On the second order derivative estimates for degenerate parabolic equations

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 $u_t=\Delta u$.