English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) withMuckenhoupt weights for the equation $$ \partial_{t}^{\alpha}u(t,x)=\Deltau(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $\alpha\in (0,2)$ and$\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha$.In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and$w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p\,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N\int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx\right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$weights. Our approach is based on the sharp function estimates of thederivatives of solutions.

Weighted $L_q(L_p)$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives