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

We study the fully degenerate second-order evolution equation 0.1 \(\begin{aligned} u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, \quad t>0, x\in \mathbb {R}^d \end{aligned}\) $$\begin{array}{c}\hfill u_t=a^{\mathit{ij}}(t)u_{x^i{x}^j}+b^i(t)u_{x^i}+c(t)u+f,t>0,x\in {\mathbb{R}}^d\end{array}$$ given with the zero initial data. Here \(a^{ij}(t)\) $a^{\mathit{ij}}(t)$ , \(b^i(t)\) $b^i(t)$ , c(t) are merely locally integrable functions, and \((a^{ij}(t))_{d \times d}\) ${(a^{\mathit{ij}}(t))}_{d\times d}$ is a nonnegative symmetric matrix with the smallest eigenvalue \(\delta (t)\ge 0\) $\delta (t)\ge 0$ . We show that there is a positive constant N such that 0.2 \(\begin{aligned}&\int _0^{T} \left( \int _{\mathbb {R}^d} \left( |u(t,x)|+|u_{xx}(t,x) |\right) ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) \delta (t) dt \nonumber \\&\le N \int _0^{T} \left( \int _{\mathbb {R}^d} \left| f\left( t,x\right) \right| ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) (\delta (t))^{1-q} dt, \end{aligned}\) $$\begin{array}{cc}& {\int }_0^T{\left({\int }_{{\mathbb{R}}^d},{\left(|u(t,x)|+|,u_{\mathit{xx}},(t,x)|\right)}^p,d,x\right)}^{q/p}{e}^{-q{\int }_0^{t}c(s)ds}w(\alpha (t))\delta (t)dt\hfill \\ \hfill & \le N{\int }_0^T{\left({\int }_{{\mathbb{R}}^d},{\left|f,\left(t,,,x\right)\right|}^p,d,x\right)}^{q/p}{e}^{-q{\int }_0^{t}c(s)ds}w(\alpha (t)){(\delta (t))}^{1-q}dt,\hfill \end{array}$$ where \(p,q \in (1,\infty )\) $p,q\in (1,\infty )$ , \(\alpha (t)=\int _0^t \delta (s)ds\) $\alpha (t)={\int }_0^t\delta (s)ds$ , and w is Muckenhoupt’s weight.