We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data $u_{ 0}$, especially the condition $\partial_{ x}^{ -1}u_{ 0} \in L^{ 1}(\mathbb{ R}^{ 2})$, it is known that the solution to this problem decays at the rate of $t^{ -\frac{ 7}{ 4}}$ in the $L^{ \infty}$-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at $t\to \infty$. Moreover, we obtain a lower bound of the $L^{ \infty}$-norm of the solution and prove that the decay rate $t^{ -\frac{ 7}{ 4}}$ of the solution given in the previous work to be optimal.
Categories: math.AP