Abstract: The existence and the uniqueness of weak solutions to a degenerate parabolic equation with variable exponents u_t=div(ρ^α|a(u))|^p(x)-2a(u))+g(x)div(b(u)) is considered, where ρ(x)=dist(x,Ω) is the distance function from the boundary, a(s) is a strictly monotone increasing function. By choosing a suitable test function, the uniqueness of weak solution of the equation is proved under the condition of no boundary value.

Key words: degenerate parabolic equation, variable exponent, boundary value condition, stability, uniqueness