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A weighted $L_q(L_p)$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients | | Ildoo Kim
; | | Date: |
2 Jan 2023 | | Abstract: | We study the fully degenerate second-order evolution equation
$u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, quad t>0, xin
mathbb{R}^d$ given with the zero initial data. Here $a^{ij}(t)$, $b^i(t)$,
$c(t)$ are merely locally integrable functions, and $(a^{ij}(t))_{d imes d}$
is a nonnegative symmetric matrix with the smallest eigenvalue $delta(t)geq
0$. We show that there is a positive constant $N$ such that
$int_0^{T} left(int_{mathbb{R}^d} left(|u|+|u_{xx} |
ight)^{p} dx
ight)^{q/p} e^{-qint_0^t c(s)ds} w(alpha(t)) delta(t) dt leq N int_0^{T}
left(int_{mathbb{R}^d} left|fleft(t,x
ight)
ight|^{p} dx
ight)^{q/p}
e^{-qint_0^t c(s)ds} w(alpha(t)) (delta(t))^{1-q} dt,$ where $p,q in
(1,infty)$, $alpha(t)=int_0^t delta(s)ds$, and $w$ is a Muckenhoupt’s
weight. | | Source: | arXiv, 2301.00492 | | Services: | Forum | Review | PDF | Favorites | |
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