A weighted $L_q(L_p)$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

Abstract

We study the fully degenerate second-order evolution equation $u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, \quad t>0, x\in \mathbb{R}^d$ given with the zero initial data. Here $a^{ij}(t)$, $b^i(t)$, $c(t)$ are merely locally integrable functions, and $(a^{ij}(t))_{d \times d}$ is a nonnegative symmetric matrix with the smallest eigenvalue $\delta(t)\geq 0$. We show that there is a positive constant $N$ such that $\int_0^{T} \left(\int_{\mathbb{R}^d} \left(|u|+|u_{xx} |\right)^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(\alpha(t)) \delta(t) dt \leq N \int_0^{T} \left(\int_{\mathbb{R}^d} \left|f\left(t,x\right)\right|^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(\alpha(t)) (\delta(t))^{1-q} dt,$ where $p,q \in (1,\infty)$, $\alpha(t)=\int_0^t \delta(s)ds$, and $w$ is a Muckenhoupt's weight.