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Stable blow‐up dynamics in the ‐critical and ‐supercritical generalized Hartree equation
Studies in Applied Mathematics ( IF 3.108 ) Pub Date : 2020-08-01 , DOI: 10.1111/sapm.12328
Kai Yang; Svetlana Roudenko; Yanxiang Zhao

We study stable blow‐up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger‐type equation with a nonlocal, convolution‐type nonlinearity:
urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0003
First, we consider the urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0004‐critical case in dimensions urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0005 and obtain that a generic blow‐up has a self‐similar structure and exhibits not only the square root blowup rate urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0006, but also the log‐log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow‐up regime in the urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0007‐critical nonlinear Schrödinger equation. In this setting, we also study blow‐up profiles and show that generic blow‐up solutions converge to the rescaled urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0008, a ground state solution of the elliptic equation urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0009.
更新日期:2020-08-01

 

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