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Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations. (English) Zbl 1149.35367

Summary: We study the Cauchy problem of semilinear heat equations
\[ u_t- \Delta u=|u|^{p-1}u-u, \quad x\in\mathbb R^n,\;t>0, \qquad u(x,0)= u_0(x), \quad x\in\mathbb R^n, \]
where \(p\) satisfies \(1<p<\infty\) if \(n=1,2\); \(1<p\leq \frac{n+2}{n-2}\) if \(n\geq 3\). By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Then we obtain a threshold result for the global existence and nonexistence of solutions. Finally we discuss the asymptotic behavior of the solution.

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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