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 consider the Cauchy problem for the Hamilton-Jacobi equation with criticaldissipation, $$ \partial_t u + (-\Delta)^{ 1/2} u = |\nabla u|^p, \quad x \in\mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where$p > 1$ and $u_0 \in B^1_{r,1}(\mathbb R^N) \cap B^1_{\infty,1} (\mathbb R^N)$with $r \in [1,\infty]$. We show that for sufficiently small $u_0 \in \dotB^1_{\infty,1}(\mathbb R^N)$, there exists a global-in-time mild solution.Furthermore, we prove that the solution behaves asymptotically like suitablemultiplies of the Poisson kernel.

Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces