Uniqueness of the solution of the first mixed problem for a two-dimensional nonlinear parabolic equation (Q1591899)

In the cylindrical domain \(Q_T=\Pi\times(0,T)\), \(\Pi=\{(x,y)\in\mathbb R^2:\kappa_1<x<\kappa_2,\kappa_3<y<\kappa_4\},\) there is considered the following parabolic Cauchy-Dirichlet problem \[ u_t=f(t,x,y,u,u_x,u_y,u_{xx},u_{yy}, u_{xy} + u_{xx} + u_{yy}), \] \[ u(0,x,y)=\Phi (x,y), \] \[ u\big|_{\partial\Pi\times(0,T)}=\varphi(t,x,y). \] Under natural smoothness conditions on the functions \(f\), \(\varphi\), \(\Phi\) and some additional structural conditions on the function \(f\) there is established the uniqueness of classical solution in the space \(C^{2,1}(Q_T)\). The proof is realized by applying the longitudinal version of the method of straight lines.