ON THE STABILITY OF THE POSITIVE RADIAL STEADY STATES FOR A SEMILINEAR CAUCHY PROBLEM INVOLVING CRITICAL EXPONENTS

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doi:10.1016/S0252-9602(08)60036-3

Abstract: This article is contributed to the Cauchy problem

{\begin{cases} \D \frac{\partial u}{\partial t} = Δ u + K (| x |) u^{p} in R^{n} \times (0, T), \\ u (x, 0) = φ (x) in R^{n}; \end{cases}

$\left\{\begin{array}{ll} \D \frac{\partial u}{\partial t}= \Delta u+K( |x|)u^p \ \ \mbox{in} \ R^n \times(0,T),\\ u(x,0)=\varphi(x) \ \ \mbox{in} \ R^n ; \end{array} \right.$
with initial function

φ ≢ 0

$\varphi \not \equiv 0$ . The stability of positive radial steady state, which are positive solutions of △u+K( | x|)u^p=0, is obtained when p is critical for general K(|x|).

Key words: Stability, Cauchy problem, asymptotic stability

CLC Number:

35J10

Deng Yinbin; Yang Fen. ON THE STABILITY OF THE POSITIVE RADIAL STEADY STATES FOR A SEMILINEAR CAUCHY PROBLEM INVOLVING CRITICAL EXPONENTS[J].Acta mathematica scientia,Series B, 2008, 28(2): 348-354.

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