Weak solutions to one initial-boundary value problem with three boundary conditions for quasilinear evolution equations of the third order

The authors prove the existence, uniqueness and continuous dependence of the weak solutions uin C_w([0,T];L_2(Omega))cap L_2(0,T;H^1(Omega)) for the nonhomogeneous quasilinear partial differential equation gather u_t-displaystylesum_{|alpha|le 3}a_{alpha}partial_x^{alpha}u+displaystylesum_{j=1}^ng_j'(u)u_{x_j}=f(t,x), (t,x)in Q_T=(0,T)times Omega,endgather with the initial condition u|_{t=0}=u_0(x),quad xinOmega, and the three boundary conditions gather u|_{x_n=0}=u_1(t,x'),quad u|_{x_n=1}=u_2(t,x'),quad u_{x_n}|_{x_n=1}=u_3(t,x'), (t,x')in S_T=(0,T)times Bbb{R}^{n-1},endgather where x=(x_1,ldots,x_n), x'=(x_1,ldots,x_{n-1}), Omega={xinBbb{R}^n,|,,x_nin (0,1)}=Bbb{R}^{n-1}times (0,1), nge 2, a_{alpha}inBbb{R}, the functions g_jin C^1(Bbb{R}),,,j=1,ldots,n, have at most quadratic rate of growth and fin L_1(0,T;L_2(Omega)).