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 $u_t=a^{ij}(t)u_{x^i{x}^j}+b^i(t)u_{x^i}+c(t)u+f,t>0,x\in {\mathbb{R}}^d$ 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 $\hat{I}´(t)\ge 0$. We show that there is a positive constant $N$ such that ${\int }_0^T{\left({\int }_{{\mathbb{R}}^d}{\left(|u|+|u_{xx}|\right)}^{p}dx\right)}^{q/p}{e}^{-q{\int }_0^{t}c(s)ds}w(\hat{I}\pm (t))\hat{I}´(t)dt\le N{\int }_0^T{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(t,x\right)\right|}^{p}dx\right)}^{q/p}{e}^{-q{\int }_0^{t}c(s)ds}w(\hat{I}\pm (t))(\hat{I}´(t){)}^{1-q}dt,$ where $p,q\in (1,\infty )$, $\hat{I}\pm (t)={\int }_0^t\hat{I}´(s)ds$, and $w$ is a Muckenhoupt's weight.