A weighted Lq(Lp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q(L_p)$$\end{document}-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

We study the fully degenerate second-order evolution equation 0.1ut=aij(t)uxixj+bi(t)uxi+c(t)u+f,t>0,x∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, \quad t>0, x\in \mathbb {R}^d \end{aligned}$$\end{document}given with the zero initial data. Here aij(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{ij}(t)$$\end{document}, bi(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^i(t)$$\end{document}, c(t) are merely locally integrable functions, and (aij(t))d×d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a^{ij}(t))_{d \times d}$$\end{document} is a nonnegative symmetric matrix with the smallest eigenvalue δ(t)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (t)\ge 0$$\end{document}. We show that there is a positive constant N such that 0.2∫0T∫Rd|u(t,x)|+|uxx(t,x)|pdxq/pe-q∫0tc(s)dsw(α(t))δ(t)dt≤N∫0T∫Rdft,xpdxq/pe-q∫0tc(s)dsw(α(t))(δ(t))1-qdt,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _0^{T} \left( \int _{\mathbb {R}^d} \left( |u(t,x)|+|u_{xx}(t,x) |\right) ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) \delta (t) dt \nonumber \\&\le N \int _0^{T} \left( \int _{\mathbb {R}^d} \left| f\left( t,x\right) \right| ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) (\delta (t))^{1-q} dt, \end{aligned}$$\end{document}where p,q∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q \in (1,\infty )$$\end{document}, α(t)=∫0tδ(s)ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)=\int _0^t \delta (s)ds$$\end{document}, and w is Muckenhoupt’s weight.