# American Institute of Mathematical Sciences

October  2017, 37(10): 5211-5252. doi: 10.3934/dcds.2017226

## Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities

 Mathematics Department, Texas A & M University, College Station, TX 77843-3368, USA

Received  December 2016 Revised  May 2017 Published  June 2017

We study the behavior as
 $t \to 0^+$
of nonnegative functions
 $u\in {{C}^{2,1}}({{\mathbb{R}}^{n}}\times (0,1))\cap {{L}^{\lambda }}({{\mathbb{R}}^{n}}\times (0,1)),\ \ n\ge 1,\ \ \ \ \ \ \ \ \ \ \ \ \left( 0.1 \right)$
satisfying the parabolic Choquard-Pekar type inequalities
 $0\le {{u}_{t}}-\Delta u\le ({{\Phi }^{\alpha /n}}*{{u}^{\lambda }}){{u}^{\sigma }}\ \ \text{ in }{{B}_{1}}\ (0)\times (0,1)\ \ \ \ \ \ \ \ \ \ \left( 0.2 \right)$
where
 $α∈(0, n+2)$
,
 $λ>0$
, and
 $σ≥0$
are constants,
 $Φ$
is the heat kernel, and
 $*$
is the convolution operation in
 $\mathbb{R}^n× (0, 1)$
. We provide optimal conditions on
 $α, λ$
, and
 $σ$
such that nonnegative solutions
 $u$
of (0.1), (0.2) satisfy pointwise bounds in compact subsets of
 $B_1(0)$
as
 $t \to0^+$
. We obtain similar results for nonnegative solutions of (0.1), (0.2) when
 $Φ^{α/n}$
in (0.2) is replaced with the fundamental solution
 $Φ_α$
of the fractional heat operator
 $(\frac{\partial}{\partial t}-Δ)^{α/2}$
.
Citation: Steven D. Taliaferro. Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5211-5252. doi: 10.3934/dcds.2017226
##### References:

show all references

##### References:
Case $\alpha\in (2,n+2)$
Case $\alpha\in (0,2]$. When $\alpha=2$ the graph on the interval $\lambda>(n+2)/n$ is the horizontal half line $\sigma=1$
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