In this paper the initial-boundary value problems of the equation uu=uxxt+σ(ux). including sec-ond boundary conditions are studied by means of Galerkin's method and energy estimates. In σ∈C1

σ1 (s) bounded from below

the existence and uniqueness of the global strong solution are obtained. Inaddition if σ(s) and the initial functions are suitably smooth and σ2? (0)=0

i=1

2

…

n

for cer-tain n

the corresponding smoothness of the strong solution is proved; finally the asymptotiic behaviorof solution and its physical meaning are investigated.