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

$$\begin{aligned} u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, \quad t>0, x\in \mathbb {R}^d \end{aligned}$$(0.1)

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 \(\delta (t)\ge 0\). We show that there is a positive constant N such that

$$\begin{aligned}&\int _0^{T} \left( \int _{\mathbb {R}^d} \left( |u(t,x)|+|u_{xx}(t,x) |\right) ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) \delta (t) dt \nonumber \\&\le N \int _0^{T} \left( \int _{\mathbb {R}^d} \left| f\left( t,x\right) \right| ^{p} dx \right) ^{q/p} e^{-q\int _0^t c(s)ds} w(\alpha (t)) (\delta (t))^{1-q} dt, \end{aligned}$$(0.2)

where \(p,q \in (1,\infty )\), \(\alpha (t)=\int _0^t \delta (s)ds\), and w is Muckenhoupt’s weight.