Abstract
We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.
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