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Asymptotic profile of solutions to the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with anisotropic dissipation in 2D

Ikki Fukuda

Abstract

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with the dissipation term $-νu_{xx}$ in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropy. In this paper, we investigate the large time behavior of the solution to this problem. Especially, we show that the $L^{\infty}$-norm of the solution decays at the rate of $t^{-7/4}$ if the initial data $u_{0}(x, y)$ satisfies $(1+|x|)u_{0}\in L^{1}(\mathbb{R}^{2})$ with the zero-mass condition and some appropriate regularity assumptions. Moreover, combining techniques used for parabolic equations and the Schrödinger equation, we also derive the detailed asymptotic profile of the solution.

Asymptotic profile of solutions to the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with anisotropic dissipation in 2D

Abstract

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with the dissipation term in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropy. In this paper, we investigate the large time behavior of the solution to this problem. Especially, we show that the -norm of the solution decays at the rate of if the initial data satisfies with the zero-mass condition and some appropriate regularity assumptions. Moreover, combining techniques used for parabolic equations and the Schrödinger equation, we also derive the detailed asymptotic profile of the solution.
Paper Structure (7 sections, 18 theorems, 141 equations)

This paper contains 7 sections, 18 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear Analysis
  4. Nolinear Analysis
  5. Proof of the Main Result
  6. Decay Estimate
  7. Asymptotic Behavior

Key Result

Theorem 1.1

Let $p\ge1$ be an integer. Assume that $u_{0}\in X^{3}(\mathbb{R}^{2})\cap L^{1}(\mathbb{R}^{2})$, $xu_{0}\in L^{1}(\mathbb{R}^{2})$ and $B(u_{0})$ is sufficiently small. Then, the solution $u(x, y, t)$ to KPB satisfies the following estimate: Moreover, there exists a remainder function $R(x, y, t)$ satisfying for each $(x, y)\in \mathbb{R}^{2}$, $t>0$, and where $K(x, y, t)$, $\mathcal{N}_{0}$

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Corollary 2.4
  • ...and 24 more