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On the asymptotic behavior for convection-diffusion equations associated to higher order elliptic operators in divergence form. (English) Zbl 1031.35052

Summary: We consider the linear convection-diffusion equation associated to higher-order elliptic operators \[ \begin{gathered} u_t+{\mathcal L}_tu= a\nabla u\quad\text{on }\mathbb{R}^n\times (0,\infty),\\ u(0)= u_0\in L^1(\mathbb{R}^n),\end{gathered}\tag{1} \] where \(a\) is a constant vector in \(\mathbb{R}^n\), \(m\in\mathbb{N}^*\), \(n\geq 1\) and \({\mathcal L}_0\) belongs to a class of higher-order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on \(\mathbb{R}^n\). The aim of this paper is to study the asymptotic behavior, in \(L^p\) \((1\leq p\leq\infty)\), of the derivatives \(D^\gamma u(t)\) of the solution of (1) when \(t\) tends to \(\infty\).

MSC:

35K30 Initial value problems for higher-order parabolic equations
35K25 Higher-order parabolic equations
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35A08 Fundamental solutions to PDEs
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