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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The Cauchy problem for the Hardy-H\'enon parabolic equation is studied in thecritical and subcritical regime in weighted Lebesgue spaces on the Euclideanspace $\mathbb{R}^d$. Well-posedness for singular initial data and existence ofnon-radial forward self-similar solution of the problem are previously shownonly for the Hardy and Fujita cases ($\gamma\le 0$) in earlier works. Theweighted spaces enable us to treat the potential $|x|^{\gamma}$ as an increaseor decrease of the weight, thereby we can prove well-posedness to the problemfor all $\gamma$ with $-\min\{2,d\}<\gamma$ including the H\'enon case($\gamma>0$). As a byproduct of the well-posedness, the self-similar solutionsto the problem are also constructed for all $\gamma$ without restrictions. Anon-existence result of local solution for supercritical data is also shown.Therefore our critical exponent $s_c$ turns out to be optimal in regards to thesolvability.

Optimal well-posedness and forward self-similar solution for the Hardy-H\'enon parabolic equation in critical weighted Lebesgue spaces