-
Previous Article
Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems
- CPAA Home
- This Issue
-
Next Article
Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities
Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions
| 1. | Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid |
| 2. | Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 MADRID, Spain |
$u_t-$ div $(|x|^{-p\gamma}|\nabla u|^{p-2}\nabla u) = \lambda f(x,u), u\ge 0$ in $\Omega\times (0,T)$,
$ B(u) = 0$ on $\partial\Omega\times (0,T),$
$ u(x,0) = \varphi (x)\quad$ if $x\in\Omega$,
where $\Omega\subset\mathbb R^N$ is a smooth bounded domain with $0\in\Omega$,
$B(u)\equiv u\chi_{\Sigma_1\times(0,T)}+|x|^{-p\gamma} |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\chi_{\Sigma_2 \times (0,T)}$
and
$-\infty<\gamma<\frac{N-p}{p}$. The boundary conditions over
$\partial\Omega\times (0,T)$ verify hypotheses that will be precised in each case.
Mainly, we will consider the second member
$f(x,u)=\frac{u^{\alpha}}{|x|^{p(\gamma+1)}}$ with $ \alpha\ge p-1$, as a model case. The
main points under analysis are some existence, nonexistence and
complete blow-up results related to some Hardy-Sobolev
inequalities and a weak version of Harnack inequality, that holds
for $p\ge 2$ and $\gamma+1>0$.
| [1] |
Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061 |
| [2] |
Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016 |
| [3] |
Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63 |
| [4] |
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 |
| [5] |
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 |
| [6] |
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 |
| [7] |
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 |
| [8] |
Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191 |
| [9] |
Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 |
| [10] |
Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 |
| [11] |
C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523 |
| [12] |
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 |
| [13] |
José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138 |
| [14] |
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 |
| [15] |
Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1 |
| [16] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 |
| [17] |
Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181 |
| [18] |
Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 |
| [19] |
Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. |
| [20] |
Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]