We investigate the evolution and propagation of the electron plasma waves (EPWs) excited by stimulated Raman scattering (SRS) in the inhomogeneous plasma theoretically and numerically with particle-in-cell (PIC) simulations. A theoretical model of EPWs in inhomogeneous plasmas is presented, which shows that the evolution of the EPW wavenumber is mainly related to the plasma density profile rather than the plasma electron temperature, in agreement with PIC simulations. When the density gradient is positive along the propagation direction of an EPW, its wavenumber decreases with time and consequently its phase velocity increases continuously, causing the trapped electrons to be accelerated to anomalous high energy. Furthermore, it is found that the Langmuir decay instability tends to reduce the levels of SRS saturation and electron acceleration and produce hot electrons in the opposite direction. This work provides a new understanding of electron heating due to SRS excitation.