Inverse parabolic problems

In this chapter we consider the second-order parabolic equation 9.0.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGHbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgkGi2+aa % daWgaaWcbaWdbiaadshaa8aabeaak8qacaWG1bGaeyOeI0Iaamizai % aadMgacaWG2bGaaiikaiaadggacqGHhis0caWG1bGaaiykaiabgUca % RiaadkgacqGHhis0caWG1bGaey4kaSIaam4yaiaadwhacqGH9aqpca % WGMbWdaiaayEW7peGaamyAaiaad6gapaGaaG5bV-qacaWGrbGaeyyp % a0JaeuyQdCLaey41aqRaaiikaiaaicdacaGGSaWdaiaayEW7peGaam % ivaiaacMcacaGGSaaaaa!5E2C! $$ {a_0}{\partial _t}u - div(a\nabla u) + b\nabla u + cu = f{\mkern 1mu} in{\mkern 1mu} Q = \Omega \times (0,{\mkern 1mu} T), $$ where Ω is a bounded domain the space ℝn with the C2-smooth boundary ∂Ω. In Section 9.5 we study the nonlinear equation 9.0.1n % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGHbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacIcacaWG % 4bGaaiilaiaadwhacaGGPaGaamyDa8aadaWgaaWcbaWdbiaadshaa8 % aabeaak8qacqGHsislcqqHuoarcaWG1bGaey4kaSIaam4yaiaacIca % caWG4bGaaiila8aacaaMh8+dbiaadshacaGGSaWdaiaayEW7peGaam % yDaiaacMcacqGH9aqpcaaIWaWdaiaayEW7peGaamyAaiaad6gapaGa % aG5bV-qacaWGrbGaaiOlaaaa!5534! $$ {a_0}(x,u){u_t} - \Delta u + c(x,{\mkern 1mu} t,{\mkern 1mu} u) = 0{\mkern 1mu} in{\mkern 1mu} Q. $$