Lifespan of solutions of a fractional evolution equation with higher order diffusion on the Heisenberg group
We consider the higher order diffusion Schrodinger equation with a time nonlocal nonlinearity $$ i\partial_tu-(-\Delta_{\mathbb{H}})^mu =\frac{\lambda}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1} | u(s)|^{p}\,ds, $$ posed in $(\eta, t) \in \mathbb{H}\times(0,+\infty)$, supplemented with an initial...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/02/abstr.html |
| Summary: | We consider the higher order diffusion Schrodinger equation with a time nonlocal
nonlinearity
$$
i\partial_tu-(-\Delta_{\mathbb{H}})^mu
=\frac{\lambda}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1} | u(s)|^{p}\,ds,
$$
posed in $(\eta, t) \in \mathbb{H}\times(0,+\infty)$, supplemented with an initial
data $u(\eta,0)=f(\eta)$, where $m>1,\,p>1,\,0<\alpha<1$, and $\Delta_{\mathbb{H}}$
is the Laplacian operator on the $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$.
Then, we prove a blow up result for its solutions. Furthermore, we give an upper
bound estimate of the life span of blow up solutions. |
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| ISSN: | 1072-6691 |