-
Previous Article
Nonlinear parabolic differential equations and inequalities
- DCDS Home
- This Issue
-
Next Article
The problem Of blow-up in nonlinear parabolic equations
O.D.E. type behavior of blow-up solutions of nonlinear heat equations
| 1. | Département de Mathématiques, Université de Cergy-Pontoise and IUF, 2 Ave. Adolphe Chauvin, BP 222, Pontoise, 95 302 Cergy-Pontoise cedex, France |
| 2. | Département de Mathématiques et Applications, CNRS École Normale Supérieure, 45 rue d'Ulm, 75 230 Paris cedex 05, France |
$\frac{\partial w}{\partial s}=\Delta w-\frac{1}{2}y \cdot \nabla w -\frac{w}{p-1}+|w|^{p-1}w$
(where $w : \mathbb R^N\times \mathbb R \to \mathbb R^M, p> 1$ and $(N - 2)p < N + 2$) are independent of space and completely explicit. We then derive from this various uniform estimates and a uniform localization property for blow-up solutions of $\partial_t u=\Delta u + |u|^{p-1}u$.
| [1] |
Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585 |
| [2] |
Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139 |
| [3] |
Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113 |
| [4] |
Francesca De Marchis, Isabella Ianni. Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 891-907. doi: 10.3934/dcds.2015.35.891 |
| [5] |
Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4847-4885. doi: 10.3934/dcds.2021060 |
| [6] |
Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016 |
| [7] |
Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021144 |
| [8] |
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 |
| [9] |
Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 |
| [10] |
Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729 |
| [11] |
Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 |
| [12] |
Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 |
| [13] |
Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905 |
| [14] |
Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248 |
| [15] |
Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Y. S. Hamed. Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021083 |
| [16] |
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 |
| [17] |
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 |
| [18] |
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 |
| [19] |
Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 |
| [20] |
Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]