arXiv subjects

Ryunosuke Kusaba

Publications and source records attributed to Ryunosuke Kusaba.

5 recordsLinked to original sources

arXiv2026

On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data

We consider the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data. We establish an improved decay estimate of the global solutions under the zero-mass condition. Moreover, we derive a self-similar asymptotic profile of the global solutions. This result provides necessary and sufficient conditions for attaining the improved decay rate and shows that the decay rate is optimal.

math.AP↗

arXiv2025

Remarks on the commutation relations between the Gauss--Weierstrass semigroup and monomial weights

We consider the commutation relations between the Gauss--Weierstrass semigroup on $\mathbb{R}^{n}$, generated by convolution with the complex Gaussian kernel, and monomial weights. We provide an explicit representation and a sharper estimate of the commutation relations with more concise proofs than those in previous works.

math.AP↗

arXiv2024

Higher order asymptotic expansions for the convection-diffusion equation in the Fujita-subcritical case

This paper is devoted to the asymptotic behavior of global solutions to the convection-diffusion equation in the Fujita-subcritical case. We improve the result by Zuazua (1993) and establish higher order asymptotic expansions with decay estimates of the remainders. We also discuss the optimality for the decay rates of the remainders.

math.AP↗

arXiv2024

Asymptotic behavior of global solutions to the complex Ginzburg--Landau type equation in the super Fujita-critical case

We present weighted estimates and higher order asymptotic expansions of global solutions to the complex Ginzburg--Landau (CGL) type equation in the super Fujita-critical case. Our approach is based on commutation relations between the CGL semigroup and monomial weights in $\mathbb{R}^{n}$ for the weighted estimates and on the Taylor expansions with respect to the both space and time variables for the asymptotic expansions. We also characterize the optimal decay rate in time of the remainder for the asymptotic expansion from the viewpoint of the moments of the initial data in space and those of the nonlinear term in spacetime.

math.AP↗

arXiv2023

Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.

math.AP↗