Optimal time decay of solution to the 𝑀₁ model

In this paper, we are concerned with the asymptotic behavior of solution to the Cauchy problem of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 1"> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">M_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> model. Under some smallness conditions, we show that the solution to this system exists globally and tends time-asymptotically to the corresponding equilibrium state at the optimal rate. The proof is based on both the time-weighted energy method and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline minus upper L Superscript q"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo> − </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^{p}-L^{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> estimates from the detailed analysis of the Green’s function. To the best of our knowledge, this may be the first result regarding the optimal decay of solution to this model in the multi-dimensional case.