# 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 - A, 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 $\sigma=1$
 [1] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [2] Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025 [3] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [4] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [5] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [6] Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 [7] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [8] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [9] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [10] Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585 [11] Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 [12] Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449 [13] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [14] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [15] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [16] Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 [17] Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 [18] Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169 [19] Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 [20] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

2018 Impact Factor: 1.143