Detection and estimation of abnormalities for distributed parameter system have wide applications in industry, e.g., battery thermal fault detection, quality monitoring of hot-rolled strip laminar cooling process. In this chapter, the abnormal spatio-temporal source detection and estimation problem for a linear unstable distributed parameter system is first studied. The proposed methodology consists of two steps: first, an abnormality detection filter which generates a residual signal for abnormality detection in the time domain is constructed using pointwise measurement; Then, an adaptive Luenberger-type partial differential equation observer including an adaptive estimation algorithm is designed and triggered only when an alarm raises from the abnormality detection filter. Theoretic analysis based on the spatial domain decomposition approach is presented to show the convergence of the estimation errors. Finally, an illustrative example is presented to show the performance of the proposed method.