On the Łojasiewicz–Simon gradient inequality

We prove a general version of the Łojasiewicz–Simon inequality, and we show how to apply the abstract result to study energy functionals E of the formE(v)=12 a(v,v)+∫ΩF(x,v),defined on a Hilbert space V↪L2(Ω). We show that in some cases it is possible to prove the Łojasiewicz–Simon inequality for such functionals without the assumption of analyticity. The results apply to study the asymptotic behaviour of parabolic and hyperbolic evolution equations.