Global existence for the nonlinear heat equation in the Fujita subcritical case with initial value having zero mean value

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Abstract

In this paper we prove for 1<p<1+2N+k, where k is an integer in 1,N, the existence of an initial value ψ, odd with respect to the k first coordinates, and with RNx1xkψdx1dxN0, such that the resulting solution of utΔu=|u|p1u is global. In the case k=1 and 1<p<1+1N+1, it is known that the solution u with the initial value u(0)=λψ blows up in finite time if λ>0 either sufficiently small or sufficiently large. The result in this paper extends a similar result of Cazenave, Dickstein, and Weissler in the case k=0, i.e. with RNψ0 and 1<p<1+2N.

Keywords

Blow-up
Local existence
Global existence
Sign-changing solutions
Nonlinear heat equation
Invariant manifold
Semiflow

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