Matches in SemOpenAlex for { <https://semopenalex.org/work/W4289744144> ?p ?o ?g. }
Showing items 1 to 51 of
51
with 100 items per page.
- W4289744144 abstract "We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: begin{equation*} begin{cases} partial_t overline{u} - y^{-a} nabla cdot(y^a nabla overline{u}) = 0 quad &text{ in } mathbb{B}_1^+ times (-1,0) -partial_y^a overline{u} = q(x,t)u quad &text{ on } B_1 times {0} times (-1,0), end{cases} end{equation*} where $ain(-1,1)$, is a fixed parameter $mathbb{B}_1^+subset mathbb{R}^{N+1}$ is the upper unit half ball and $B_1$ is the unit ball in $mathbb{R}^N$. Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator begin{equation*} H^su(x,t) = frac{1}{|Gamma(-s)|} int_{-infty}^t int_{mathbb{R}^N} left[u(x,t) - u(z,tau)right] frac{G_N(x-z,t-tau)}{(t-tau)^{1+s}} dzdtau. end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in $mathbb{R}^Ntimesmathbb{R}$, and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer's reduction principle and the parabolic Whitney's extension." @default.
- W4289744144 created "2022-08-04" @default.
- W4289744144 creator A5042598267 @default.
- W4289744144 creator A5056021577 @default.
- W4289744144 date "2018-07-26" @default.
- W4289744144 modified "2023-10-16" @default.
- W4289744144 title "On the nodal set of solutions to a class of nonlocal parabolic equations" @default.
- W4289744144 doi "https://doi.org/10.48550/arxiv.1807.10135" @default.
- W4289744144 hasPublicationYear "2018" @default.
- W4289744144 type Work @default.
- W4289744144 citedByCount "0" @default.
- W4289744144 crossrefType "posted-content" @default.
- W4289744144 hasAuthorship W4289744144A5042598267 @default.
- W4289744144 hasAuthorship W4289744144A5056021577 @default.
- W4289744144 hasBestOaLocation W42897441441 @default.
- W4289744144 hasConcept C114614502 @default.
- W4289744144 hasConcept C121332964 @default.
- W4289744144 hasConcept C122041747 @default.
- W4289744144 hasConcept C134306372 @default.
- W4289744144 hasConcept C191948623 @default.
- W4289744144 hasConcept C2779557605 @default.
- W4289744144 hasConcept C33923547 @default.
- W4289744144 hasConcept C54207081 @default.
- W4289744144 hasConcept C62520636 @default.
- W4289744144 hasConcept C83979697 @default.
- W4289744144 hasConceptScore W4289744144C114614502 @default.
- W4289744144 hasConceptScore W4289744144C121332964 @default.
- W4289744144 hasConceptScore W4289744144C122041747 @default.
- W4289744144 hasConceptScore W4289744144C134306372 @default.
- W4289744144 hasConceptScore W4289744144C191948623 @default.
- W4289744144 hasConceptScore W4289744144C2779557605 @default.
- W4289744144 hasConceptScore W4289744144C33923547 @default.
- W4289744144 hasConceptScore W4289744144C54207081 @default.
- W4289744144 hasConceptScore W4289744144C62520636 @default.
- W4289744144 hasConceptScore W4289744144C83979697 @default.
- W4289744144 hasLocation W42897441441 @default.
- W4289744144 hasOpenAccess W4289744144 @default.
- W4289744144 hasPrimaryLocation W42897441441 @default.
- W4289744144 hasRelatedWork W1970843405 @default.
- W4289744144 hasRelatedWork W2016092567 @default.
- W4289744144 hasRelatedWork W2056977145 @default.
- W4289744144 hasRelatedWork W2086637669 @default.
- W4289744144 hasRelatedWork W2222086116 @default.
- W4289744144 hasRelatedWork W2949874568 @default.
- W4289744144 hasRelatedWork W2952893691 @default.
- W4289744144 hasRelatedWork W2979832966 @default.
- W4289744144 hasRelatedWork W4300816254 @default.
- W4289744144 hasRelatedWork W4308112727 @default.
- W4289744144 isParatext "false" @default.
- W4289744144 isRetracted "false" @default.
- W4289744144 workType "article" @default.