Some decay results for biparabolic equation

Abstract

This paper investigates the initial value problem for a fourth-order bi-parabolic equation in $\mathbb{R}^d$ with slowly decaying initial data.
We focus on the asymptotic behavior and decay properties of global solutions in both linear and nonlinear settings.
The main results establish optimal time-decay estimates under minimal assumptions, showing that additional moment conditions on the initial data yield systematically faster decay rates.
In particular, we provide sharp Gaussian kernel bounds with explicit constants, gradient decay estimates, and fractional-order smoothing effects.
The approach is based on a transparent frequency-splitting method at the natural parabolic scale: low frequencies are handled through Taylor expansion combined with moment cancellation, while high frequencies are controlled by elementary Gaussian estimates.
Our analysis requires only basic tools such as scaling arguments, kernel inequalities, and the Hausdorff--Young inequality, making the proofs both simple and effective.
The results contribute to a unified understanding of higher-order dissipative systems and highlight their robustness in physical models arising from fluid mechanics, heat conduction, and material science.
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\noindent {\bf Keywords and phrases:} {nonlinear fourth order parabolic equation; memory term; Cahn-Hilliard equation; well-posedness.}\\

\noindent {\bf Mathematics Subject Classification {2020}:} \red{35K10, 35K55, 35Q99, 35K30, 35A01 }