A Threshold Result for a Non-local Parabolic Equation |
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Authors: | Yong-hui Wu |
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Abstract: | In this paper, we consider the Cauchy problem: (ECP) ut−Δu+p(x)u=u(x,t)∫ u2(y,t)/∣x−y∣dy; x∈ℝ3, t>0, u(x, 0)=u0(x)⩾0 x∈ℝ3, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ3, p(x)⩾ā>0, and lim∣x∣→∞p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u0(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u0(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd. |
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