# 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), 431-449.  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), 547-562.  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), 237-246.  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., Springer, 2000. Google Scholar [8] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. Journal, 57 (2008), 3365-3396. 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), 431-449.  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), 547-562.  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), 237-246.  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., Springer, 2000. Google Scholar [8] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. Journal, 57 (2008), 3365-3396. Google Scholar
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