285 research outputs found
Blowup solutions of Grushin's operator
In this note, we consider the blowup phenomenon of Grushin's operator. By
using the knowledge of probability, we first get expression of heat kernel of
Grushin's operator. Then by using the properties of heat kernel and suitable
auxiliary function, we get that the solutions will blow up in finite time.Comment:
Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincar\'e inequality, established in [38] for stratified groups
Critical Keller-Segel meets Burgers on : large-time smooth solutions
We show that solutions to the parabolic-elliptic Keller-Segel system on
with critical fractional diffusion
remain smooth for any initial data and any positive time. This disproves, at
least in the periodic setting, the large-data-blowup conjecture by Bournaveas
and Calvez. As a tool, we show smoothness of solutions to a modified critical
Burgers equation via a generalization of the method of moduli of continuity by
Kiselev, Nazarov and Shterenberg. over a setting where the considered equation
has no scaling. This auxiliary result may be interesting by itself. Finally, we
study the asymptotic behavior of global solutions, improving the existing
results.Comment: 17 page
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