Hyperbolic Problems

In this chapter we are interested in finding coefficients of the second-order hyperbolic operator 8.0.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa % aaleaacaaIWaaabeaakiabgkGi2oaaDaaaleaacaWG0baabaGaaGOm % aaaakiaadwhacqGHRaWkcaWGbbGaamyDaiabg2da9iaadAgacaaMe8 % UaamyAaiaad6gacaaMe8Uaamyuaiabg2da9iabgM6axjabgEna0kaa % cIcacaaIWaGaaiilaiaadsfacaGGPaaaaa!4EC2! $$ {a_0}\partial _t^2u + Au = f\;in\;Q = \Omega \times (0,T) $$ given the initial data 8.0.2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 % da9iaadwhadaWgaaWcbaGaaGimaaqabaGccaGGSaGaaGjbVlabgkGi % 2oaaBaaaleaacaWG0baabeaakiaadwhacqGH9aqpcaWG1bWaaSbaaS % qaaiaaigdaaeqaaOGaaGjbVlaad+gacaWGUbGaaGjbVlabgM6axjab % gEna0oaacmaabaGaaGimaaGaay5Eaiaaw2haaiaacYcaaaa!4EDD! $$ u = {u_0},\;{\partial _t}u = {u_1}\;on\;\Omega \times \left\{ 0 \right\}, $$ the Neumann lateral data 8.0.3 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadA % hacqGHflY1cqGHhis0caWG1bGaeyypa0JaamiAaiaaysW7caqGVbGa % aeOBaiaaysW7cqqHtoWrdaWgaaWcbaGaaGymaaqabaGccqGHxdaTca % GGOaGaaGimaiaacYcacaWGubGaaiykaiaacYcaaaa!4C4B! $$ av \cdot \nabla u = h\;{\text{on}}\;{\Gamma _1} \times (0,T), $$ and the additional lateral data 8.0.4 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 % da9iaadEgacaaMe8Uaae4Baiaab6gacaaMe8Uaeu4KdC0aaSbaaSqa % aiaaicdaaeqaaOGaey41aqRaaiikaiaaicdacaGGSaGaamivaiaacM % cacaGGUaaaaa!469A! $$ u = g\;{\text{on}}\;{\Gamma _0} \times (0,T). $$