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 and 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), 431-449.

[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), 547-562.

[3]

V. A. Galaktionov and J. L. Vazquez, A stability technique for evolution partial differential equations, a dynamical systems approach, 2004. Buch. XIX, 377 S.: 10 s/w-Abbildungen. Birkhauser, ISBN 978-0-8176-4146-7.

[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), 237-246.

[5]

Li Ma, Boundary value problem for a classical semilinear parabolic equation, to appear in Chinese Ann. Math., 2012.

[6]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States," Birkhauser. Advanced text, 2007.

[7]

M. Struwe, "Variational Methods," third ed., Springer, 2000.

[8]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. Journal, 57 (2008), 3365-3396.

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), 431-449.

[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), 547-562.

[3]

V. A. Galaktionov and J. L. Vazquez, A stability technique for evolution partial differential equations, a dynamical systems approach, 2004. Buch. XIX, 377 S.: 10 s/w-Abbildungen. Birkhauser, ISBN 978-0-8176-4146-7.

[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), 237-246.

[5]

Li Ma, Boundary value problem for a classical semilinear parabolic equation, to appear in Chinese Ann. Math., 2012.

[6]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States," Birkhauser. Advanced text, 2007.

[7]

M. Struwe, "Variational Methods," third ed., Springer, 2000.

[8]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. Journal, 57 (2008), 3365-3396.

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