# American Institute of Mathematical Sciences

March  2013, 12(2): 1103-1110. doi: 10.3934/cpaa.2013.12.1103

## Blow-up for semilinear parabolic equations with critical Sobolev exponent

 1 Department of Mathematics, Henan Normal University, Xinxiang 453007, China

Received  July 2011 Revised  March 2012 Published  September 2012

In this paper, we study the global existence and blow-up results of semilinear parabolic equations with critical Sobolev exponent \begin{eqnarray*} u_t-\Delta u=|u|^{p-1}u, in \Omega\times (0,T) \end{eqnarray*} with the Dirichlet boundary condition $u=0$ on the boundary $\partial\Omega\times [0,T)$ and $u=\phi$ at $t=0$, where $\Omega\subset R^n$, $n\geq 3$, is a compact $C^1$ domain, $p=p_S=\frac{n+2}{n-2}$ is the critical Sobolev exponent, and $0 ≨ \phi \in C^1_0(\Omega)$ is a given smooth function. We show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $\phi\in\tilde{W}$, there is a global positive solution $u(t)\in \tilde{W}$ with $H^1$ omega limit $\{0\}$ and for $\phi\in \tilde{Z}$, the solution blows up at finite time.
Citation: 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
##### References:
 [1] Th. Cazenave, F. Dickstein and F. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball,, Math. Ann., 344 (2009), 431.   Google Scholar [2] F. Dickstein, N. Mizoguchi, P. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for semilinear heat equation,, Calculus of Variations and Partial Differential Equations, 42 (2011), 547.   Google Scholar [3] V. A. Galaktionov and J. L. Vazquez, A stability technique for evolution partial differential equations, a dynamical systems approach,, 2004. Buch. XIX, (): 978.   Google Scholar [4] Li Ma, Chong Li and Lin Zhao, Monotone solutions to a class of elliptic and diffusion equations,, Communications on Pure and Applied Analysis, 6 (2007), 237.   Google Scholar [5] Li Ma, Boundary value problem for a classical semilinear parabolic equation,, to appear in Chinese Ann. Math., (2012).   Google Scholar [6] P. Quittner and P. Souplet, "Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,", Birkhauser. Advanced text, (2007).   Google Scholar [7] M. Struwe, "Variational Methods,", third ed., (2000).   Google Scholar [8] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent,, Indiana Univ. Math. Journal, 57 (2008), 3365.   Google Scholar

show all references

##### References:
 [1] Th. Cazenave, F. Dickstein and F. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball,, Math. Ann., 344 (2009), 431.   Google Scholar [2] F. Dickstein, N. Mizoguchi, P. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for semilinear heat equation,, Calculus of Variations and Partial Differential Equations, 42 (2011), 547.   Google Scholar [3] V. A. Galaktionov and J. L. Vazquez, A stability technique for evolution partial differential equations, a dynamical systems approach,, 2004. Buch. XIX, (): 978.   Google Scholar [4] Li Ma, Chong Li and Lin Zhao, Monotone solutions to a class of elliptic and diffusion equations,, Communications on Pure and Applied Analysis, 6 (2007), 237.   Google Scholar [5] Li Ma, Boundary value problem for a classical semilinear parabolic equation,, to appear in Chinese Ann. Math., (2012).   Google Scholar [6] P. Quittner and P. Souplet, "Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,", Birkhauser. Advanced text, (2007).   Google Scholar [7] M. Struwe, "Variational Methods,", third ed., (2000).   Google Scholar [8] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent,, Indiana Univ. Math. Journal, 57 (2008), 3365.   Google Scholar
 [1] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [2] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [3] Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 [4] Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077 [5] Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043 [6] 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 [7] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [8] 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 [9] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [10] Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 [11] Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535 [12] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [13] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [14] Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093 [15] István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134 [16] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [17] Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 [18] Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 [19] 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 [20] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

2019 Impact Factor: 1.105