$ where ψ is a smooth function with nonvanishing differential on ω⊂ℝd, U 0(x,θ) and U 1(x,θ) are smooth, 2π-periodic in θ with mean 0 and compactly supported in ψ x and. In this problem, the scales of the wavelength and of the amplitude of the oscillations are chosen so that the nonlinear effects are expected to appear in time O(1).