On the second order derivative estimates for degenerate parabolic equations

TL;DR

This paper establishes second order derivative estimates for solutions to degenerate parabolic equations with time-dependent coefficients, showing regularity results even when coefficients degenerate, using Besov spaces and asymptotic behavior of degeneracy.

Contribution

It proves new second order derivative estimates for degenerate parabolic equations with full degeneracy, extending regularity theory to cases with time-dependent degeneracy.

Findings

Solutions have second derivatives in Lp where coefficients are non-degenerate.

Derived integral estimates involving the degeneracy function δ(t).

Established regularity results in Besov spaces under degeneracy conditions.

Abstract

We study the parabolic equation \begin{align} \notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \quad (t,x) \in [0,T] \times \mathbf{R}^d \\ &u(0,x)=u_0(x) \label{main eqn} \end{align} with the full degeneracy of the leading coefficients, that is, \begin{align} (a^{ij}(t)) \geq \delta(t)I_{d\times d} \geq 0. \end{align} It is well known that if $f$ and $u_0$ are not smooth enough, say $f\in \mathbb{L}_p(T):=L_p([0,T] ; L_p(\mathbf{R}^d))$ and $u_0\in L_p(\mathbf{R}^d)$, then in general the solution is only in $C([0,T];L_p(\mathbf{R}^d))$, and thus derivative estimates are not possible. In this article we prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ on the set $\{t: \delta(t)>0 \}$ and \begin{align*} \int^T_0 \|u_{xx}(t)\|^p_{L_p} \delta(t)dt\leq N(d,p) \left(\int^T_0 \|f(t)\|^p_{L_p}\delta^{1-p}(t)dt + \|u_0\|^p_{B^{2-2/ p}_p} \right), \end{align*} where $B^{2-2/ p}_p$ is the Besov space of order $2-2/p$. We also prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ for all $t>0$ and \begin{equation} \label{10.13.3} \int^T_0 \|u_{xx}\|^p_{L_p(\mathbf{R}^d)}\,dt \leq N \|u_0\|^p_{B^{2-2/(\beta p)}_p}, \end{equation} if $f=0$, $\int^t_0 \delta(s)ds>0$ for each $t>0$, and a certain asymptotic behavior of $\delta(t)$ holds near $t=0$ (see (1.3)). Here $\beta>0$ is the constant related to the asymptotic behavior in (1.3). For instance, if $d=1$ and $a^{11}(t)=\delta(t)=1+\sin(1/t)$, then the estimate holds with $\beta=1$, which actually equals the maximal regularity of the heat equation $u_t=\Delta u$.